Termination w.r.t. Q proof of /tmp/tmpDBL9Tc/Ex5_7_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)))
a__dbl1(0) → 01
a__dbl1(s(X)) → s1(s1(a__dbl1(mark(X))))
a__sel1(0, cons(X, Y)) → mark(X)
a__sel1(s(X), cons(Y, Z)) → a__sel1(mark(X), mark(Z))
a__quote(0) → 01
a__quote(s(X)) → s1(a__quote(mark(X)))
a__quote(dbl(X)) → a__dbl1(mark(X))
a__quote(sel(X, Y)) → a__sel1(mark(X), mark(Y))
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(dbl1(X)) → a__dbl1(mark(X))
mark(sel1(X1, X2)) → a__sel1(mark(X1), mark(X2))
mark(quote(X)) → a__quote(mark(X))
mark(0) → 0
mark(s(X)) → s(X)
mark(nil) → nil
mark(cons(X1, X2)) → cons(X1, X2)
mark(01) → 01
mark(s1(X)) → s1(mark(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__from(X) → from(X)
a__dbl1(X) → dbl1(X)
a__sel1(X1, X2) → sel1(X1, X2)
a__quote(X) → quote(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)))
a__dbl1(0) → 01
a__dbl1(s(X)) → s1(s1(a__dbl1(mark(X))))
a__sel1(0, cons(X, Y)) → mark(X)
a__sel1(s(X), cons(Y, Z)) → a__sel1(mark(X), mark(Z))
a__quote(0) → 01
a__quote(s(X)) → s1(a__quote(mark(X)))
a__quote(dbl(X)) → a__dbl1(mark(X))
a__quote(sel(X, Y)) → a__sel1(mark(X), mark(Y))
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(dbl1(X)) → a__dbl1(mark(X))
mark(sel1(X1, X2)) → a__sel1(mark(X1), mark(X2))
mark(quote(X)) → a__quote(mark(X))
mark(0) → 0
mark(s(X)) → s(X)
mark(nil) → nil
mark(cons(X1, X2)) → cons(X1, X2)
mark(01) → 01
mark(s1(X)) → s1(mark(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__from(X) → from(X)
a__dbl1(X) → dbl1(X)
a__sel1(X1, X2) → sel1(X1, X2)
a__quote(X) → quote(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(indx(X1, X2)) → MARK(X1)
A__QUOTE(s(X)) → MARK(X)
A__QUOTE(dbl(X)) → MARK(X)
MARK(dbl1(X)) → A__DBL1(mark(X))
A__QUOTE(sel(X, Y)) → MARK(X)
MARK(dbls(X)) → MARK(X)
A__SEL(0, cons(X, Y)) → MARK(X)
MARK(indx(X1, X2)) → A__INDX(mark(X1), X2)
A__SEL(s(X), cons(Y, Z)) → MARK(Z)
A__QUOTE(sel(X, Y)) → MARK(Y)
MARK(quote(X)) → A__QUOTE(mark(X))
MARK(from(X)) → A__FROM(X)
MARK(quote(X)) → MARK(X)
A__SEL1(s(X), cons(Y, Z)) → MARK(X)
A__SEL(s(X), cons(Y, Z)) → MARK(X)
A__SEL1(s(X), cons(Y, Z)) → A__SEL1(mark(X), mark(Z))
A__QUOTE(sel(X, Y)) → A__SEL1(mark(X), mark(Y))
MARK(sel(X1, X2)) → MARK(X1)
MARK(sel(X1, X2)) → A__SEL(mark(X1), mark(X2))
A__DBL1(s(X)) → MARK(X)
A__SEL(s(X), cons(Y, Z)) → A__SEL(mark(X), mark(Z))
MARK(dbl(X)) → MARK(X)
MARK(s1(X)) → MARK(X)
A__QUOTE(dbl(X)) → A__DBL1(mark(X))
A__SEL1(0, cons(X, Y)) → MARK(X)
MARK(sel1(X1, X2)) → MARK(X1)
MARK(dbls(X)) → A__DBLS(mark(X))
A__SEL1(s(X), cons(Y, Z)) → MARK(Z)
MARK(sel1(X1, X2)) → MARK(X2)
MARK(dbl1(X)) → MARK(X)
MARK(dbl(X)) → A__DBL(mark(X))
MARK(sel1(X1, X2)) → A__SEL1(mark(X1), mark(X2))
A__QUOTE(s(X)) → A__QUOTE(mark(X))
A__DBL1(s(X)) → A__DBL1(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)))
a__dbl1(0) → 01
a__dbl1(s(X)) → s1(s1(a__dbl1(mark(X))))
a__sel1(0, cons(X, Y)) → mark(X)
a__sel1(s(X), cons(Y, Z)) → a__sel1(mark(X), mark(Z))
a__quote(0) → 01
a__quote(s(X)) → s1(a__quote(mark(X)))
a__quote(dbl(X)) → a__dbl1(mark(X))
a__quote(sel(X, Y)) → a__sel1(mark(X), mark(Y))
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(dbl1(X)) → a__dbl1(mark(X))
mark(sel1(X1, X2)) → a__sel1(mark(X1), mark(X2))
mark(quote(X)) → a__quote(mark(X))
mark(0) → 0
mark(s(X)) → s(X)
mark(nil) → nil
mark(cons(X1, X2)) → cons(X1, X2)
mark(01) → 01
mark(s1(X)) → s1(mark(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__from(X) → from(X)
a__dbl1(X) → dbl1(X)
a__sel1(X1, X2) → sel1(X1, X2)
a__quote(X) → quote(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(indx(X1, X2)) → MARK(X1)
A__QUOTE(s(X)) → MARK(X)
A__QUOTE(dbl(X)) → MARK(X)
MARK(dbl1(X)) → A__DBL1(mark(X))
A__QUOTE(sel(X, Y)) → MARK(X)
MARK(dbls(X)) → MARK(X)
A__SEL(0, cons(X, Y)) → MARK(X)
MARK(indx(X1, X2)) → A__INDX(mark(X1), X2)
A__SEL(s(X), cons(Y, Z)) → MARK(Z)
A__QUOTE(sel(X, Y)) → MARK(Y)
MARK(quote(X)) → A__QUOTE(mark(X))
MARK(from(X)) → A__FROM(X)
MARK(quote(X)) → MARK(X)
A__SEL1(s(X), cons(Y, Z)) → MARK(X)
A__SEL(s(X), cons(Y, Z)) → MARK(X)
A__SEL1(s(X), cons(Y, Z)) → A__SEL1(mark(X), mark(Z))
A__QUOTE(sel(X, Y)) → A__SEL1(mark(X), mark(Y))
MARK(sel(X1, X2)) → MARK(X1)
MARK(sel(X1, X2)) → A__SEL(mark(X1), mark(X2))
A__DBL1(s(X)) → MARK(X)
A__SEL(s(X), cons(Y, Z)) → A__SEL(mark(X), mark(Z))
MARK(dbl(X)) → MARK(X)
MARK(s1(X)) → MARK(X)
A__QUOTE(dbl(X)) → A__DBL1(mark(X))
A__SEL1(0, cons(X, Y)) → MARK(X)
MARK(sel1(X1, X2)) → MARK(X1)
MARK(dbls(X)) → A__DBLS(mark(X))
A__SEL1(s(X), cons(Y, Z)) → MARK(Z)
MARK(sel1(X1, X2)) → MARK(X2)
MARK(dbl1(X)) → MARK(X)
MARK(dbl(X)) → A__DBL(mark(X))
MARK(sel1(X1, X2)) → A__SEL1(mark(X1), mark(X2))
A__QUOTE(s(X)) → A__QUOTE(mark(X))
A__DBL1(s(X)) → A__DBL1(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)))
a__dbl1(0) → 01
a__dbl1(s(X)) → s1(s1(a__dbl1(mark(X))))
a__sel1(0, cons(X, Y)) → mark(X)
a__sel1(s(X), cons(Y, Z)) → a__sel1(mark(X), mark(Z))
a__quote(0) → 01
a__quote(s(X)) → s1(a__quote(mark(X)))
a__quote(dbl(X)) → a__dbl1(mark(X))
a__quote(sel(X, Y)) → a__sel1(mark(X), mark(Y))
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(dbl1(X)) → a__dbl1(mark(X))
mark(sel1(X1, X2)) → a__sel1(mark(X1), mark(X2))
mark(quote(X)) → a__quote(mark(X))
mark(0) → 0
mark(s(X)) → s(X)
mark(nil) → nil
mark(cons(X1, X2)) → cons(X1, X2)
mark(01) → 01
mark(s1(X)) → s1(mark(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__from(X) → from(X)
a__dbl1(X) → dbl1(X)
a__sel1(X1, X2) → sel1(X1, X2)
a__quote(X) → quote(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(indx(X1, X2)) → MARK(X1)
A__QUOTE(s(X)) → MARK(X)
A__QUOTE(dbl(X)) → MARK(X)
MARK(dbl1(X)) → A__DBL1(mark(X))
A__QUOTE(sel(X, Y)) → MARK(X)
MARK(dbls(X)) → MARK(X)
A__SEL(0, cons(X, Y)) → MARK(X)
A__SEL(s(X), cons(Y, Z)) → MARK(Z)
A__QUOTE(sel(X, Y)) → MARK(Y)
MARK(quote(X)) → A__QUOTE(mark(X))
MARK(quote(X)) → MARK(X)
A__SEL(s(X), cons(Y, Z)) → MARK(X)
A__SEL1(s(X), cons(Y, Z)) → MARK(X)
MARK(sel(X1, X2)) → MARK(X1)
A__QUOTE(sel(X, Y)) → A__SEL1(mark(X), mark(Y))
A__SEL1(s(X), cons(Y, Z)) → A__SEL1(mark(X), 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__DBL1(s(X)) → MARK(X)
MARK(dbl(X)) → MARK(X)
MARK(s1(X)) → MARK(X)
A__QUOTE(dbl(X)) → A__DBL1(mark(X))
A__SEL1(0, cons(X, Y)) → MARK(X)
MARK(sel1(X1, X2)) → MARK(X1)
A__SEL1(s(X), cons(Y, Z)) → MARK(Z)
MARK(sel1(X1, X2)) → MARK(X2)
MARK(dbl1(X)) → MARK(X)
MARK(sel1(X1, X2)) → A__SEL1(mark(X1), mark(X2))
A__QUOTE(s(X)) → A__QUOTE(mark(X))
A__DBL1(s(X)) → A__DBL1(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)))
a__dbl1(0) → 01
a__dbl1(s(X)) → s1(s1(a__dbl1(mark(X))))
a__sel1(0, cons(X, Y)) → mark(X)
a__sel1(s(X), cons(Y, Z)) → a__sel1(mark(X), mark(Z))
a__quote(0) → 01
a__quote(s(X)) → s1(a__quote(mark(X)))
a__quote(dbl(X)) → a__dbl1(mark(X))
a__quote(sel(X, Y)) → a__sel1(mark(X), mark(Y))
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(dbl1(X)) → a__dbl1(mark(X))
mark(sel1(X1, X2)) → a__sel1(mark(X1), mark(X2))
mark(quote(X)) → a__quote(mark(X))
mark(0) → 0
mark(s(X)) → s(X)
mark(nil) → nil
mark(cons(X1, X2)) → cons(X1, X2)
mark(01) → 01
mark(s1(X)) → s1(mark(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__from(X) → from(X)
a__dbl1(X) → dbl1(X)
a__sel1(X1, X2) → sel1(X1, X2)
a__quote(X) → quote(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)
A__SEL1(0, cons(X, Y)) → MARK(X)

The remaining pairs can at least be oriented weakly.

MARK(indx(X1, X2)) → MARK(X1)
A__QUOTE(s(X)) → MARK(X)
A__QUOTE(dbl(X)) → MARK(X)
MARK(dbl1(X)) → A__DBL1(mark(X))
A__QUOTE(sel(X, Y)) → MARK(X)
MARK(dbls(X)) → MARK(X)
A__SEL(s(X), cons(Y, Z)) → MARK(Z)
A__QUOTE(sel(X, Y)) → MARK(Y)
MARK(quote(X)) → A__QUOTE(mark(X))
MARK(quote(X)) → MARK(X)
A__SEL(s(X), cons(Y, Z)) → MARK(X)
A__SEL1(s(X), cons(Y, Z)) → MARK(X)
MARK(sel(X1, X2)) → MARK(X1)
A__QUOTE(sel(X, Y)) → A__SEL1(mark(X), mark(Y))
A__SEL1(s(X), cons(Y, Z)) → A__SEL1(mark(X), 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__DBL1(s(X)) → MARK(X)
MARK(dbl(X)) → MARK(X)
MARK(s1(X)) → MARK(X)
A__QUOTE(dbl(X)) → A__DBL1(mark(X))
MARK(sel1(X1, X2)) → MARK(X1)
A__SEL1(s(X), cons(Y, Z)) → MARK(Z)
MARK(sel1(X1, X2)) → MARK(X2)
MARK(dbl1(X)) → MARK(X)
MARK(sel1(X1, X2)) → A__SEL1(mark(X1), mark(X2))
A__QUOTE(s(X)) → A__QUOTE(mark(X))
A__DBL1(s(X)) → A__DBL1(mark(X))
MARK(sel(X1, X2)) → MARK(X2)

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

POL(0) = 1
POL(01) = 0
POL(A__DBL1(x₁)) = x₁
POL(A__QUOTE(x₁)) = x₁
POL(A__SEL(x₁, x₂)) = x₁ + x₂
POL(A__SEL1(x₁, x₂)) = x₁ + x₂
POL(MARK(x₁)) = x₁
POL(a__dbl(x₁)) = x₁
POL(a__dbl1(x₁)) = x₁
POL(a__dbls(x₁)) = x₁
POL(a__from(x₁)) = x₁
POL(a__indx(x₁, x₂)) = x₁ + x₂
POL(a__quote(x₁)) = x₁
POL(a__sel(x₁, x₂)) = x₁ + x₂
POL(a__sel1(x₁, x₂)) = x₁ + x₂
POL(cons(x₁, x₂)) = max(x₁, x₂)
POL(dbl(x₁)) = x₁
POL(dbl1(x₁)) = x₁
POL(dbls(x₁)) = x₁
POL(from(x₁)) = x₁
POL(indx(x₁, x₂)) = x₁ + x₂
POL(mark(x₁)) = x₁
POL(nil) = 0
POL(quote(x₁)) = x₁
POL(s(x₁)) = x₁
POL(s1(x₁)) = x₁
POL(sel(x₁, x₂)) = x₁ + x₂
POL(sel1(x₁, x₂)) = x₁ + x₂

The following usable rules [17] were oriented:

a__dbls(cons(X, Y)) → cons(dbl(X), dbls(Y))
a__dbls(nil) → nil
a__dbl(s(X)) → s(s(dbl(X)))
a__dbl(0) → 0
a__dbl1(s(X)) → s1(s1(a__dbl1(mark(X))))
a__quote(0) → 01
a__indx(cons(X, Y), Z) → cons(sel(X, Z), indx(Y, Z))
a__indx(nil, X) → nil
a__dbl1(0) → 01
a__from(X) → cons(X, from(s(X)))
mark(indx(X1, X2)) → a__indx(mark(X1), X2)
mark(from(X)) → a__from(X)
mark(dbls(X)) → a__dbls(mark(X))
mark(sel1(X1, X2)) → a__sel1(mark(X1), mark(X2))
a__sel(s(X), cons(Y, Z)) → a__sel(mark(X), mark(Z))
a__sel1(0, cons(X, Y)) → mark(X)
a__sel(0, cons(X, Y)) → mark(X)
a__sel1(s(X), cons(Y, Z)) → a__sel1(mark(X), mark(Z))
mark(quote(X)) → a__quote(mark(X))
a__quote(sel(X, Y)) → a__sel1(mark(X), mark(Y))
mark(sel(X1, X2)) → a__sel(mark(X1), mark(X2))
mark(dbl(X)) → a__dbl(mark(X))
a__quote(s(X)) → s1(a__quote(mark(X)))
a__quote(dbl(X)) → a__dbl1(mark(X))
mark(cons(X1, X2)) → cons(X1, X2)
mark(01) → 01
mark(s(X)) → s(X)
mark(nil) → nil
mark(0) → 0
mark(dbl1(X)) → a__dbl1(mark(X))
a__sel1(X1, X2) → sel1(X1, X2)
a__dbl1(X) → dbl1(X)
a__from(X) → from(X)
a__indx(X1, X2) → indx(X1, X2)
a__sel(X1, X2) → sel(X1, X2)
a__dbls(X) → dbls(X)
a__dbl(X) → dbl(X)
mark(s1(X)) → s1(mark(X))
a__quote(X) → quote(X)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ QDPOrderProof

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

MARK(indx(X1, X2)) → MARK(X1)
A__QUOTE(s(X)) → MARK(X)
A__QUOTE(dbl(X)) → MARK(X)
MARK(dbl1(X)) → A__DBL1(mark(X))
A__QUOTE(sel(X, Y)) → MARK(X)
MARK(dbls(X)) → MARK(X)
A__SEL(s(X), cons(Y, Z)) → MARK(Z)
A__QUOTE(sel(X, Y)) → MARK(Y)
MARK(quote(X)) → A__QUOTE(mark(X))
MARK(quote(X)) → MARK(X)
A__SEL1(s(X), cons(Y, Z)) → MARK(X)
A__SEL(s(X), cons(Y, Z)) → MARK(X)
MARK(sel(X1, X2)) → MARK(X1)
A__QUOTE(sel(X, Y)) → A__SEL1(mark(X), mark(Y))
A__SEL1(s(X), cons(Y, Z)) → A__SEL1(mark(X), 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__DBL1(s(X)) → MARK(X)
MARK(dbl(X)) → MARK(X)
MARK(s1(X)) → MARK(X)
A__QUOTE(dbl(X)) → A__DBL1(mark(X))
MARK(sel1(X1, X2)) → MARK(X1)
A__SEL1(s(X), cons(Y, Z)) → MARK(Z)
MARK(sel1(X1, X2)) → MARK(X2)
MARK(dbl1(X)) → MARK(X)
A__QUOTE(s(X)) → A__QUOTE(mark(X))
MARK(sel1(X1, X2)) → A__SEL1(mark(X1), mark(X2))
A__DBL1(s(X)) → A__DBL1(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)))
a__dbl1(0) → 01
a__dbl1(s(X)) → s1(s1(a__dbl1(mark(X))))
a__sel1(0, cons(X, Y)) → mark(X)
a__sel1(s(X), cons(Y, Z)) → a__sel1(mark(X), mark(Z))
a__quote(0) → 01
a__quote(s(X)) → s1(a__quote(mark(X)))
a__quote(dbl(X)) → a__dbl1(mark(X))
a__quote(sel(X, Y)) → a__sel1(mark(X), mark(Y))
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(dbl1(X)) → a__dbl1(mark(X))
mark(sel1(X1, X2)) → a__sel1(mark(X1), mark(X2))
mark(quote(X)) → a__quote(mark(X))
mark(0) → 0
mark(s(X)) → s(X)
mark(nil) → nil
mark(cons(X1, X2)) → cons(X1, X2)
mark(01) → 01
mark(s1(X)) → s1(mark(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__from(X) → from(X)
a__dbl1(X) → dbl1(X)
a__sel1(X1, X2) → sel1(X1, X2)
a__quote(X) → quote(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__QUOTE(s(X)) → MARK(X)
A__QUOTE(dbl(X)) → MARK(X)
A__QUOTE(sel(X, Y)) → MARK(X)
A__QUOTE(sel(X, Y)) → MARK(Y)
MARK(quote(X)) → MARK(X)
A__QUOTE(sel(X, Y)) → A__SEL1(mark(X), mark(Y))
A__QUOTE(dbl(X)) → A__DBL1(mark(X))

The remaining pairs can at least be oriented weakly.

MARK(indx(X1, X2)) → MARK(X1)
MARK(dbl1(X)) → A__DBL1(mark(X))
MARK(dbls(X)) → MARK(X)
A__SEL(s(X), cons(Y, Z)) → MARK(Z)
MARK(quote(X)) → A__QUOTE(mark(X))
A__SEL1(s(X), cons(Y, Z)) → MARK(X)
A__SEL(s(X), cons(Y, Z)) → MARK(X)
MARK(sel(X1, X2)) → MARK(X1)
A__SEL1(s(X), cons(Y, Z)) → A__SEL1(mark(X), 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__DBL1(s(X)) → MARK(X)
MARK(dbl(X)) → MARK(X)
MARK(s1(X)) → MARK(X)
MARK(sel1(X1, X2)) → MARK(X1)
A__SEL1(s(X), cons(Y, Z)) → MARK(Z)
MARK(sel1(X1, X2)) → MARK(X2)
MARK(dbl1(X)) → MARK(X)
A__QUOTE(s(X)) → A__QUOTE(mark(X))
MARK(sel1(X1, X2)) → A__SEL1(mark(X1), mark(X2))
A__DBL1(s(X)) → A__DBL1(mark(X))
MARK(sel(X1, X2)) → MARK(X2)

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

POL(0) = 0
POL(01) = 0
POL(A__DBL1(x₁)) = x₁
POL(A__QUOTE(x₁)) = 1 + x₁
POL(A__SEL(x₁, x₂)) = x₁ + x₂
POL(A__SEL1(x₁, x₂)) = x₁ + x₂
POL(MARK(x₁)) = x₁
POL(a__dbl(x₁)) = x₁
POL(a__dbl1(x₁)) = x₁
POL(a__dbls(x₁)) = x₁
POL(a__from(x₁)) = x₁
POL(a__indx(x₁, x₂)) = x₁ + x₂
POL(a__quote(x₁)) = 1 + x₁
POL(a__sel(x₁, x₂)) = x₁ + x₂
POL(a__sel1(x₁, x₂)) = x₁ + x₂
POL(cons(x₁, x₂)) = max(x₁, x₂)
POL(dbl(x₁)) = x₁
POL(dbl1(x₁)) = x₁
POL(dbls(x₁)) = x₁
POL(from(x₁)) = x₁
POL(indx(x₁, x₂)) = x₁ + x₂
POL(mark(x₁)) = x₁
POL(nil) = 0
POL(quote(x₁)) = 1 + x₁
POL(s(x₁)) = x₁
POL(s1(x₁)) = x₁
POL(sel(x₁, x₂)) = x₁ + x₂
POL(sel1(x₁, x₂)) = x₁ + x₂

The following usable rules [17] were oriented:

a__dbls(cons(X, Y)) → cons(dbl(X), dbls(Y))
a__dbls(nil) → nil
a__dbl(s(X)) → s(s(dbl(X)))
a__dbl(0) → 0
a__dbl1(s(X)) → s1(s1(a__dbl1(mark(X))))
a__quote(0) → 01
a__indx(cons(X, Y), Z) → cons(sel(X, Z), indx(Y, Z))
a__indx(nil, X) → nil
a__dbl1(0) → 01
a__from(X) → cons(X, from(s(X)))
mark(indx(X1, X2)) → a__indx(mark(X1), X2)
mark(from(X)) → a__from(X)
mark(dbls(X)) → a__dbls(mark(X))
mark(sel1(X1, X2)) → a__sel1(mark(X1), mark(X2))
a__sel(s(X), cons(Y, Z)) → a__sel(mark(X), mark(Z))
a__sel1(0, cons(X, Y)) → mark(X)
a__sel(0, cons(X, Y)) → mark(X)
a__sel1(s(X), cons(Y, Z)) → a__sel1(mark(X), mark(Z))
mark(quote(X)) → a__quote(mark(X))
a__quote(sel(X, Y)) → a__sel1(mark(X), mark(Y))
mark(sel(X1, X2)) → a__sel(mark(X1), mark(X2))
mark(dbl(X)) → a__dbl(mark(X))
a__quote(s(X)) → s1(a__quote(mark(X)))
a__quote(dbl(X)) → a__dbl1(mark(X))
mark(cons(X1, X2)) → cons(X1, X2)
mark(01) → 01
mark(s(X)) → s(X)
mark(nil) → nil
mark(0) → 0
mark(dbl1(X)) → a__dbl1(mark(X))
a__sel1(X1, X2) → sel1(X1, X2)
a__dbl1(X) → dbl1(X)
a__from(X) → from(X)
a__indx(X1, X2) → indx(X1, X2)
a__sel(X1, X2) → sel(X1, X2)
a__dbls(X) → dbls(X)
a__dbl(X) → dbl(X)
mark(s1(X)) → s1(mark(X))
a__quote(X) → quote(X)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ QDPOrderProof

                ↳ QDP

                  ↳ DependencyGraphProof

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

MARK(sel(X1, X2)) → MARK(X1)
A__SEL1(s(X), cons(Y, Z)) → A__SEL1(mark(X), 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))
A__DBL1(s(X)) → MARK(X)
MARK(dbl(X)) → MARK(X)
MARK(dbl1(X)) → A__DBL1(mark(X))
MARK(s1(X)) → MARK(X)
MARK(dbls(X)) → MARK(X)
MARK(sel1(X1, X2)) → MARK(X1)
A__SEL(s(X), cons(Y, Z)) → MARK(Z)
A__SEL1(s(X), cons(Y, Z)) → MARK(Z)
MARK(sel1(X1, X2)) → MARK(X2)
MARK(quote(X)) → A__QUOTE(mark(X))
MARK(dbl1(X)) → MARK(X)
MARK(sel1(X1, X2)) → A__SEL1(mark(X1), mark(X2))
A__QUOTE(s(X)) → A__QUOTE(mark(X))
A__DBL1(s(X)) → A__DBL1(mark(X))
MARK(sel(X1, X2)) → MARK(X2)
A__SEL1(s(X), cons(Y, Z)) → MARK(X)
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)))
a__dbl1(0) → 01
a__dbl1(s(X)) → s1(s1(a__dbl1(mark(X))))
a__sel1(0, cons(X, Y)) → mark(X)
a__sel1(s(X), cons(Y, Z)) → a__sel1(mark(X), mark(Z))
a__quote(0) → 01
a__quote(s(X)) → s1(a__quote(mark(X)))
a__quote(dbl(X)) → a__dbl1(mark(X))
a__quote(sel(X, Y)) → a__sel1(mark(X), mark(Y))
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(dbl1(X)) → a__dbl1(mark(X))
mark(sel1(X1, X2)) → a__sel1(mark(X1), mark(X2))
mark(quote(X)) → a__quote(mark(X))
mark(0) → 0
mark(s(X)) → s(X)
mark(nil) → nil
mark(cons(X1, X2)) → cons(X1, X2)
mark(01) → 01
mark(s1(X)) → s1(mark(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__from(X) → from(X)
a__dbl1(X) → dbl1(X)
a__sel1(X1, X2) → sel1(X1, X2)
a__quote(X) → quote(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

                  ↳ DependencyGraphProof

                    ↳ AND

                      ↳ QDP

                        ↳ Narrowing

                      ↳ QDP

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

A__QUOTE(s(X)) → A__QUOTE(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)))
a__dbl1(0) → 01
a__dbl1(s(X)) → s1(s1(a__dbl1(mark(X))))
a__sel1(0, cons(X, Y)) → mark(X)
a__sel1(s(X), cons(Y, Z)) → a__sel1(mark(X), mark(Z))
a__quote(0) → 01
a__quote(s(X)) → s1(a__quote(mark(X)))
a__quote(dbl(X)) → a__dbl1(mark(X))
a__quote(sel(X, Y)) → a__sel1(mark(X), mark(Y))
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(dbl1(X)) → a__dbl1(mark(X))
mark(sel1(X1, X2)) → a__sel1(mark(X1), mark(X2))
mark(quote(X)) → a__quote(mark(X))
mark(0) → 0
mark(s(X)) → s(X)
mark(nil) → nil
mark(cons(X1, X2)) → cons(X1, X2)
mark(01) → 01
mark(s1(X)) → s1(mark(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__from(X) → from(X)
a__dbl1(X) → dbl1(X)
a__sel1(X1, X2) → sel1(X1, X2)
a__quote(X) → quote(X)

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

A__QUOTE(s(from(x0))) → A__QUOTE(a__from(x0))
A__QUOTE(s(quote(x0))) → A__QUOTE(a__quote(mark(x0)))
A__QUOTE(s(sel(x0, x1))) → A__QUOTE(a__sel(mark(x0), mark(x1)))
A__QUOTE(s(indx(x0, x1))) → A__QUOTE(a__indx(mark(x0), x1))
A__QUOTE(s(dbl(x0))) → A__QUOTE(a__dbl(mark(x0)))
A__QUOTE(s(cons(x0, x1))) → A__QUOTE(cons(x0, x1))
A__QUOTE(s(s(x0))) → A__QUOTE(s(x0))
A__QUOTE(s(dbls(x0))) → A__QUOTE(a__dbls(mark(x0)))
A__QUOTE(s(dbl1(x0))) → A__QUOTE(a__dbl1(mark(x0)))
A__QUOTE(s(0)) → A__QUOTE(0)
A__QUOTE(s(01)) → A__QUOTE(01)
A__QUOTE(s(s1(x0))) → A__QUOTE(s1(mark(x0)))
A__QUOTE(s(sel1(x0, x1))) → A__QUOTE(a__sel1(mark(x0), mark(x1)))
A__QUOTE(s(nil)) → A__QUOTE(nil)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ QDPOrderProof

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ AND

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                      ↳ QDP

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

A__QUOTE(s(from(x0))) → A__QUOTE(a__from(x0))
A__QUOTE(s(quote(x0))) → A__QUOTE(a__quote(mark(x0)))
A__QUOTE(s(sel(x0, x1))) → A__QUOTE(a__sel(mark(x0), mark(x1)))
A__QUOTE(s(indx(x0, x1))) → A__QUOTE(a__indx(mark(x0), x1))
A__QUOTE(s(dbl(x0))) → A__QUOTE(a__dbl(mark(x0)))
A__QUOTE(s(cons(x0, x1))) → A__QUOTE(cons(x0, x1))
A__QUOTE(s(s(x0))) → A__QUOTE(s(x0))
A__QUOTE(s(dbls(x0))) → A__QUOTE(a__dbls(mark(x0)))
A__QUOTE(s(dbl1(x0))) → A__QUOTE(a__dbl1(mark(x0)))
A__QUOTE(s(0)) → A__QUOTE(0)
A__QUOTE(s(01)) → A__QUOTE(01)
A__QUOTE(s(s1(x0))) → A__QUOTE(s1(mark(x0)))
A__QUOTE(s(nil)) → A__QUOTE(nil)
A__QUOTE(s(sel1(x0, x1))) → A__QUOTE(a__sel1(mark(x0), mark(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)))
a__dbl1(0) → 01
a__dbl1(s(X)) → s1(s1(a__dbl1(mark(X))))
a__sel1(0, cons(X, Y)) → mark(X)
a__sel1(s(X), cons(Y, Z)) → a__sel1(mark(X), mark(Z))
a__quote(0) → 01
a__quote(s(X)) → s1(a__quote(mark(X)))
a__quote(dbl(X)) → a__dbl1(mark(X))
a__quote(sel(X, Y)) → a__sel1(mark(X), mark(Y))
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(dbl1(X)) → a__dbl1(mark(X))
mark(sel1(X1, X2)) → a__sel1(mark(X1), mark(X2))
mark(quote(X)) → a__quote(mark(X))
mark(0) → 0
mark(s(X)) → s(X)
mark(nil) → nil
mark(cons(X1, X2)) → cons(X1, X2)
mark(01) → 01
mark(s1(X)) → s1(mark(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__from(X) → from(X)
a__dbl1(X) → dbl1(X)
a__sel1(X1, X2) → sel1(X1, X2)
a__quote(X) → quote(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 5 less nodes.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ QDPOrderProof

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ AND

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ Narrowing

                      ↳ QDP

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

A__QUOTE(s(from(x0))) → A__QUOTE(a__from(x0))
A__QUOTE(s(quote(x0))) → A__QUOTE(a__quote(mark(x0)))
A__QUOTE(s(sel(x0, x1))) → A__QUOTE(a__sel(mark(x0), mark(x1)))
A__QUOTE(s(dbl(x0))) → A__QUOTE(a__dbl(mark(x0)))
A__QUOTE(s(indx(x0, x1))) → A__QUOTE(a__indx(mark(x0), x1))
A__QUOTE(s(sel1(x0, x1))) → A__QUOTE(a__sel1(mark(x0), mark(x1)))
A__QUOTE(s(s(x0))) → A__QUOTE(s(x0))
A__QUOTE(s(dbls(x0))) → A__QUOTE(a__dbls(mark(x0)))
A__QUOTE(s(dbl1(x0))) → A__QUOTE(a__dbl1(mark(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)))
a__dbl1(0) → 01
a__dbl1(s(X)) → s1(s1(a__dbl1(mark(X))))
a__sel1(0, cons(X, Y)) → mark(X)
a__sel1(s(X), cons(Y, Z)) → a__sel1(mark(X), mark(Z))
a__quote(0) → 01
a__quote(s(X)) → s1(a__quote(mark(X)))
a__quote(dbl(X)) → a__dbl1(mark(X))
a__quote(sel(X, Y)) → a__sel1(mark(X), mark(Y))
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(dbl1(X)) → a__dbl1(mark(X))
mark(sel1(X1, X2)) → a__sel1(mark(X1), mark(X2))
mark(quote(X)) → a__quote(mark(X))
mark(0) → 0
mark(s(X)) → s(X)
mark(nil) → nil
mark(cons(X1, X2)) → cons(X1, X2)
mark(01) → 01
mark(s1(X)) → s1(mark(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__from(X) → from(X)
a__dbl1(X) → dbl1(X)
a__sel1(X1, X2) → sel1(X1, X2)
a__quote(X) → quote(X)

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

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ QDPOrderProof

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ AND

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ Narrowing

                                  ↳ QDP

                                    ↳ DependencyGraphProof

                      ↳ QDP

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

A__QUOTE(s(from(x0))) → A__QUOTE(cons(x0, from(s(x0))))
A__QUOTE(s(quote(x0))) → A__QUOTE(a__quote(mark(x0)))
A__QUOTE(s(sel(x0, x1))) → A__QUOTE(a__sel(mark(x0), mark(x1)))
A__QUOTE(s(indx(x0, x1))) → A__QUOTE(a__indx(mark(x0), x1))
A__QUOTE(s(dbl(x0))) → A__QUOTE(a__dbl(mark(x0)))
A__QUOTE(s(from(x0))) → A__QUOTE(from(x0))
A__QUOTE(s(sel1(x0, x1))) → A__QUOTE(a__sel1(mark(x0), mark(x1)))
A__QUOTE(s(s(x0))) → A__QUOTE(s(x0))
A__QUOTE(s(dbls(x0))) → A__QUOTE(a__dbls(mark(x0)))
A__QUOTE(s(dbl1(x0))) → A__QUOTE(a__dbl1(mark(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)))
a__dbl1(0) → 01
a__dbl1(s(X)) → s1(s1(a__dbl1(mark(X))))
a__sel1(0, cons(X, Y)) → mark(X)
a__sel1(s(X), cons(Y, Z)) → a__sel1(mark(X), mark(Z))
a__quote(0) → 01
a__quote(s(X)) → s1(a__quote(mark(X)))
a__quote(dbl(X)) → a__dbl1(mark(X))
a__quote(sel(X, Y)) → a__sel1(mark(X), mark(Y))
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(dbl1(X)) → a__dbl1(mark(X))
mark(sel1(X1, X2)) → a__sel1(mark(X1), mark(X2))
mark(quote(X)) → a__quote(mark(X))
mark(0) → 0
mark(s(X)) → s(X)
mark(nil) → nil
mark(cons(X1, X2)) → cons(X1, X2)
mark(01) → 01
mark(s1(X)) → s1(mark(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__from(X) → from(X)
a__dbl1(X) → dbl1(X)
a__sel1(X1, X2) → sel1(X1, X2)
a__quote(X) → quote(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

                  ↳ DependencyGraphProof

                    ↳ AND

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ Narrowing

                                  ↳ QDP

                                    ↳ DependencyGraphProof

                                      ↳ QDP

                                        ↳ QDPOrderProof

                      ↳ QDP

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

A__QUOTE(s(quote(x0))) → A__QUOTE(a__quote(mark(x0)))
A__QUOTE(s(sel(x0, x1))) → A__QUOTE(a__sel(mark(x0), mark(x1)))
A__QUOTE(s(dbl(x0))) → A__QUOTE(a__dbl(mark(x0)))
A__QUOTE(s(indx(x0, x1))) → A__QUOTE(a__indx(mark(x0), x1))
A__QUOTE(s(sel1(x0, x1))) → A__QUOTE(a__sel1(mark(x0), mark(x1)))
A__QUOTE(s(s(x0))) → A__QUOTE(s(x0))
A__QUOTE(s(dbls(x0))) → A__QUOTE(a__dbls(mark(x0)))
A__QUOTE(s(dbl1(x0))) → A__QUOTE(a__dbl1(mark(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)))
a__dbl1(0) → 01
a__dbl1(s(X)) → s1(s1(a__dbl1(mark(X))))
a__sel1(0, cons(X, Y)) → mark(X)
a__sel1(s(X), cons(Y, Z)) → a__sel1(mark(X), mark(Z))
a__quote(0) → 01
a__quote(s(X)) → s1(a__quote(mark(X)))
a__quote(dbl(X)) → a__dbl1(mark(X))
a__quote(sel(X, Y)) → a__sel1(mark(X), mark(Y))
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(dbl1(X)) → a__dbl1(mark(X))
mark(sel1(X1, X2)) → a__sel1(mark(X1), mark(X2))
mark(quote(X)) → a__quote(mark(X))
mark(0) → 0
mark(s(X)) → s(X)
mark(nil) → nil
mark(cons(X1, X2)) → cons(X1, X2)
mark(01) → 01
mark(s1(X)) → s1(mark(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__from(X) → from(X)
a__dbl1(X) → dbl1(X)
a__sel1(X1, X2) → sel1(X1, X2)
a__quote(X) → quote(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__QUOTE(s(dbl1(x0))) → A__QUOTE(a__dbl1(mark(x0)))

The remaining pairs can at least be oriented weakly.

A__QUOTE(s(quote(x0))) → A__QUOTE(a__quote(mark(x0)))
A__QUOTE(s(sel(x0, x1))) → A__QUOTE(a__sel(mark(x0), mark(x1)))
A__QUOTE(s(dbl(x0))) → A__QUOTE(a__dbl(mark(x0)))
A__QUOTE(s(indx(x0, x1))) → A__QUOTE(a__indx(mark(x0), x1))
A__QUOTE(s(sel1(x0, x1))) → A__QUOTE(a__sel1(mark(x0), mark(x1)))
A__QUOTE(s(s(x0))) → A__QUOTE(s(x0))
A__QUOTE(s(dbls(x0))) → A__QUOTE(a__dbls(mark(x0)))

Used ordering: Polynomial interpretation [25]:

POL(0) = 0
POL(01) = 0
POL(A__QUOTE(x₁)) = x₁
POL(a__dbl(x₁)) = x₁
POL(a__dbl1(x₁)) = 0
POL(a__dbls(x₁)) = x₁
POL(a__from(x₁)) = 1
POL(a__indx(x₁, x₂)) = 1
POL(a__quote(x₁)) = 1
POL(a__sel(x₁, x₂)) = x₂
POL(a__sel1(x₁, x₂)) = x₂
POL(cons(x₁, x₂)) = 1
POL(dbl(x₁)) = 0
POL(dbl1(x₁)) = 0
POL(dbls(x₁)) = x₁
POL(from(x₁)) = 0
POL(indx(x₁, x₂)) = 0
POL(mark(x₁)) = 1
POL(nil) = 0
POL(quote(x₁)) = 1
POL(s(x₁)) = 1
POL(s1(x₁)) = 0
POL(sel(x₁, x₂)) = 0
POL(sel1(x₁, x₂)) = 0

The following usable rules [17] were oriented:

a__dbls(cons(X, Y)) → cons(dbl(X), dbls(Y))
a__dbls(nil) → nil
a__dbl(s(X)) → s(s(dbl(X)))
a__dbl(0) → 0
a__dbl1(s(X)) → s1(s1(a__dbl1(mark(X))))
a__quote(0) → 01
a__indx(cons(X, Y), Z) → cons(sel(X, Z), indx(Y, Z))
a__indx(nil, X) → nil
a__dbl1(0) → 01
a__from(X) → cons(X, from(s(X)))
mark(indx(X1, X2)) → a__indx(mark(X1), X2)
mark(from(X)) → a__from(X)
mark(dbls(X)) → a__dbls(mark(X))
mark(sel1(X1, X2)) → a__sel1(mark(X1), mark(X2))
a__sel(s(X), cons(Y, Z)) → a__sel(mark(X), mark(Z))
a__sel1(0, cons(X, Y)) → mark(X)
a__sel(0, cons(X, Y)) → mark(X)
a__sel1(s(X), cons(Y, Z)) → a__sel1(mark(X), mark(Z))
mark(quote(X)) → a__quote(mark(X))
a__quote(sel(X, Y)) → a__sel1(mark(X), mark(Y))
mark(sel(X1, X2)) → a__sel(mark(X1), mark(X2))
mark(dbl(X)) → a__dbl(mark(X))
a__quote(s(X)) → s1(a__quote(mark(X)))
a__quote(dbl(X)) → a__dbl1(mark(X))
mark(cons(X1, X2)) → cons(X1, X2)
mark(01) → 01
mark(s(X)) → s(X)
mark(nil) → nil
mark(0) → 0
mark(dbl1(X)) → a__dbl1(mark(X))
a__sel1(X1, X2) → sel1(X1, X2)
a__dbl1(X) → dbl1(X)
a__from(X) → from(X)
a__indx(X1, X2) → indx(X1, X2)
a__sel(X1, X2) → sel(X1, X2)
a__dbls(X) → dbls(X)
a__dbl(X) → dbl(X)
mark(s1(X)) → s1(mark(X))
a__quote(X) → quote(X)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ QDPOrderProof

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ AND

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ Narrowing

                                  ↳ QDP

                                    ↳ DependencyGraphProof

                                      ↳ QDP

                                        ↳ QDPOrderProof

                                          ↳ QDP

                                            ↳ QDPOrderProof

                      ↳ QDP

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

A__QUOTE(s(quote(x0))) → A__QUOTE(a__quote(mark(x0)))
A__QUOTE(s(sel(x0, x1))) → A__QUOTE(a__sel(mark(x0), mark(x1)))
A__QUOTE(s(indx(x0, x1))) → A__QUOTE(a__indx(mark(x0), x1))
A__QUOTE(s(dbl(x0))) → A__QUOTE(a__dbl(mark(x0)))
A__QUOTE(s(sel1(x0, x1))) → A__QUOTE(a__sel1(mark(x0), mark(x1)))
A__QUOTE(s(s(x0))) → A__QUOTE(s(x0))
A__QUOTE(s(dbls(x0))) → A__QUOTE(a__dbls(mark(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)))
a__dbl1(0) → 01
a__dbl1(s(X)) → s1(s1(a__dbl1(mark(X))))
a__sel1(0, cons(X, Y)) → mark(X)
a__sel1(s(X), cons(Y, Z)) → a__sel1(mark(X), mark(Z))
a__quote(0) → 01
a__quote(s(X)) → s1(a__quote(mark(X)))
a__quote(dbl(X)) → a__dbl1(mark(X))
a__quote(sel(X, Y)) → a__sel1(mark(X), mark(Y))
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(dbl1(X)) → a__dbl1(mark(X))
mark(sel1(X1, X2)) → a__sel1(mark(X1), mark(X2))
mark(quote(X)) → a__quote(mark(X))
mark(0) → 0
mark(s(X)) → s(X)
mark(nil) → nil
mark(cons(X1, X2)) → cons(X1, X2)
mark(01) → 01
mark(s1(X)) → s1(mark(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__from(X) → from(X)
a__dbl1(X) → dbl1(X)
a__sel1(X1, X2) → sel1(X1, X2)
a__quote(X) → quote(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__QUOTE(s(indx(x0, x1))) → A__QUOTE(a__indx(mark(x0), x1))
A__QUOTE(s(dbls(x0))) → A__QUOTE(a__dbls(mark(x0)))

The remaining pairs can at least be oriented weakly.

A__QUOTE(s(quote(x0))) → A__QUOTE(a__quote(mark(x0)))
A__QUOTE(s(sel(x0, x1))) → A__QUOTE(a__sel(mark(x0), mark(x1)))
A__QUOTE(s(dbl(x0))) → A__QUOTE(a__dbl(mark(x0)))
A__QUOTE(s(sel1(x0, x1))) → A__QUOTE(a__sel1(mark(x0), mark(x1)))
A__QUOTE(s(s(x0))) → A__QUOTE(s(x0))

Used ordering: Polynomial interpretation [25]:

POL(0) = 0
POL(01) = 1
POL(A__QUOTE(x₁)) = x₁
POL(a__dbl(x₁)) = x₁
POL(a__dbl1(x₁)) = 1
POL(a__dbls(x₁)) = 0
POL(a__from(x₁)) = 1
POL(a__indx(x₁, x₂)) = 0
POL(a__quote(x₁)) = 1
POL(a__sel(x₁, x₂)) = 1
POL(a__sel1(x₁, x₂)) = 1
POL(cons(x₁, x₂)) = 0
POL(dbl(x₁)) = x₁
POL(dbl1(x₁)) = 1
POL(dbls(x₁)) = 0
POL(from(x₁)) = 1
POL(indx(x₁, x₂)) = 0
POL(mark(x₁)) = 1
POL(nil) = 0
POL(quote(x₁)) = 0
POL(s(x₁)) = 1
POL(s1(x₁)) = 0
POL(sel(x₁, x₂)) = 0
POL(sel1(x₁, x₂)) = 0

The following usable rules [17] were oriented:

a__dbls(cons(X, Y)) → cons(dbl(X), dbls(Y))
a__dbls(nil) → nil
a__dbl(s(X)) → s(s(dbl(X)))
a__dbl(0) → 0
a__dbl1(s(X)) → s1(s1(a__dbl1(mark(X))))
a__quote(0) → 01
a__indx(cons(X, Y), Z) → cons(sel(X, Z), indx(Y, Z))
a__indx(nil, X) → nil
a__dbl1(0) → 01
a__from(X) → cons(X, from(s(X)))
mark(indx(X1, X2)) → a__indx(mark(X1), X2)
mark(from(X)) → a__from(X)
mark(dbls(X)) → a__dbls(mark(X))
mark(sel1(X1, X2)) → a__sel1(mark(X1), mark(X2))
a__sel(s(X), cons(Y, Z)) → a__sel(mark(X), mark(Z))
a__sel1(0, cons(X, Y)) → mark(X)
a__sel(0, cons(X, Y)) → mark(X)
a__sel1(s(X), cons(Y, Z)) → a__sel1(mark(X), mark(Z))
mark(quote(X)) → a__quote(mark(X))
a__quote(sel(X, Y)) → a__sel1(mark(X), mark(Y))
mark(sel(X1, X2)) → a__sel(mark(X1), mark(X2))
mark(dbl(X)) → a__dbl(mark(X))
a__quote(s(X)) → s1(a__quote(mark(X)))
a__quote(dbl(X)) → a__dbl1(mark(X))
mark(cons(X1, X2)) → cons(X1, X2)
mark(01) → 01
mark(s(X)) → s(X)
mark(nil) → nil
mark(0) → 0
mark(dbl1(X)) → a__dbl1(mark(X))
a__sel1(X1, X2) → sel1(X1, X2)
a__dbl1(X) → dbl1(X)
a__from(X) → from(X)
a__indx(X1, X2) → indx(X1, X2)
a__sel(X1, X2) → sel(X1, X2)
a__dbls(X) → dbls(X)
a__dbl(X) → dbl(X)
mark(s1(X)) → s1(mark(X))
a__quote(X) → quote(X)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ QDPOrderProof

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ AND

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ Narrowing

                                  ↳ QDP

                                    ↳ DependencyGraphProof

                                      ↳ QDP

                                        ↳ QDPOrderProof

                                          ↳ QDP

                                            ↳ QDPOrderProof

                                              ↳ QDP

                      ↳ QDP

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

A__QUOTE(s(quote(x0))) → A__QUOTE(a__quote(mark(x0)))
A__QUOTE(s(sel(x0, x1))) → A__QUOTE(a__sel(mark(x0), mark(x1)))
A__QUOTE(s(dbl(x0))) → A__QUOTE(a__dbl(mark(x0)))
A__QUOTE(s(sel1(x0, x1))) → A__QUOTE(a__sel1(mark(x0), mark(x1)))
A__QUOTE(s(s(x0))) → A__QUOTE(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)))
a__dbl1(0) → 01
a__dbl1(s(X)) → s1(s1(a__dbl1(mark(X))))
a__sel1(0, cons(X, Y)) → mark(X)
a__sel1(s(X), cons(Y, Z)) → a__sel1(mark(X), mark(Z))
a__quote(0) → 01
a__quote(s(X)) → s1(a__quote(mark(X)))
a__quote(dbl(X)) → a__dbl1(mark(X))
a__quote(sel(X, Y)) → a__sel1(mark(X), mark(Y))
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(dbl1(X)) → a__dbl1(mark(X))
mark(sel1(X1, X2)) → a__sel1(mark(X1), mark(X2))
mark(quote(X)) → a__quote(mark(X))
mark(0) → 0
mark(s(X)) → s(X)
mark(nil) → nil
mark(cons(X1, X2)) → cons(X1, X2)
mark(01) → 01
mark(s1(X)) → s1(mark(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__from(X) → from(X)
a__dbl1(X) → dbl1(X)
a__sel1(X1, X2) → sel1(X1, X2)
a__quote(X) → quote(X)

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ QDPOrderProof

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ AND

                      ↳ QDP

                      ↳ QDP

                        ↳ QDPOrderProof

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

A__SEL1(s(X), cons(Y, Z)) → A__SEL1(mark(X), mark(Z))
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))
A__DBL1(s(X)) → MARK(X)
MARK(dbl(X)) → MARK(X)
MARK(dbl1(X)) → A__DBL1(mark(X))
MARK(s1(X)) → MARK(X)
MARK(dbls(X)) → MARK(X)
MARK(sel1(X1, X2)) → MARK(X1)
A__SEL(s(X), cons(Y, Z)) → MARK(Z)
A__SEL1(s(X), cons(Y, Z)) → MARK(Z)
MARK(sel1(X1, X2)) → MARK(X2)
MARK(dbl1(X)) → MARK(X)
MARK(sel1(X1, X2)) → A__SEL1(mark(X1), mark(X2))
A__DBL1(s(X)) → A__DBL1(mark(X))
A__SEL1(s(X), cons(Y, Z)) → 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)))
a__dbl1(0) → 01
a__dbl1(s(X)) → s1(s1(a__dbl1(mark(X))))
a__sel1(0, cons(X, Y)) → mark(X)
a__sel1(s(X), cons(Y, Z)) → a__sel1(mark(X), mark(Z))
a__quote(0) → 01
a__quote(s(X)) → s1(a__quote(mark(X)))
a__quote(dbl(X)) → a__dbl1(mark(X))
a__quote(sel(X, Y)) → a__sel1(mark(X), mark(Y))
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(dbl1(X)) → a__dbl1(mark(X))
mark(sel1(X1, X2)) → a__sel1(mark(X1), mark(X2))
mark(quote(X)) → a__quote(mark(X))
mark(0) → 0
mark(s(X)) → s(X)
mark(nil) → nil
mark(cons(X1, X2)) → cons(X1, X2)
mark(01) → 01
mark(s1(X)) → s1(mark(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__from(X) → from(X)
a__dbl1(X) → dbl1(X)
a__sel1(X1, X2) → sel1(X1, X2)
a__quote(X) → quote(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)
MARK(sel1(X1, X2)) → MARK(X1)
MARK(sel1(X1, X2)) → MARK(X2)
MARK(sel1(X1, X2)) → A__SEL1(mark(X1), mark(X2))

The remaining pairs can at least be oriented weakly.

A__SEL1(s(X), cons(Y, Z)) → A__SEL1(mark(X), mark(Z))
MARK(sel(X1, X2)) → MARK(X1)
MARK(sel(X1, X2)) → A__SEL(mark(X1), mark(X2))
A__SEL(s(X), cons(Y, Z)) → A__SEL(mark(X), mark(Z))
A__DBL1(s(X)) → MARK(X)
MARK(dbl(X)) → MARK(X)
MARK(dbl1(X)) → A__DBL1(mark(X))
MARK(s1(X)) → MARK(X)
MARK(dbls(X)) → MARK(X)
A__SEL(s(X), cons(Y, Z)) → MARK(Z)
A__SEL1(s(X), cons(Y, Z)) → MARK(Z)
MARK(dbl1(X)) → MARK(X)
A__DBL1(s(X)) → A__DBL1(mark(X))
A__SEL1(s(X), cons(Y, Z)) → 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(01) = 0
POL(A__DBL1(x₁)) = x₁
POL(A__SEL(x₁, x₂)) = x₁ + x₂
POL(A__SEL1(x₁, x₂)) = x₁ + x₂
POL(MARK(x₁)) = x₁
POL(a__dbl(x₁)) = x₁
POL(a__dbl1(x₁)) = x₁
POL(a__dbls(x₁)) = x₁
POL(a__from(x₁)) = x₁
POL(a__indx(x₁, x₂)) = 1 + x₁ + x₂
POL(a__quote(x₁)) = 1 + x₁
POL(a__sel(x₁, x₂)) = x₁ + x₂
POL(a__sel1(x₁, x₂)) = 1 + x₁ + x₂
POL(cons(x₁, x₂)) = max(x₁, x₂)
POL(dbl(x₁)) = x₁
POL(dbl1(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(quote(x₁)) = 1 + x₁
POL(s(x₁)) = x₁
POL(s1(x₁)) = x₁
POL(sel(x₁, x₂)) = x₁ + x₂
POL(sel1(x₁, x₂)) = 1 + x₁ + x₂

The following usable rules [17] were oriented:

a__dbls(cons(X, Y)) → cons(dbl(X), dbls(Y))
a__dbls(nil) → nil
a__dbl(s(X)) → s(s(dbl(X)))
a__dbl(0) → 0
a__dbl1(s(X)) → s1(s1(a__dbl1(mark(X))))
a__quote(0) → 01
a__indx(cons(X, Y), Z) → cons(sel(X, Z), indx(Y, Z))
a__indx(nil, X) → nil
a__dbl1(0) → 01
a__from(X) → cons(X, from(s(X)))
mark(indx(X1, X2)) → a__indx(mark(X1), X2)
mark(from(X)) → a__from(X)
mark(dbls(X)) → a__dbls(mark(X))
mark(sel1(X1, X2)) → a__sel1(mark(X1), mark(X2))
a__sel(s(X), cons(Y, Z)) → a__sel(mark(X), mark(Z))
a__sel1(0, cons(X, Y)) → mark(X)
a__sel(0, cons(X, Y)) → mark(X)
a__sel1(s(X), cons(Y, Z)) → a__sel1(mark(X), mark(Z))
mark(quote(X)) → a__quote(mark(X))
a__quote(sel(X, Y)) → a__sel1(mark(X), mark(Y))
mark(sel(X1, X2)) → a__sel(mark(X1), mark(X2))
mark(dbl(X)) → a__dbl(mark(X))
a__quote(s(X)) → s1(a__quote(mark(X)))
a__quote(dbl(X)) → a__dbl1(mark(X))
mark(cons(X1, X2)) → cons(X1, X2)
mark(01) → 01
mark(s(X)) → s(X)
mark(nil) → nil
mark(0) → 0
mark(dbl1(X)) → a__dbl1(mark(X))
a__sel1(X1, X2) → sel1(X1, X2)
a__dbl1(X) → dbl1(X)
a__from(X) → from(X)
a__indx(X1, X2) → indx(X1, X2)
a__sel(X1, X2) → sel(X1, X2)
a__dbls(X) → dbls(X)
a__dbl(X) → dbl(X)
mark(s1(X)) → s1(mark(X))
a__quote(X) → quote(X)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ QDPOrderProof

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ AND

                      ↳ QDP

                      ↳ QDP

                        ↳ QDPOrderProof

                          ↳ QDP

                            ↳ DependencyGraphProof

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

A__SEL1(s(X), cons(Y, Z)) → A__SEL1(mark(X), mark(Z))
MARK(sel(X1, X2)) → MARK(X1)
MARK(sel(X1, X2)) → A__SEL(mark(X1), mark(X2))
A__SEL(s(X), cons(Y, Z)) → A__SEL(mark(X), mark(Z))
A__DBL1(s(X)) → MARK(X)
MARK(dbl(X)) → MARK(X)
MARK(dbl1(X)) → A__DBL1(mark(X))
MARK(s1(X)) → MARK(X)
MARK(dbls(X)) → MARK(X)
A__SEL(s(X), cons(Y, Z)) → MARK(Z)
A__SEL1(s(X), cons(Y, Z)) → MARK(Z)
MARK(dbl1(X)) → MARK(X)
A__DBL1(s(X)) → A__DBL1(mark(X))
MARK(sel(X1, X2)) → MARK(X2)
A__SEL(s(X), cons(Y, Z)) → MARK(X)
A__SEL1(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)))
a__dbl1(0) → 01
a__dbl1(s(X)) → s1(s1(a__dbl1(mark(X))))
a__sel1(0, cons(X, Y)) → mark(X)
a__sel1(s(X), cons(Y, Z)) → a__sel1(mark(X), mark(Z))
a__quote(0) → 01
a__quote(s(X)) → s1(a__quote(mark(X)))
a__quote(dbl(X)) → a__dbl1(mark(X))
a__quote(sel(X, Y)) → a__sel1(mark(X), mark(Y))
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(dbl1(X)) → a__dbl1(mark(X))
mark(sel1(X1, X2)) → a__sel1(mark(X1), mark(X2))
mark(quote(X)) → a__quote(mark(X))
mark(0) → 0
mark(s(X)) → s(X)
mark(nil) → nil
mark(cons(X1, X2)) → cons(X1, X2)
mark(01) → 01
mark(s1(X)) → s1(mark(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__from(X) → from(X)
a__dbl1(X) → dbl1(X)
a__sel1(X1, X2) → sel1(X1, X2)
a__quote(X) → quote(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 2 less nodes.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ QDPOrderProof

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ AND

                      ↳ QDP

                      ↳ QDP

                        ↳ QDPOrderProof

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ AND

                                ↳ QDP

                                  ↳ QDPOrderProof

                                ↳ QDP

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)) → A__SEL(mark(X), mark(Z))
A__DBL1(s(X)) → MARK(X)
MARK(dbl(X)) → MARK(X)
MARK(dbl1(X)) → A__DBL1(mark(X))
MARK(s1(X)) → MARK(X)
MARK(dbls(X)) → MARK(X)
A__SEL(s(X), cons(Y, Z)) → MARK(Z)
MARK(dbl1(X)) → MARK(X)
A__DBL1(s(X)) → A__DBL1(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)))
a__dbl1(0) → 01
a__dbl1(s(X)) → s1(s1(a__dbl1(mark(X))))
a__sel1(0, cons(X, Y)) → mark(X)
a__sel1(s(X), cons(Y, Z)) → a__sel1(mark(X), mark(Z))
a__quote(0) → 01
a__quote(s(X)) → s1(a__quote(mark(X)))
a__quote(dbl(X)) → a__dbl1(mark(X))
a__quote(sel(X, Y)) → a__sel1(mark(X), mark(Y))
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(dbl1(X)) → a__dbl1(mark(X))
mark(sel1(X1, X2)) → a__sel1(mark(X1), mark(X2))
mark(quote(X)) → a__quote(mark(X))
mark(0) → 0
mark(s(X)) → s(X)
mark(nil) → nil
mark(cons(X1, X2)) → cons(X1, X2)
mark(01) → 01
mark(s1(X)) → s1(mark(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__from(X) → from(X)
a__dbl1(X) → dbl1(X)
a__sel1(X1, X2) → sel1(X1, X2)
a__quote(X) → quote(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(dbl1(X)) → A__DBL1(mark(X))
MARK(dbls(X)) → MARK(X)
MARK(dbl1(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)) → A__SEL(mark(X), mark(Z))
A__DBL1(s(X)) → MARK(X)
MARK(dbl(X)) → MARK(X)
MARK(s1(X)) → MARK(X)
A__SEL(s(X), cons(Y, Z)) → MARK(Z)
A__DBL1(s(X)) → A__DBL1(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(01) = 0
POL(A__DBL1(x₁)) = x₁
POL(A__SEL(x₁, x₂)) = x₁ + x₂
POL(MARK(x₁)) = x₁
POL(a__dbl(x₁)) = x₁
POL(a__dbl1(x₁)) = 1 + x₁
POL(a__dbls(x₁)) = 1 + x₁
POL(a__from(x₁)) = x₁
POL(a__indx(x₁, x₂)) = x₁ + x₂
POL(a__quote(x₁)) = 1 + x₁
POL(a__sel(x₁, x₂)) = x₁ + x₂
POL(a__sel1(x₁, x₂)) = 1 + x₂
POL(cons(x₁, x₂)) = max(x₁, x₂)
POL(dbl(x₁)) = x₁
POL(dbl1(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(quote(x₁)) = 1 + x₁
POL(s(x₁)) = x₁
POL(s1(x₁)) = x₁
POL(sel(x₁, x₂)) = x₁ + x₂
POL(sel1(x₁, x₂)) = 1 + x₂

The following usable rules [17] were oriented:

a__dbls(cons(X, Y)) → cons(dbl(X), dbls(Y))
a__dbls(nil) → nil
a__dbl(s(X)) → s(s(dbl(X)))
a__dbl(0) → 0
a__dbl1(s(X)) → s1(s1(a__dbl1(mark(X))))
a__quote(0) → 01
a__indx(cons(X, Y), Z) → cons(sel(X, Z), indx(Y, Z))
a__indx(nil, X) → nil
a__dbl1(0) → 01
a__from(X) → cons(X, from(s(X)))
mark(indx(X1, X2)) → a__indx(mark(X1), X2)
mark(from(X)) → a__from(X)
mark(dbls(X)) → a__dbls(mark(X))
mark(sel1(X1, X2)) → a__sel1(mark(X1), mark(X2))
a__sel(s(X), cons(Y, Z)) → a__sel(mark(X), mark(Z))
a__sel1(0, cons(X, Y)) → mark(X)
a__sel(0, cons(X, Y)) → mark(X)
a__sel1(s(X), cons(Y, Z)) → a__sel1(mark(X), mark(Z))
mark(quote(X)) → a__quote(mark(X))
a__quote(sel(X, Y)) → a__sel1(mark(X), mark(Y))
mark(sel(X1, X2)) → a__sel(mark(X1), mark(X2))
mark(dbl(X)) → a__dbl(mark(X))
a__quote(s(X)) → s1(a__quote(mark(X)))
a__quote(dbl(X)) → a__dbl1(mark(X))
mark(cons(X1, X2)) → cons(X1, X2)
mark(01) → 01
mark(s(X)) → s(X)
mark(nil) → nil
mark(0) → 0
mark(dbl1(X)) → a__dbl1(mark(X))
a__sel1(X1, X2) → sel1(X1, X2)
a__dbl1(X) → dbl1(X)
a__from(X) → from(X)
a__indx(X1, X2) → indx(X1, X2)
a__sel(X1, X2) → sel(X1, X2)
a__dbls(X) → dbls(X)
a__dbl(X) → dbl(X)
mark(s1(X)) → s1(mark(X))
a__quote(X) → quote(X)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ QDPOrderProof

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ AND

                      ↳ QDP

                      ↳ QDP

                        ↳ QDPOrderProof

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ AND

                                ↳ QDP

                                  ↳ QDPOrderProof

                                    ↳ QDP

                                      ↳ DependencyGraphProof

                                ↳ QDP

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__DBL1(s(X)) → MARK(X)
A__SEL(s(X), cons(Y, Z)) → A__SEL(mark(X), mark(Z))
MARK(dbl(X)) → MARK(X)
MARK(s1(X)) → MARK(X)
A__DBL1(s(X)) → A__DBL1(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)))
a__dbl1(0) → 01
a__dbl1(s(X)) → s1(s1(a__dbl1(mark(X))))
a__sel1(0, cons(X, Y)) → mark(X)
a__sel1(s(X), cons(Y, Z)) → a__sel1(mark(X), mark(Z))
a__quote(0) → 01
a__quote(s(X)) → s1(a__quote(mark(X)))
a__quote(dbl(X)) → a__dbl1(mark(X))
a__quote(sel(X, Y)) → a__sel1(mark(X), mark(Y))
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(dbl1(X)) → a__dbl1(mark(X))
mark(sel1(X1, X2)) → a__sel1(mark(X1), mark(X2))
mark(quote(X)) → a__quote(mark(X))
mark(0) → 0
mark(s(X)) → s(X)
mark(nil) → nil
mark(cons(X1, X2)) → cons(X1, X2)
mark(01) → 01
mark(s1(X)) → s1(mark(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__from(X) → from(X)
a__dbl1(X) → dbl1(X)
a__sel1(X1, X2) → sel1(X1, X2)
a__quote(X) → quote(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

                  ↳ DependencyGraphProof

                    ↳ AND

                      ↳ QDP

                      ↳ QDP

                        ↳ QDPOrderProof

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ AND

                                ↳ QDP

                                  ↳ QDPOrderProof

                                    ↳ QDP

                                      ↳ DependencyGraphProof

                                        ↳ AND

                                          ↳ QDP

                                            ↳ QDPOrderProof

                                          ↳ QDP

                                ↳ QDP

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))
MARK(dbl(X)) → MARK(X)
MARK(s1(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)))
a__dbl1(0) → 01
a__dbl1(s(X)) → s1(s1(a__dbl1(mark(X))))
a__sel1(0, cons(X, Y)) → mark(X)
a__sel1(s(X), cons(Y, Z)) → a__sel1(mark(X), mark(Z))
a__quote(0) → 01
a__quote(s(X)) → s1(a__quote(mark(X)))
a__quote(dbl(X)) → a__dbl1(mark(X))
a__quote(sel(X, Y)) → a__sel1(mark(X), mark(Y))
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(dbl1(X)) → a__dbl1(mark(X))
mark(sel1(X1, X2)) → a__sel1(mark(X1), mark(X2))
mark(quote(X)) → a__quote(mark(X))
mark(0) → 0
mark(s(X)) → s(X)
mark(nil) → nil
mark(cons(X1, X2)) → cons(X1, X2)
mark(01) → 01
mark(s1(X)) → s1(mark(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__from(X) → from(X)
a__dbl1(X) → dbl1(X)
a__sel1(X1, X2) → sel1(X1, X2)
a__quote(X) → quote(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))
MARK(dbl(X)) → MARK(X)
MARK(s1(X)) → MARK(X)
A__SEL(s(X), cons(Y, Z)) → MARK(X)

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

POL(0) = 0
POL(01) = 0
POL(A__SEL(x₁, x₂)) = x₁ + x₂
POL(MARK(x₁)) = x₁
POL(a__dbl(x₁)) = x₁
POL(a__dbl1(x₁)) = 0
POL(a__dbls(x₁)) = x₁
POL(a__from(x₁)) = x₁
POL(a__indx(x₁, x₂)) = 1 + x₁ + x₂
POL(a__quote(x₁)) = x₁
POL(a__sel(x₁, x₂)) = 1 + x₁ + x₂
POL(a__sel1(x₁, x₂)) = x₂
POL(cons(x₁, x₂)) = max(x₁, x₂)
POL(dbl(x₁)) = x₁
POL(dbl1(x₁)) = 0
POL(dbls(x₁)) = x₁
POL(from(x₁)) = x₁
POL(indx(x₁, x₂)) = 1 + x₁ + x₂
POL(mark(x₁)) = x₁
POL(nil) = 0
POL(quote(x₁)) = x₁
POL(s(x₁)) = x₁
POL(s1(x₁)) = x₁
POL(sel(x₁, x₂)) = 1 + x₁ + x₂
POL(sel1(x₁, x₂)) = x₂

The following usable rules [17] were oriented:

a__dbls(cons(X, Y)) → cons(dbl(X), dbls(Y))
a__dbls(nil) → nil
a__dbl(s(X)) → s(s(dbl(X)))
a__dbl(0) → 0
a__dbl1(s(X)) → s1(s1(a__dbl1(mark(X))))
a__quote(0) → 01
a__indx(cons(X, Y), Z) → cons(sel(X, Z), indx(Y, Z))
a__indx(nil, X) → nil
a__dbl1(0) → 01
a__from(X) → cons(X, from(s(X)))
mark(indx(X1, X2)) → a__indx(mark(X1), X2)
mark(from(X)) → a__from(X)
mark(dbls(X)) → a__dbls(mark(X))
mark(sel1(X1, X2)) → a__sel1(mark(X1), mark(X2))
a__sel(s(X), cons(Y, Z)) → a__sel(mark(X), mark(Z))
a__sel1(0, cons(X, Y)) → mark(X)
a__sel(0, cons(X, Y)) → mark(X)
a__sel1(s(X), cons(Y, Z)) → a__sel1(mark(X), mark(Z))
mark(quote(X)) → a__quote(mark(X))
a__quote(sel(X, Y)) → a__sel1(mark(X), mark(Y))
mark(sel(X1, X2)) → a__sel(mark(X1), mark(X2))
mark(dbl(X)) → a__dbl(mark(X))
a__quote(s(X)) → s1(a__quote(mark(X)))
a__quote(dbl(X)) → a__dbl1(mark(X))
mark(cons(X1, X2)) → cons(X1, X2)
mark(01) → 01
mark(s(X)) → s(X)
mark(nil) → nil
mark(0) → 0
mark(dbl1(X)) → a__dbl1(mark(X))
a__sel1(X1, X2) → sel1(X1, X2)
a__dbl1(X) → dbl1(X)
a__from(X) → from(X)
a__indx(X1, X2) → indx(X1, X2)
a__sel(X1, X2) → sel(X1, X2)
a__dbls(X) → dbls(X)
a__dbl(X) → dbl(X)
mark(s1(X)) → s1(mark(X))
a__quote(X) → quote(X)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ QDPOrderProof

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ AND

                      ↳ QDP

                      ↳ QDP

                        ↳ QDPOrderProof

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ AND

                                ↳ QDP

                                  ↳ QDPOrderProof

                                    ↳ QDP

                                      ↳ DependencyGraphProof

                                        ↳ AND

                                          ↳ QDP

                                            ↳ QDPOrderProof

                                              ↳ QDP

                                                ↳ DependencyGraphProof

                                          ↳ QDP

                                ↳ QDP

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))
MARK(dbl(X)) → MARK(X)
MARK(s1(X)) → MARK(X)
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)))
a__dbl1(0) → 01
a__dbl1(s(X)) → s1(s1(a__dbl1(mark(X))))
a__sel1(0, cons(X, Y)) → mark(X)
a__sel1(s(X), cons(Y, Z)) → a__sel1(mark(X), mark(Z))
a__quote(0) → 01
a__quote(s(X)) → s1(a__quote(mark(X)))
a__quote(dbl(X)) → a__dbl1(mark(X))
a__quote(sel(X, Y)) → a__sel1(mark(X), mark(Y))
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(dbl1(X)) → a__dbl1(mark(X))
mark(sel1(X1, X2)) → a__sel1(mark(X1), mark(X2))
mark(quote(X)) → a__quote(mark(X))
mark(0) → 0
mark(s(X)) → s(X)
mark(nil) → nil
mark(cons(X1, X2)) → cons(X1, X2)
mark(01) → 01
mark(s1(X)) → s1(mark(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__from(X) → from(X)
a__dbl1(X) → dbl1(X)
a__sel1(X1, X2) → sel1(X1, X2)
a__quote(X) → quote(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 2 less nodes.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ QDPOrderProof

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ AND

                      ↳ QDP

                      ↳ QDP

                        ↳ QDPOrderProof

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ AND

                                ↳ QDP

                                  ↳ QDPOrderProof

                                    ↳ QDP

                                      ↳ DependencyGraphProof

                                        ↳ AND

                                          ↳ QDP

                                            ↳ QDPOrderProof

                                              ↳ QDP

                                                ↳ DependencyGraphProof

                                                  ↳ AND

                                                    ↳ QDP

                                                      ↳ UsableRulesProof

                                                    ↳ QDP

                                          ↳ QDP

                                ↳ QDP

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

MARK(dbl(X)) → MARK(X)
MARK(s1(X)) → 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)))
a__dbl1(0) → 01
a__dbl1(s(X)) → s1(s1(a__dbl1(mark(X))))
a__sel1(0, cons(X, Y)) → mark(X)
a__sel1(s(X), cons(Y, Z)) → a__sel1(mark(X), mark(Z))
a__quote(0) → 01
a__quote(s(X)) → s1(a__quote(mark(X)))
a__quote(dbl(X)) → a__dbl1(mark(X))
a__quote(sel(X, Y)) → a__sel1(mark(X), mark(Y))
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(dbl1(X)) → a__dbl1(mark(X))
mark(sel1(X1, X2)) → a__sel1(mark(X1), mark(X2))
mark(quote(X)) → a__quote(mark(X))
mark(0) → 0
mark(s(X)) → s(X)
mark(nil) → nil
mark(cons(X1, X2)) → cons(X1, X2)
mark(01) → 01
mark(s1(X)) → s1(mark(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__from(X) → from(X)
a__dbl1(X) → dbl1(X)
a__sel1(X1, X2) → sel1(X1, X2)
a__quote(X) → quote(X)

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ QDPOrderProof

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ AND

                      ↳ QDP

                      ↳ QDP

                        ↳ QDPOrderProof

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ AND

                                ↳ QDP

                                  ↳ QDPOrderProof

                                    ↳ QDP

                                      ↳ DependencyGraphProof

                                        ↳ AND

                                          ↳ QDP

                                            ↳ QDPOrderProof

                                              ↳ QDP

                                                ↳ DependencyGraphProof

                                                  ↳ AND

                                                    ↳ QDP

                                                      ↳ UsableRulesProof

                                                        ↳ QDP

                                                          ↳ QDPSizeChangeProof

                                                    ↳ QDP

                                          ↳ QDP

                                ↳ QDP

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

MARK(dbl(X)) → MARK(X)
MARK(s1(X)) → MARK(X)

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

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

MARK(dbl(X)) → MARK(X)
The graph contains the following edges 1 > 1
MARK(s1(X)) → MARK(X)
The graph contains the following edges 1 > 1

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ QDPOrderProof

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ AND

                      ↳ QDP

                      ↳ QDP

                        ↳ QDPOrderProof

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ AND

                                ↳ QDP

                                  ↳ QDPOrderProof

                                    ↳ QDP

                                      ↳ DependencyGraphProof

                                        ↳ AND

                                          ↳ QDP

                                            ↳ QDPOrderProof

                                              ↳ QDP

                                                ↳ DependencyGraphProof

                                                  ↳ AND

                                                    ↳ QDP

                                                    ↳ QDP

                                                      ↳ Narrowing

                                          ↳ QDP

                                ↳ QDP

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)))
a__dbl1(0) → 01
a__dbl1(s(X)) → s1(s1(a__dbl1(mark(X))))
a__sel1(0, cons(X, Y)) → mark(X)
a__sel1(s(X), cons(Y, Z)) → a__sel1(mark(X), mark(Z))
a__quote(0) → 01
a__quote(s(X)) → s1(a__quote(mark(X)))
a__quote(dbl(X)) → a__dbl1(mark(X))
a__quote(sel(X, Y)) → a__sel1(mark(X), mark(Y))
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(dbl1(X)) → a__dbl1(mark(X))
mark(sel1(X1, X2)) → a__sel1(mark(X1), mark(X2))
mark(quote(X)) → a__quote(mark(X))
mark(0) → 0
mark(s(X)) → s(X)
mark(nil) → nil
mark(cons(X1, X2)) → cons(X1, X2)
mark(01) → 01
mark(s1(X)) → s1(mark(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__from(X) → from(X)
a__dbl1(X) → dbl1(X)
a__sel1(X1, X2) → sel1(X1, X2)
a__quote(X) → quote(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(01), cons(y1, y2)) → A__SEL(01, mark(y2))
A__SEL(s(dbl(x0)), cons(y1, y2)) → A__SEL(a__dbl(mark(x0)), mark(y2))
A__SEL(s(cons(x0, x1)), cons(y1, y2)) → A__SEL(cons(x0, x1), mark(y2))
A__SEL(s(sel1(x0, x1)), cons(y1, y2)) → A__SEL(a__sel1(mark(x0), mark(x1)), mark(y2))
A__SEL(s(dbl1(x0)), cons(y1, y2)) → A__SEL(a__dbl1(mark(x0)), mark(y2))
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(s1(x0)), cons(y1, y2)) → A__SEL(s1(mark(x0)), mark(y2))
A__SEL(s(from(x0)), cons(y1, y2)) → A__SEL(a__from(x0), mark(y2))
A__SEL(s(quote(x0)), cons(y1, y2)) → A__SEL(a__quote(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(dbls(x0)), cons(y1, y2)) → A__SEL(a__dbls(mark(x0)), mark(y2))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ QDPOrderProof

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ AND

                      ↳ QDP

                      ↳ QDP

                        ↳ QDPOrderProof

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ AND

                                ↳ QDP

                                  ↳ QDPOrderProof

                                    ↳ QDP

                                      ↳ DependencyGraphProof

                                        ↳ AND

                                          ↳ QDP

                                            ↳ QDPOrderProof

                                              ↳ QDP

                                                ↳ DependencyGraphProof

                                                  ↳ AND

                                                    ↳ QDP

                                                    ↳ QDP

                                                      ↳ Narrowing

                                                        ↳ QDP

                                                          ↳ DependencyGraphProof

                                          ↳ QDP

                                ↳ QDP

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

A__SEL(s(01), cons(y1, y2)) → A__SEL(01, mark(y2))
A__SEL(s(dbl(x0)), cons(y1, y2)) → A__SEL(a__dbl(mark(x0)), mark(y2))
A__SEL(s(cons(x0, x1)), cons(y1, y2)) → A__SEL(cons(x0, x1), mark(y2))
A__SEL(s(sel1(x0, x1)), cons(y1, y2)) → A__SEL(a__sel1(mark(x0), mark(x1)), mark(y2))
A__SEL(s(dbl1(x0)), cons(y1, y2)) → A__SEL(a__dbl1(mark(x0)), mark(y2))
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(s1(x0)), cons(y1, y2)) → A__SEL(s1(mark(x0)), mark(y2))
A__SEL(s(from(x0)), cons(y1, y2)) → A__SEL(a__from(x0), mark(y2))
A__SEL(s(quote(x0)), cons(y1, y2)) → A__SEL(a__quote(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(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)))
a__dbl1(0) → 01
a__dbl1(s(X)) → s1(s1(a__dbl1(mark(X))))
a__sel1(0, cons(X, Y)) → mark(X)
a__sel1(s(X), cons(Y, Z)) → a__sel1(mark(X), mark(Z))
a__quote(0) → 01
a__quote(s(X)) → s1(a__quote(mark(X)))
a__quote(dbl(X)) → a__dbl1(mark(X))
a__quote(sel(X, Y)) → a__sel1(mark(X), mark(Y))
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(dbl1(X)) → a__dbl1(mark(X))
mark(sel1(X1, X2)) → a__sel1(mark(X1), mark(X2))
mark(quote(X)) → a__quote(mark(X))
mark(0) → 0
mark(s(X)) → s(X)
mark(nil) → nil
mark(cons(X1, X2)) → cons(X1, X2)
mark(01) → 01
mark(s1(X)) → s1(mark(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__from(X) → from(X)
a__dbl1(X) → dbl1(X)
a__sel1(X1, X2) → sel1(X1, X2)
a__quote(X) → quote(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 5 less nodes.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ QDPOrderProof

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ AND

                      ↳ QDP

                      ↳ QDP

                        ↳ QDPOrderProof

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ AND

                                ↳ QDP

                                  ↳ QDPOrderProof

                                    ↳ QDP

                                      ↳ DependencyGraphProof

                                        ↳ AND

                                          ↳ QDP

                                            ↳ QDPOrderProof

                                              ↳ QDP

                                                ↳ DependencyGraphProof

                                                  ↳ AND

                                                    ↳ QDP

                                                    ↳ QDP

                                                      ↳ Narrowing

                                                        ↳ QDP

                                                          ↳ DependencyGraphProof

                                                            ↳ QDP

                                                              ↳ Narrowing

                                          ↳ QDP

                                ↳ QDP

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

A__SEL(s(dbl1(x0)), cons(y1, y2)) → A__SEL(a__dbl1(mark(x0)), 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(quote(x0)), cons(y1, y2)) → A__SEL(a__quote(mark(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))
A__SEL(s(sel1(x0, x1)), cons(y1, y2)) → A__SEL(a__sel1(mark(x0), mark(x1)), 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)))
a__dbl1(0) → 01
a__dbl1(s(X)) → s1(s1(a__dbl1(mark(X))))
a__sel1(0, cons(X, Y)) → mark(X)
a__sel1(s(X), cons(Y, Z)) → a__sel1(mark(X), mark(Z))
a__quote(0) → 01
a__quote(s(X)) → s1(a__quote(mark(X)))
a__quote(dbl(X)) → a__dbl1(mark(X))
a__quote(sel(X, Y)) → a__sel1(mark(X), mark(Y))
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(dbl1(X)) → a__dbl1(mark(X))
mark(sel1(X1, X2)) → a__sel1(mark(X1), mark(X2))
mark(quote(X)) → a__quote(mark(X))
mark(0) → 0
mark(s(X)) → s(X)
mark(nil) → nil
mark(cons(X1, X2)) → cons(X1, X2)
mark(01) → 01
mark(s1(X)) → s1(mark(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__from(X) → from(X)
a__dbl1(X) → dbl1(X)
a__sel1(X1, X2) → sel1(X1, X2)
a__quote(X) → quote(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, dbl1(x0))) → A__SEL(a__dbl(mark(y0)), a__dbl1(mark(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))
A__SEL(s(dbl(y0)), cons(y1, sel1(x0, x1))) → A__SEL(a__dbl(mark(y0)), a__sel1(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, 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, quote(x0))) → A__SEL(a__dbl(mark(y0)), a__quote(mark(x0)))
A__SEL(s(dbl(y0)), cons(y1, 01)) → A__SEL(a__dbl(mark(y0)), 01)
A__SEL(s(dbl(y0)), cons(y1, s1(x0))) → A__SEL(a__dbl(mark(y0)), s1(mark(x0)))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ QDPOrderProof

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ AND

                      ↳ QDP

                      ↳ QDP

                        ↳ QDPOrderProof

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ AND

                                ↳ QDP

                                  ↳ QDPOrderProof

                                    ↳ QDP

                                      ↳ DependencyGraphProof

                                        ↳ AND

                                          ↳ QDP

                                            ↳ QDPOrderProof

                                              ↳ QDP

                                                ↳ DependencyGraphProof

                                                  ↳ AND

                                                    ↳ QDP

                                                    ↳ QDP

                                                      ↳ Narrowing

                                                        ↳ QDP

                                                          ↳ DependencyGraphProof

                                                            ↳ QDP

                                                              ↳ Narrowing

                                                                ↳ QDP

                                                                  ↳ DependencyGraphProof

                                          ↳ QDP

                                ↳ QDP

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

A__SEL(s(dbl(y0)), cons(y1, dbl1(x0))) → A__SEL(a__dbl(mark(y0)), a__dbl1(mark(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, 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, sel1(x0, x1))) → A__SEL(a__dbl(mark(y0)), a__sel1(mark(x0), mark(x1)))
A__SEL(s(sel1(x0, x1)), cons(y1, y2)) → A__SEL(a__sel1(mark(x0), mark(x1)), mark(y2))
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(dbl1(x0)), cons(y1, y2)) → A__SEL(a__dbl1(mark(x0)), mark(y2))
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(x0, x1)), cons(y1, y2)) → A__SEL(a__sel(mark(x0), mark(x1)), mark(y2))
A__SEL(s(dbl(y0)), cons(y1, s(x0))) → A__SEL(a__dbl(mark(y0)), s(x0))
A__SEL(s(from(x0)), cons(y1, y2)) → A__SEL(a__from(x0), mark(y2))
A__SEL(s(dbl(y0)), cons(y1, quote(x0))) → A__SEL(a__dbl(mark(y0)), a__quote(mark(x0)))
A__SEL(s(quote(x0)), cons(y1, y2)) → A__SEL(a__quote(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(dbl(y0)), cons(y1, 01)) → A__SEL(a__dbl(mark(y0)), 01)
A__SEL(s(s(x0)), cons(y1, y2)) → A__SEL(s(x0), mark(y2))
A__SEL(s(dbl(y0)), cons(y1, s1(x0))) → A__SEL(a__dbl(mark(y0)), s1(mark(x0)))
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)))
a__dbl1(0) → 01
a__dbl1(s(X)) → s1(s1(a__dbl1(mark(X))))
a__sel1(0, cons(X, Y)) → mark(X)
a__sel1(s(X), cons(Y, Z)) → a__sel1(mark(X), mark(Z))
a__quote(0) → 01
a__quote(s(X)) → s1(a__quote(mark(X)))
a__quote(dbl(X)) → a__dbl1(mark(X))
a__quote(sel(X, Y)) → a__sel1(mark(X), mark(Y))
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(dbl1(X)) → a__dbl1(mark(X))
mark(sel1(X1, X2)) → a__sel1(mark(X1), mark(X2))
mark(quote(X)) → a__quote(mark(X))
mark(0) → 0
mark(s(X)) → s(X)
mark(nil) → nil
mark(cons(X1, X2)) → cons(X1, X2)
mark(01) → 01
mark(s1(X)) → s1(mark(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__from(X) → from(X)
a__dbl1(X) → dbl1(X)
a__sel1(X1, X2) → sel1(X1, X2)
a__quote(X) → quote(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 5 less nodes.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ QDPOrderProof

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ AND

                      ↳ QDP

                      ↳ QDP

                        ↳ QDPOrderProof

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ AND

                                ↳ QDP

                                  ↳ QDPOrderProof

                                    ↳ QDP

                                      ↳ DependencyGraphProof

                                        ↳ AND

                                          ↳ QDP

                                            ↳ QDPOrderProof

                                              ↳ QDP

                                                ↳ DependencyGraphProof

                                                  ↳ AND

                                                    ↳ QDP

                                                    ↳ QDP

                                                      ↳ Narrowing

                                                        ↳ QDP

                                                          ↳ DependencyGraphProof

                                                            ↳ QDP

                                                              ↳ Narrowing

                                                                ↳ QDP

                                                                  ↳ DependencyGraphProof

                                                                    ↳ QDP

                                                                      ↳ Narrowing

                                          ↳ QDP

                                ↳ QDP

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

A__SEL(s(dbl(y0)), cons(y1, dbl1(x0))) → A__SEL(a__dbl(mark(y0)), a__dbl1(mark(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, from(x0))) → A__SEL(a__dbl(mark(y0)), a__from(x0))
A__SEL(s(sel1(x0, x1)), cons(y1, y2)) → A__SEL(a__sel1(mark(x0), mark(x1)), mark(y2))
A__SEL(s(dbl(y0)), cons(y1, sel1(x0, x1))) → A__SEL(a__dbl(mark(y0)), a__sel1(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(dbl1(x0)), cons(y1, y2)) → A__SEL(a__dbl1(mark(x0)), mark(y2))
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(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(y0)), cons(y1, quote(x0))) → A__SEL(a__dbl(mark(y0)), a__quote(mark(x0)))
A__SEL(s(quote(x0)), cons(y1, y2)) → A__SEL(a__quote(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)))
a__dbl1(0) → 01
a__dbl1(s(X)) → s1(s1(a__dbl1(mark(X))))
a__sel1(0, cons(X, Y)) → mark(X)
a__sel1(s(X), cons(Y, Z)) → a__sel1(mark(X), mark(Z))
a__quote(0) → 01
a__quote(s(X)) → s1(a__quote(mark(X)))
a__quote(dbl(X)) → a__dbl1(mark(X))
a__quote(sel(X, Y)) → a__sel1(mark(X), mark(Y))
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(dbl1(X)) → a__dbl1(mark(X))
mark(sel1(X1, X2)) → a__sel1(mark(X1), mark(X2))
mark(quote(X)) → a__quote(mark(X))
mark(0) → 0
mark(s(X)) → s(X)
mark(nil) → nil
mark(cons(X1, X2)) → cons(X1, X2)
mark(01) → 01
mark(s1(X)) → s1(mark(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__from(X) → from(X)
a__dbl1(X) → dbl1(X)
a__sel1(X1, X2) → sel1(X1, X2)
a__quote(X) → quote(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule A__SEL(s(sel1(x0, x1)), cons(y1, y2)) → A__SEL(a__sel1(mark(x0), mark(x1)), mark(y2)) at position [1] we obtained the following new rules:

A__SEL(s(sel1(y0, y1)), cons(y2, indx(x0, x1))) → A__SEL(a__sel1(mark(y0), mark(y1)), a__indx(mark(x0), x1))
A__SEL(s(sel1(y0, y1)), cons(y2, dbl1(x0))) → A__SEL(a__sel1(mark(y0), mark(y1)), a__dbl1(mark(x0)))
A__SEL(s(sel1(y0, y1)), cons(y2, dbl(x0))) → A__SEL(a__sel1(mark(y0), mark(y1)), a__dbl(mark(x0)))
A__SEL(s(sel1(y0, y1)), cons(y2, quote(x0))) → A__SEL(a__sel1(mark(y0), mark(y1)), a__quote(mark(x0)))
A__SEL(s(sel1(y0, y1)), cons(y2, sel(x0, x1))) → A__SEL(a__sel1(mark(y0), mark(y1)), a__sel(mark(x0), mark(x1)))
A__SEL(s(sel1(y0, y1)), cons(y2, nil)) → A__SEL(a__sel1(mark(y0), mark(y1)), nil)
A__SEL(s(sel1(y0, y1)), cons(y2, sel1(x0, x1))) → A__SEL(a__sel1(mark(y0), mark(y1)), a__sel1(mark(x0), mark(x1)))
A__SEL(s(sel1(y0, y1)), cons(y2, s1(x0))) → A__SEL(a__sel1(mark(y0), mark(y1)), s1(mark(x0)))
A__SEL(s(sel1(y0, y1)), cons(y2, dbls(x0))) → A__SEL(a__sel1(mark(y0), mark(y1)), a__dbls(mark(x0)))
A__SEL(s(sel1(y0, y1)), cons(y2, 0)) → A__SEL(a__sel1(mark(y0), mark(y1)), 0)
A__SEL(s(sel1(y0, y1)), cons(y2, cons(x0, x1))) → A__SEL(a__sel1(mark(y0), mark(y1)), cons(x0, x1))
A__SEL(s(sel1(y0, y1)), cons(y2, from(x0))) → A__SEL(a__sel1(mark(y0), mark(y1)), a__from(x0))
A__SEL(s(sel1(y0, y1)), cons(y2, s(x0))) → A__SEL(a__sel1(mark(y0), mark(y1)), s(x0))
A__SEL(s(sel1(y0, y1)), cons(y2, 01)) → A__SEL(a__sel1(mark(y0), mark(y1)), 01)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ QDPOrderProof

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ AND

                      ↳ QDP

                      ↳ QDP

                        ↳ QDPOrderProof

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ AND

                                ↳ QDP

                                  ↳ QDPOrderProof

                                    ↳ QDP

                                      ↳ DependencyGraphProof

                                        ↳ AND

                                          ↳ QDP

                                            ↳ QDPOrderProof

                                              ↳ QDP

                                                ↳ DependencyGraphProof

                                                  ↳ AND

                                                    ↳ QDP

                                                    ↳ QDP

                                                      ↳ Narrowing

                                                        ↳ QDP

                                                          ↳ DependencyGraphProof

                                                            ↳ QDP

                                                              ↳ Narrowing

                                                                ↳ QDP

                                                                  ↳ DependencyGraphProof

                                                                    ↳ QDP

                                                                      ↳ Narrowing

                                                                        ↳ QDP

                                                                          ↳ DependencyGraphProof

                                          ↳ QDP

                                ↳ QDP

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

A__SEL(s(sel1(y0, y1)), cons(y2, indx(x0, x1))) → A__SEL(a__sel1(mark(y0), mark(y1)), a__indx(mark(x0), x1))
A__SEL(s(dbl(y0)), cons(y1, dbl1(x0))) → A__SEL(a__dbl(mark(y0)), a__dbl1(mark(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, from(x0))) → A__SEL(a__dbl(mark(y0)), a__from(x0))
A__SEL(s(dbl(y0)), cons(y1, sel1(x0, x1))) → A__SEL(a__dbl(mark(y0)), a__sel1(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(sel1(y0, y1)), cons(y2, dbl1(x0))) → A__SEL(a__sel1(mark(y0), mark(y1)), a__dbl1(mark(x0)))
A__SEL(s(dbl(y0)), cons(y1, dbl(x0))) → A__SEL(a__dbl(mark(y0)), a__dbl(mark(x0)))
A__SEL(s(sel(x0, x1)), cons(y1, y2)) → A__SEL(a__sel(mark(x0), mark(x1)), mark(y2))
A__SEL(s(sel1(y0, y1)), cons(y2, dbl(x0))) → A__SEL(a__sel1(mark(y0), mark(y1)), a__dbl(mark(x0)))
A__SEL(s(from(x0)), cons(y1, y2)) → A__SEL(a__from(x0), mark(y2))
A__SEL(s(indx(x0, x1)), cons(y1, y2)) → A__SEL(a__indx(mark(x0), x1), mark(y2))
A__SEL(s(sel1(y0, y1)), cons(y2, quote(x0))) → A__SEL(a__sel1(mark(y0), mark(y1)), a__quote(mark(x0)))
A__SEL(s(sel1(y0, y1)), cons(y2, sel(x0, x1))) → A__SEL(a__sel1(mark(y0), mark(y1)), a__sel(mark(x0), mark(x1)))
A__SEL(s(s(x0)), cons(y1, y2)) → A__SEL(s(x0), mark(y2))
A__SEL(s(sel1(y0, y1)), cons(y2, nil)) → A__SEL(a__sel1(mark(y0), mark(y1)), nil)
A__SEL(s(sel1(y0, y1)), cons(y2, sel1(x0, x1))) → A__SEL(a__sel1(mark(y0), mark(y1)), a__sel1(mark(x0), mark(x1)))
A__SEL(s(sel1(y0, y1)), cons(y2, s1(x0))) → A__SEL(a__sel1(mark(y0), mark(y1)), s1(mark(x0)))
A__SEL(s(sel1(y0, y1)), cons(y2, dbls(x0))) → A__SEL(a__sel1(mark(y0), mark(y1)), a__dbls(mark(x0)))
A__SEL(s(sel1(y0, y1)), cons(y2, 0)) → A__SEL(a__sel1(mark(y0), mark(y1)), 0)
A__SEL(s(dbl1(x0)), cons(y1, y2)) → A__SEL(a__dbl1(mark(x0)), mark(y2))
A__SEL(s(dbl(y0)), cons(y1, indx(x0, x1))) → A__SEL(a__dbl(mark(y0)), a__indx(mark(x0), x1))
A__SEL(s(sel1(y0, y1)), cons(y2, cons(x0, x1))) → A__SEL(a__sel1(mark(y0), mark(y1)), cons(x0, x1))
A__SEL(s(dbl(y0)), cons(y1, quote(x0))) → A__SEL(a__dbl(mark(y0)), a__quote(mark(x0)))
A__SEL(s(quote(x0)), cons(y1, y2)) → A__SEL(a__quote(mark(x0)), mark(y2))
A__SEL(s(sel1(y0, y1)), cons(y2, from(x0))) → A__SEL(a__sel1(mark(y0), mark(y1)), a__from(x0))
A__SEL(s(sel1(y0, y1)), cons(y2, s(x0))) → A__SEL(a__sel1(mark(y0), mark(y1)), s(x0))
A__SEL(s(sel1(y0, y1)), cons(y2, 01)) → A__SEL(a__sel1(mark(y0), mark(y1)), 01)
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)))
a__dbl1(0) → 01
a__dbl1(s(X)) → s1(s1(a__dbl1(mark(X))))
a__sel1(0, cons(X, Y)) → mark(X)
a__sel1(s(X), cons(Y, Z)) → a__sel1(mark(X), mark(Z))
a__quote(0) → 01
a__quote(s(X)) → s1(a__quote(mark(X)))
a__quote(dbl(X)) → a__dbl1(mark(X))
a__quote(sel(X, Y)) → a__sel1(mark(X), mark(Y))
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(dbl1(X)) → a__dbl1(mark(X))
mark(sel1(X1, X2)) → a__sel1(mark(X1), mark(X2))
mark(quote(X)) → a__quote(mark(X))
mark(0) → 0
mark(s(X)) → s(X)
mark(nil) → nil
mark(cons(X1, X2)) → cons(X1, X2)
mark(01) → 01
mark(s1(X)) → s1(mark(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__from(X) → from(X)
a__dbl1(X) → dbl1(X)
a__sel1(X1, X2) → sel1(X1, X2)
a__quote(X) → quote(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 5 less nodes.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ QDPOrderProof

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ AND

                      ↳ QDP

                      ↳ QDP

                        ↳ QDPOrderProof

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ AND

                                ↳ QDP

                                  ↳ QDPOrderProof

                                    ↳ QDP

                                      ↳ DependencyGraphProof

                                        ↳ AND

                                          ↳ QDP

                                            ↳ QDPOrderProof

                                              ↳ QDP

                                                ↳ DependencyGraphProof

                                                  ↳ AND

                                                    ↳ QDP

                                                    ↳ QDP

                                                      ↳ Narrowing

                                                        ↳ QDP

                                                          ↳ DependencyGraphProof

                                                            ↳ QDP

                                                              ↳ Narrowing

                                                                ↳ QDP

                                                                  ↳ DependencyGraphProof

                                                                    ↳ QDP

                                                                      ↳ Narrowing

                                                                        ↳ QDP

                                                                          ↳ DependencyGraphProof

                                                                            ↳ QDP

                                                                              ↳ Narrowing

                                          ↳ QDP

                                ↳ QDP

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

A__SEL(s(sel1(y0, y1)), cons(y2, indx(x0, x1))) → A__SEL(a__sel1(mark(y0), mark(y1)), a__indx(mark(x0), x1))
A__SEL(s(dbl(y0)), cons(y1, dbl1(x0))) → A__SEL(a__dbl(mark(y0)), a__dbl1(mark(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, from(x0))) → A__SEL(a__dbl(mark(y0)), a__from(x0))
A__SEL(s(dbl(y0)), cons(y1, sel1(x0, x1))) → A__SEL(a__dbl(mark(y0)), a__sel1(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(sel1(y0, y1)), cons(y2, dbl1(x0))) → A__SEL(a__sel1(mark(y0), mark(y1)), a__dbl1(mark(x0)))
A__SEL(s(sel(x0, x1)), cons(y1, y2)) → A__SEL(a__sel(mark(x0), mark(x1)), mark(y2))
A__SEL(s(dbl(y0)), cons(y1, dbl(x0))) → A__SEL(a__dbl(mark(y0)), a__dbl(mark(x0)))
A__SEL(s(sel1(y0, y1)), cons(y2, dbl(x0))) → A__SEL(a__sel1(mark(y0), mark(y1)), a__dbl(mark(x0)))
A__SEL(s(from(x0)), cons(y1, y2)) → A__SEL(a__from(x0), mark(y2))
A__SEL(s(indx(x0, x1)), cons(y1, y2)) → A__SEL(a__indx(mark(x0), x1), mark(y2))
A__SEL(s(sel1(y0, y1)), cons(y2, quote(x0))) → A__SEL(a__sel1(mark(y0), mark(y1)), a__quote(mark(x0)))
A__SEL(s(sel1(y0, y1)), cons(y2, sel(x0, x1))) → A__SEL(a__sel1(mark(y0), mark(y1)), a__sel(mark(x0), mark(x1)))
A__SEL(s(s(x0)), cons(y1, y2)) → A__SEL(s(x0), mark(y2))
A__SEL(s(sel1(y0, y1)), cons(y2, sel1(x0, x1))) → A__SEL(a__sel1(mark(y0), mark(y1)), a__sel1(mark(x0), mark(x1)))
A__SEL(s(sel1(y0, y1)), cons(y2, dbls(x0))) → A__SEL(a__sel1(mark(y0), mark(y1)), a__dbls(mark(x0)))
A__SEL(s(dbl1(x0)), cons(y1, y2)) → A__SEL(a__dbl1(mark(x0)), mark(y2))
A__SEL(s(dbl(y0)), cons(y1, indx(x0, x1))) → A__SEL(a__dbl(mark(y0)), a__indx(mark(x0), x1))
A__SEL(s(sel1(y0, y1)), cons(y2, cons(x0, x1))) → A__SEL(a__sel1(mark(y0), mark(y1)), cons(x0, x1))
A__SEL(s(quote(x0)), cons(y1, y2)) → A__SEL(a__quote(mark(x0)), mark(y2))
A__SEL(s(dbl(y0)), cons(y1, quote(x0))) → A__SEL(a__dbl(mark(y0)), a__quote(mark(x0)))
A__SEL(s(sel1(y0, y1)), cons(y2, from(x0))) → A__SEL(a__sel1(mark(y0), mark(y1)), a__from(x0))
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)))
a__dbl1(0) → 01
a__dbl1(s(X)) → s1(s1(a__dbl1(mark(X))))
a__sel1(0, cons(X, Y)) → mark(X)
a__sel1(s(X), cons(Y, Z)) → a__sel1(mark(X), mark(Z))
a__quote(0) → 01
a__quote(s(X)) → s1(a__quote(mark(X)))
a__quote(dbl(X)) → a__dbl1(mark(X))
a__quote(sel(X, Y)) → a__sel1(mark(X), mark(Y))
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(dbl1(X)) → a__dbl1(mark(X))
mark(sel1(X1, X2)) → a__sel1(mark(X1), mark(X2))
mark(quote(X)) → a__quote(mark(X))
mark(0) → 0
mark(s(X)) → s(X)
mark(nil) → nil
mark(cons(X1, X2)) → cons(X1, X2)
mark(01) → 01
mark(s1(X)) → s1(mark(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__from(X) → from(X)
a__dbl1(X) → dbl1(X)
a__sel1(X1, X2) → sel1(X1, X2)
a__quote(X) → quote(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule A__SEL(s(dbl1(x0)), cons(y1, y2)) → A__SEL(a__dbl1(mark(x0)), mark(y2)) at position [1] we obtained the following new rules:

A__SEL(s(dbl1(y0)), cons(y1, nil)) → A__SEL(a__dbl1(mark(y0)), nil)
A__SEL(s(dbl1(y0)), cons(y1, dbl(x0))) → A__SEL(a__dbl1(mark(y0)), a__dbl(mark(x0)))
A__SEL(s(dbl1(y0)), cons(y1, sel(x0, x1))) → A__SEL(a__dbl1(mark(y0)), a__sel(mark(x0), mark(x1)))
A__SEL(s(dbl1(y0)), cons(y1, 01)) → A__SEL(a__dbl1(mark(y0)), 01)
A__SEL(s(dbl1(y0)), cons(y1, 0)) → A__SEL(a__dbl1(mark(y0)), 0)
A__SEL(s(dbl1(y0)), cons(y1, s1(x0))) → A__SEL(a__dbl1(mark(y0)), s1(mark(x0)))
A__SEL(s(dbl1(y0)), cons(y1, sel1(x0, x1))) → A__SEL(a__dbl1(mark(y0)), a__sel1(mark(x0), mark(x1)))
A__SEL(s(dbl1(y0)), cons(y1, cons(x0, x1))) → A__SEL(a__dbl1(mark(y0)), cons(x0, x1))
A__SEL(s(dbl1(y0)), cons(y1, dbls(x0))) → A__SEL(a__dbl1(mark(y0)), a__dbls(mark(x0)))
A__SEL(s(dbl1(y0)), cons(y1, quote(x0))) → A__SEL(a__dbl1(mark(y0)), a__quote(mark(x0)))
A__SEL(s(dbl1(y0)), cons(y1, s(x0))) → A__SEL(a__dbl1(mark(y0)), s(x0))
A__SEL(s(dbl1(y0)), cons(y1, from(x0))) → A__SEL(a__dbl1(mark(y0)), a__from(x0))
A__SEL(s(dbl1(y0)), cons(y1, dbl1(x0))) → A__SEL(a__dbl1(mark(y0)), a__dbl1(mark(x0)))
A__SEL(s(dbl1(y0)), cons(y1, indx(x0, x1))) → A__SEL(a__dbl1(mark(y0)), a__indx(mark(x0), x1))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ QDPOrderProof

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ AND

                      ↳ QDP

                      ↳ QDP

                        ↳ QDPOrderProof

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ AND

                                ↳ QDP

                                  ↳ QDPOrderProof

                                    ↳ QDP

                                      ↳ DependencyGraphProof

                                        ↳ AND

                                          ↳ QDP

                                            ↳ QDPOrderProof

                                              ↳ QDP

                                                ↳ DependencyGraphProof

                                                  ↳ AND

                                                    ↳ QDP

                                                    ↳ QDP

                                                      ↳ Narrowing

                                                        ↳ QDP

                                                          ↳ DependencyGraphProof

                                                            ↳ QDP

                                                              ↳ Narrowing

                                                                ↳ QDP

                                                                  ↳ DependencyGraphProof

                                                                    ↳ QDP

                                                                      ↳ Narrowing

                                                                        ↳ QDP

                                                                          ↳ DependencyGraphProof

                                                                            ↳ QDP

                                                                              ↳ Narrowing

                                                                                ↳ QDP

                                                                                  ↳ DependencyGraphProof

                                          ↳ QDP

                                ↳ QDP

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

A__SEL(s(sel1(y0, y1)), cons(y2, indx(x0, x1))) → A__SEL(a__sel1(mark(y0), mark(y1)), a__indx(mark(x0), x1))
A__SEL(s(dbl1(y0)), cons(y1, dbl(x0))) → A__SEL(a__dbl1(mark(y0)), a__dbl(mark(x0)))
A__SEL(s(dbl(y0)), cons(y1, dbl1(x0))) → A__SEL(a__dbl(mark(y0)), a__dbl1(mark(x0)))
A__SEL(s(dbl(y0)), cons(y1, dbls(x0))) → A__SEL(a__dbl(mark(y0)), a__dbls(mark(x0)))
A__SEL(s(dbl1(y0)), cons(y1, 0)) → A__SEL(a__dbl1(mark(y0)), 0)
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(dbl1(y0)), cons(y1, cons(x0, x1))) → A__SEL(a__dbl1(mark(y0)), cons(x0, x1))
A__SEL(s(dbl(y0)), cons(y1, sel1(x0, x1))) → A__SEL(a__dbl(mark(y0)), a__sel1(mark(x0), mark(x1)))
A__SEL(s(sel1(y0, y1)), cons(y2, dbl1(x0))) → A__SEL(a__sel1(mark(y0), mark(y1)), a__dbl1(mark(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(x0, x1)), cons(y1, y2)) → A__SEL(a__sel(mark(x0), mark(x1)), mark(y2))
A__SEL(s(sel1(y0, y1)), cons(y2, dbl(x0))) → A__SEL(a__sel1(mark(y0), mark(y1)), a__dbl(mark(x0)))
A__SEL(s(from(x0)), cons(y1, y2)) → A__SEL(a__from(x0), mark(y2))
A__SEL(s(indx(x0, x1)), cons(y1, y2)) → A__SEL(a__indx(mark(x0), x1), mark(y2))
A__SEL(s(sel1(y0, y1)), cons(y2, quote(x0))) → A__SEL(a__sel1(mark(y0), mark(y1)), a__quote(mark(x0)))
A__SEL(s(sel1(y0, y1)), cons(y2, sel(x0, x1))) → A__SEL(a__sel1(mark(y0), mark(y1)), a__sel(mark(x0), mark(x1)))
A__SEL(s(s(x0)), cons(y1, y2)) → A__SEL(s(x0), mark(y2))
A__SEL(s(sel1(y0, y1)), cons(y2, sel1(x0, x1))) → A__SEL(a__sel1(mark(y0), mark(y1)), a__sel1(mark(x0), mark(x1)))
A__SEL(s(dbl1(y0)), cons(y1, nil)) → A__SEL(a__dbl1(mark(y0)), nil)
A__SEL(s(sel1(y0, y1)), cons(y2, dbls(x0))) → A__SEL(a__sel1(mark(y0), mark(y1)), a__dbls(mark(x0)))
A__SEL(s(dbl1(y0)), cons(y1, sel(x0, x1))) → A__SEL(a__dbl1(mark(y0)), a__sel(mark(x0), mark(x1)))
A__SEL(s(dbl1(y0)), cons(y1, 01)) → A__SEL(a__dbl1(mark(y0)), 01)
A__SEL(s(dbl1(y0)), cons(y1, sel1(x0, x1))) → A__SEL(a__dbl1(mark(y0)), a__sel1(mark(x0), mark(x1)))
A__SEL(s(dbl1(y0)), cons(y1, s1(x0))) → A__SEL(a__dbl1(mark(y0)), s1(mark(x0)))
A__SEL(s(dbl1(y0)), cons(y1, dbls(x0))) → A__SEL(a__dbl1(mark(y0)), a__dbls(mark(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(dbl1(y0)), cons(y1, quote(x0))) → A__SEL(a__dbl1(mark(y0)), a__quote(mark(x0)))
A__SEL(s(sel1(y0, y1)), cons(y2, cons(x0, x1))) → A__SEL(a__sel1(mark(y0), mark(y1)), cons(x0, x1))
A__SEL(s(dbl(y0)), cons(y1, quote(x0))) → A__SEL(a__dbl(mark(y0)), a__quote(mark(x0)))
A__SEL(s(quote(x0)), cons(y1, y2)) → A__SEL(a__quote(mark(x0)), mark(y2))
A__SEL(s(sel1(y0, y1)), cons(y2, from(x0))) → A__SEL(a__sel1(mark(y0), mark(y1)), a__from(x0))
A__SEL(s(dbl1(y0)), cons(y1, from(x0))) → A__SEL(a__dbl1(mark(y0)), a__from(x0))
A__SEL(s(dbl1(y0)), cons(y1, s(x0))) → A__SEL(a__dbl1(mark(y0)), s(x0))
A__SEL(s(dbl1(y0)), cons(y1, indx(x0, x1))) → A__SEL(a__dbl1(mark(y0)), a__indx(mark(x0), x1))
A__SEL(s(dbl1(y0)), cons(y1, dbl1(x0))) → A__SEL(a__dbl1(mark(y0)), a__dbl1(mark(x0)))
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)))
a__dbl1(0) → 01
a__dbl1(s(X)) → s1(s1(a__dbl1(mark(X))))
a__sel1(0, cons(X, Y)) → mark(X)
a__sel1(s(X), cons(Y, Z)) → a__sel1(mark(X), mark(Z))
a__quote(0) → 01
a__quote(s(X)) → s1(a__quote(mark(X)))
a__quote(dbl(X)) → a__dbl1(mark(X))
a__quote(sel(X, Y)) → a__sel1(mark(X), mark(Y))
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(dbl1(X)) → a__dbl1(mark(X))
mark(sel1(X1, X2)) → a__sel1(mark(X1), mark(X2))
mark(quote(X)) → a__quote(mark(X))
mark(0) → 0
mark(s(X)) → s(X)
mark(nil) → nil
mark(cons(X1, X2)) → cons(X1, X2)
mark(01) → 01
mark(s1(X)) → s1(mark(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__from(X) → from(X)
a__dbl1(X) → dbl1(X)
a__sel1(X1, X2) → sel1(X1, X2)
a__quote(X) → quote(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 5 less nodes.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ QDPOrderProof

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ AND

                      ↳ QDP

                      ↳ QDP

                        ↳ QDPOrderProof

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ AND

                                ↳ QDP

                                  ↳ QDPOrderProof

                                    ↳ QDP

                                      ↳ DependencyGraphProof

                                        ↳ AND

                                          ↳ QDP

                                            ↳ QDPOrderProof

                                              ↳ QDP

                                                ↳ DependencyGraphProof

                                                  ↳ AND

                                                    ↳ QDP

                                                    ↳ QDP

                                                      ↳ Narrowing

                                                        ↳ QDP

                                                          ↳ DependencyGraphProof

                                                            ↳ QDP

                                                              ↳ Narrowing

                                                                ↳ QDP

                                                                  ↳ DependencyGraphProof

                                                                    ↳ QDP

                                                                      ↳ Narrowing

                                                                        ↳ QDP

                                                                          ↳ DependencyGraphProof

                                                                            ↳ QDP

                                                                              ↳ Narrowing

                                                                                ↳ QDP

                                                                                  ↳ DependencyGraphProof

                                                                                    ↳ QDP

                                                                                      ↳ Narrowing

                                          ↳ QDP

                                ↳ QDP

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

A__SEL(s(sel1(y0, y1)), cons(y2, indx(x0, x1))) → A__SEL(a__sel1(mark(y0), mark(y1)), a__indx(mark(x0), x1))
A__SEL(s(dbl(y0)), cons(y1, dbl1(x0))) → A__SEL(a__dbl(mark(y0)), a__dbl1(mark(x0)))
A__SEL(s(dbl1(y0)), cons(y1, dbl(x0))) → A__SEL(a__dbl1(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(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(dbl1(y0)), cons(y1, cons(x0, x1))) → A__SEL(a__dbl1(mark(y0)), cons(x0, x1))
A__SEL(s(dbl(y0)), cons(y1, sel1(x0, x1))) → A__SEL(a__dbl(mark(y0)), a__sel1(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(sel1(y0, y1)), cons(y2, dbl1(x0))) → A__SEL(a__sel1(mark(y0), mark(y1)), a__dbl1(mark(x0)))
A__SEL(s(sel(x0, x1)), cons(y1, y2)) → A__SEL(a__sel(mark(x0), mark(x1)), mark(y2))
A__SEL(s(dbl(y0)), cons(y1, dbl(x0))) → A__SEL(a__dbl(mark(y0)), a__dbl(mark(x0)))
A__SEL(s(sel1(y0, y1)), cons(y2, dbl(x0))) → A__SEL(a__sel1(mark(y0), mark(y1)), a__dbl(mark(x0)))
A__SEL(s(from(x0)), cons(y1, y2)) → A__SEL(a__from(x0), mark(y2))
A__SEL(s(indx(x0, x1)), cons(y1, y2)) → A__SEL(a__indx(mark(x0), x1), mark(y2))
A__SEL(s(sel1(y0, y1)), cons(y2, quote(x0))) → A__SEL(a__sel1(mark(y0), mark(y1)), a__quote(mark(x0)))
A__SEL(s(sel1(y0, y1)), cons(y2, sel(x0, x1))) → A__SEL(a__sel1(mark(y0), mark(y1)), a__sel(mark(x0), mark(x1)))
A__SEL(s(s(x0)), cons(y1, y2)) → A__SEL(s(x0), mark(y2))
A__SEL(s(sel1(y0, y1)), cons(y2, sel1(x0, x1))) → A__SEL(a__sel1(mark(y0), mark(y1)), a__sel1(mark(x0), mark(x1)))
A__SEL(s(sel1(y0, y1)), cons(y2, dbls(x0))) → A__SEL(a__sel1(mark(y0), mark(y1)), a__dbls(mark(x0)))
A__SEL(s(dbl1(y0)), cons(y1, sel(x0, x1))) → A__SEL(a__dbl1(mark(y0)), a__sel(mark(x0), mark(x1)))
A__SEL(s(dbl1(y0)), cons(y1, sel1(x0, x1))) → A__SEL(a__dbl1(mark(y0)), a__sel1(mark(x0), mark(x1)))
A__SEL(s(dbl1(y0)), cons(y1, dbls(x0))) → A__SEL(a__dbl1(mark(y0)), a__dbls(mark(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(dbl1(y0)), cons(y1, quote(x0))) → A__SEL(a__dbl1(mark(y0)), a__quote(mark(x0)))
A__SEL(s(sel1(y0, y1)), cons(y2, cons(x0, x1))) → A__SEL(a__sel1(mark(y0), mark(y1)), cons(x0, x1))
A__SEL(s(quote(x0)), cons(y1, y2)) → A__SEL(a__quote(mark(x0)), mark(y2))
A__SEL(s(dbl(y0)), cons(y1, quote(x0))) → A__SEL(a__dbl(mark(y0)), a__quote(mark(x0)))
A__SEL(s(sel1(y0, y1)), cons(y2, from(x0))) → A__SEL(a__sel1(mark(y0), mark(y1)), a__from(x0))
A__SEL(s(dbl1(y0)), cons(y1, from(x0))) → A__SEL(a__dbl1(mark(y0)), a__from(x0))
A__SEL(s(dbl1(y0)), cons(y1, indx(x0, x1))) → A__SEL(a__dbl1(mark(y0)), a__indx(mark(x0), x1))
A__SEL(s(dbl1(y0)), cons(y1, dbl1(x0))) → A__SEL(a__dbl1(mark(y0)), a__dbl1(mark(x0)))
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)))
a__dbl1(0) → 01
a__dbl1(s(X)) → s1(s1(a__dbl1(mark(X))))
a__sel1(0, cons(X, Y)) → mark(X)
a__sel1(s(X), cons(Y, Z)) → a__sel1(mark(X), mark(Z))
a__quote(0) → 01
a__quote(s(X)) → s1(a__quote(mark(X)))
a__quote(dbl(X)) → a__dbl1(mark(X))
a__quote(sel(X, Y)) → a__sel1(mark(X), mark(Y))
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(dbl1(X)) → a__dbl1(mark(X))
mark(sel1(X1, X2)) → a__sel1(mark(X1), mark(X2))
mark(quote(X)) → a__quote(mark(X))
mark(0) → 0
mark(s(X)) → s(X)
mark(nil) → nil
mark(cons(X1, X2)) → cons(X1, X2)
mark(01) → 01
mark(s1(X)) → s1(mark(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__from(X) → from(X)
a__dbl1(X) → dbl1(X)
a__sel1(X1, X2) → sel1(X1, X2)
a__quote(X) → quote(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, dbl(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__dbl(mark(x0)))
A__SEL(s(sel(y0, y1)), cons(y2, s1(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), s1(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, 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, sel1(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), a__sel1(mark(x0), mark(x1)))
A__SEL(s(sel(y0, y1)), cons(y2, 01)) → A__SEL(a__sel(mark(y0), mark(y1)), 01)
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, dbl1(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__dbl1(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, quote(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__quote(mark(x0)))
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, cons(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), cons(x0, x1))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ QDPOrderProof

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ AND

                      ↳ QDP

                      ↳ QDP

                        ↳ QDPOrderProof

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ AND

                                ↳ QDP

                                  ↳ QDPOrderProof

                                    ↳ QDP

                                      ↳ DependencyGraphProof

                                        ↳ AND

                                          ↳ QDP

                                            ↳ QDPOrderProof

                                              ↳ QDP

                                                ↳ DependencyGraphProof

                                                  ↳ AND

                                                    ↳ QDP

                                                    ↳ QDP

                                                      ↳ Narrowing

                                                        ↳ QDP

                                                          ↳ DependencyGraphProof

                                                            ↳ QDP

                                                              ↳ Narrowing

                                                                ↳ QDP

                                                                  ↳ DependencyGraphProof

                                                                    ↳ QDP

                                                                      ↳ Narrowing

                                                                        ↳ QDP

                                                                          ↳ DependencyGraphProof

                                                                            ↳ QDP

                                                                              ↳ Narrowing

                                                                                ↳ QDP

                                                                                  ↳ DependencyGraphProof

                                                                                    ↳ QDP

                                                                                      ↳ Narrowing

                                                                                        ↳ QDP

                                                                                          ↳ DependencyGraphProof

                                          ↳ QDP

                                ↳ QDP

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

A__SEL(s(sel1(y0, y1)), cons(y2, indx(x0, x1))) → A__SEL(a__sel1(mark(y0), mark(y1)), a__indx(mark(x0), x1))
A__SEL(s(dbl1(y0)), cons(y1, dbl(x0))) → A__SEL(a__dbl1(mark(y0)), a__dbl(mark(x0)))
A__SEL(s(dbl(y0)), cons(y1, dbl1(x0))) → A__SEL(a__dbl(mark(y0)), a__dbl1(mark(x0)))
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(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(sel(y0, y1)), cons(y2, s1(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), s1(mark(x0)))
A__SEL(s(dbl1(y0)), cons(y1, cons(x0, x1))) → A__SEL(a__dbl1(mark(y0)), cons(x0, x1))
A__SEL(s(dbl(y0)), cons(y1, sel1(x0, x1))) → A__SEL(a__dbl(mark(y0)), a__sel1(mark(x0), mark(x1)))
A__SEL(s(sel(y0, y1)), cons(y2, 0)) → A__SEL(a__sel(mark(y0), mark(y1)), 0)
A__SEL(s(sel1(y0, y1)), cons(y2, dbl1(x0))) → A__SEL(a__sel1(mark(y0), mark(y1)), a__dbl1(mark(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(sel(y0, y1)), cons(y2, dbls(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__dbls(mark(x0)))
A__SEL(s(sel1(y0, y1)), cons(y2, dbl(x0))) → A__SEL(a__sel1(mark(y0), mark(y1)), a__dbl(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(sel1(y0, y1)), cons(y2, quote(x0))) → A__SEL(a__sel1(mark(y0), mark(y1)), a__quote(mark(x0)))
A__SEL(s(sel(y0, y1)), cons(y2, sel1(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), a__sel1(mark(x0), mark(x1)))
A__SEL(s(sel1(y0, y1)), cons(y2, sel(x0, x1))) → A__SEL(a__sel1(mark(y0), mark(y1)), a__sel(mark(x0), mark(x1)))
A__SEL(s(s(x0)), cons(y1, y2)) → A__SEL(s(x0), mark(y2))
A__SEL(s(sel(y0, y1)), cons(y2, 01)) → A__SEL(a__sel(mark(y0), mark(y1)), 01)
A__SEL(s(sel1(y0, y1)), cons(y2, sel1(x0, x1))) → A__SEL(a__sel1(mark(y0), mark(y1)), a__sel1(mark(x0), mark(x1)))
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(sel1(y0, y1)), cons(y2, dbls(x0))) → A__SEL(a__sel1(mark(y0), mark(y1)), a__dbls(mark(x0)))
A__SEL(s(dbl1(y0)), cons(y1, sel(x0, x1))) → A__SEL(a__dbl1(mark(y0)), a__sel(mark(x0), mark(x1)))
A__SEL(s(sel(y0, y1)), cons(y2, dbl1(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__dbl1(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(dbl1(y0)), cons(y1, sel1(x0, x1))) → A__SEL(a__dbl1(mark(y0)), a__sel1(mark(x0), mark(x1)))
A__SEL(s(sel(y0, y1)), cons(y2, quote(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__quote(mark(x0)))
A__SEL(s(dbl1(y0)), cons(y1, dbls(x0))) → A__SEL(a__dbl1(mark(y0)), a__dbls(mark(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(sel(y0, y1)), cons(y2, nil)) → A__SEL(a__sel(mark(y0), mark(y1)), nil)
A__SEL(s(dbl1(y0)), cons(y1, quote(x0))) → A__SEL(a__dbl1(mark(y0)), a__quote(mark(x0)))
A__SEL(s(sel1(y0, y1)), cons(y2, cons(x0, x1))) → A__SEL(a__sel1(mark(y0), mark(y1)), cons(x0, x1))
A__SEL(s(dbl(y0)), cons(y1, quote(x0))) → A__SEL(a__dbl(mark(y0)), a__quote(mark(x0)))
A__SEL(s(quote(x0)), cons(y1, y2)) → A__SEL(a__quote(mark(x0)), mark(y2))
A__SEL(s(sel1(y0, y1)), cons(y2, from(x0))) → A__SEL(a__sel1(mark(y0), mark(y1)), a__from(x0))
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(dbl1(y0)), cons(y1, from(x0))) → A__SEL(a__dbl1(mark(y0)), a__from(x0))
A__SEL(s(dbl1(y0)), cons(y1, dbl1(x0))) → A__SEL(a__dbl1(mark(y0)), a__dbl1(mark(x0)))
A__SEL(s(dbl1(y0)), cons(y1, indx(x0, x1))) → A__SEL(a__dbl1(mark(y0)), a__indx(mark(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)))
a__dbl1(0) → 01
a__dbl1(s(X)) → s1(s1(a__dbl1(mark(X))))
a__sel1(0, cons(X, Y)) → mark(X)
a__sel1(s(X), cons(Y, Z)) → a__sel1(mark(X), mark(Z))
a__quote(0) → 01
a__quote(s(X)) → s1(a__quote(mark(X)))
a__quote(dbl(X)) → a__dbl1(mark(X))
a__quote(sel(X, Y)) → a__sel1(mark(X), mark(Y))
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(dbl1(X)) → a__dbl1(mark(X))
mark(sel1(X1, X2)) → a__sel1(mark(X1), mark(X2))
mark(quote(X)) → a__quote(mark(X))
mark(0) → 0
mark(s(X)) → s(X)
mark(nil) → nil
mark(cons(X1, X2)) → cons(X1, X2)
mark(01) → 01
mark(s1(X)) → s1(mark(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__from(X) → from(X)
a__dbl1(X) → dbl1(X)
a__sel1(X1, X2) → sel1(X1, X2)
a__quote(X) → quote(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 5 less nodes.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ QDPOrderProof

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ AND

                      ↳ QDP

                      ↳ QDP

                        ↳ QDPOrderProof

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ AND

                                ↳ QDP

                                  ↳ QDPOrderProof

                                    ↳ QDP

                                      ↳ DependencyGraphProof

                                        ↳ AND

                                          ↳ QDP

                                            ↳ QDPOrderProof

                                              ↳ QDP

                                                ↳ DependencyGraphProof

                                                  ↳ AND

                                                    ↳ QDP

                                                    ↳ QDP

                                                      ↳ Narrowing

                                                        ↳ QDP

                                                          ↳ DependencyGraphProof

                                                            ↳ QDP

                                                              ↳ Narrowing

                                                                ↳ QDP

                                                                  ↳ DependencyGraphProof

                                                                    ↳ QDP

                                                                      ↳ Narrowing

                                                                        ↳ QDP

                                                                          ↳ DependencyGraphProof

                                                                            ↳ QDP

                                                                              ↳ Narrowing

                                                                                ↳ QDP

                                                                                  ↳ DependencyGraphProof

                                                                                    ↳ QDP

                                                                                      ↳ Narrowing

                                                                                        ↳ QDP

                                                                                          ↳ DependencyGraphProof

                                                                                            ↳ QDP

                                                                                              ↳ Narrowing

                                          ↳ QDP

                                ↳ QDP

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

A__SEL(s(sel1(y0, y1)), cons(y2, indx(x0, x1))) → A__SEL(a__sel1(mark(y0), mark(y1)), a__indx(mark(x0), x1))
A__SEL(s(dbl(y0)), cons(y1, dbl1(x0))) → A__SEL(a__dbl(mark(y0)), a__dbl1(mark(x0)))
A__SEL(s(dbl1(y0)), cons(y1, dbl(x0))) → A__SEL(a__dbl1(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(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(dbl1(y0)), cons(y1, cons(x0, x1))) → A__SEL(a__dbl1(mark(y0)), cons(x0, x1))
A__SEL(s(dbl(y0)), cons(y1, sel1(x0, x1))) → A__SEL(a__dbl(mark(y0)), a__sel1(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(sel1(y0, y1)), cons(y2, dbl1(x0))) → A__SEL(a__sel1(mark(y0), mark(y1)), a__dbl1(mark(x0)))
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(sel1(y0, y1)), cons(y2, dbl(x0))) → A__SEL(a__sel1(mark(y0), mark(y1)), a__dbl(mark(x0)))
A__SEL(s(from(x0)), cons(y1, y2)) → A__SEL(a__from(x0), mark(y2))
A__SEL(s(indx(x0, x1)), cons(y1, y2)) → A__SEL(a__indx(mark(x0), x1), mark(y2))
A__SEL(s(sel1(y0, y1)), cons(y2, quote(x0))) → A__SEL(a__sel1(mark(y0), mark(y1)), a__quote(mark(x0)))
A__SEL(s(sel(y0, y1)), cons(y2, sel1(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), a__sel1(mark(x0), mark(x1)))
A__SEL(s(sel1(y0, y1)), cons(y2, sel(x0, x1))) → A__SEL(a__sel1(mark(y0), mark(y1)), a__sel(mark(x0), mark(x1)))
A__SEL(s(s(x0)), cons(y1, y2)) → A__SEL(s(x0), mark(y2))
A__SEL(s(sel1(y0, y1)), cons(y2, sel1(x0, x1))) → A__SEL(a__sel1(mark(y0), mark(y1)), a__sel1(mark(x0), mark(x1)))
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(sel1(y0, y1)), cons(y2, dbls(x0))) → A__SEL(a__sel1(mark(y0), mark(y1)), a__dbls(mark(x0)))
A__SEL(s(dbl1(y0)), cons(y1, sel(x0, x1))) → A__SEL(a__dbl1(mark(y0)), a__sel(mark(x0), mark(x1)))
A__SEL(s(sel(y0, y1)), cons(y2, dbl1(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__dbl1(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(dbl1(y0)), cons(y1, sel1(x0, x1))) → A__SEL(a__dbl1(mark(y0)), a__sel1(mark(x0), mark(x1)))
A__SEL(s(sel(y0, y1)), cons(y2, quote(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__quote(mark(x0)))
A__SEL(s(dbl1(y0)), cons(y1, dbls(x0))) → A__SEL(a__dbl1(mark(y0)), a__dbls(mark(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(dbl1(y0)), cons(y1, quote(x0))) → A__SEL(a__dbl1(mark(y0)), a__quote(mark(x0)))
A__SEL(s(sel1(y0, y1)), cons(y2, cons(x0, x1))) → A__SEL(a__sel1(mark(y0), mark(y1)), cons(x0, x1))
A__SEL(s(quote(x0)), cons(y1, y2)) → A__SEL(a__quote(mark(x0)), mark(y2))
A__SEL(s(dbl(y0)), cons(y1, quote(x0))) → A__SEL(a__dbl(mark(y0)), a__quote(mark(x0)))
A__SEL(s(sel1(y0, y1)), cons(y2, from(x0))) → A__SEL(a__sel1(mark(y0), mark(y1)), a__from(x0))
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(dbl1(y0)), cons(y1, from(x0))) → A__SEL(a__dbl1(mark(y0)), a__from(x0))
A__SEL(s(dbl1(y0)), cons(y1, indx(x0, x1))) → A__SEL(a__dbl1(mark(y0)), a__indx(mark(x0), x1))
A__SEL(s(dbl1(y0)), cons(y1, dbl1(x0))) → A__SEL(a__dbl1(mark(y0)), a__dbl1(mark(x0)))
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)))
a__dbl1(0) → 01
a__dbl1(s(X)) → s1(s1(a__dbl1(mark(X))))
a__sel1(0, cons(X, Y)) → mark(X)
a__sel1(s(X), cons(Y, Z)) → a__sel1(mark(X), mark(Z))
a__quote(0) → 01
a__quote(s(X)) → s1(a__quote(mark(X)))
a__quote(dbl(X)) → a__dbl1(mark(X))
a__quote(sel(X, Y)) → a__sel1(mark(X), mark(Y))
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(dbl1(X)) → a__dbl1(mark(X))
mark(sel1(X1, X2)) → a__sel1(mark(X1), mark(X2))
mark(quote(X)) → a__quote(mark(X))
mark(0) → 0
mark(s(X)) → s(X)
mark(nil) → nil
mark(cons(X1, X2)) → cons(X1, X2)
mark(01) → 01
mark(s1(X)) → s1(mark(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__from(X) → from(X)
a__dbl1(X) → dbl1(X)
a__sel1(X1, X2) → sel1(X1, X2)
a__quote(X) → quote(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

                  ↳ DependencyGraphProof

                    ↳ AND

                      ↳ QDP

                      ↳ QDP

                        ↳ QDPOrderProof

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ AND

                                ↳ QDP

                                  ↳ QDPOrderProof

                                    ↳ QDP

                                      ↳ DependencyGraphProof

                                        ↳ AND

                                          ↳ QDP

                                            ↳ QDPOrderProof

                                              ↳ QDP

                                                ↳ DependencyGraphProof

                                                  ↳ AND

                                                    ↳ QDP

                                                    ↳ QDP

                                                      ↳ Narrowing

                                                        ↳ QDP

                                                          ↳ DependencyGraphProof

                                                            ↳ QDP

                                                              ↳ Narrowing

                                                                ↳ QDP

                                                                  ↳ DependencyGraphProof

                                                                    ↳ QDP

                                                                      ↳ Narrowing

                                                                        ↳ QDP

                                                                          ↳ DependencyGraphProof

                                                                            ↳ QDP

                                                                              ↳ Narrowing

                                                                                ↳ QDP

                                                                                  ↳ DependencyGraphProof

                                                                                    ↳ QDP

                                                                                      ↳ Narrowing

                                                                                        ↳ QDP

                                                                                          ↳ DependencyGraphProof

                                                                                            ↳ QDP

                                                                                              ↳ Narrowing

                                                                                                ↳ QDP

                                                                                                  ↳ DependencyGraphProof

                                          ↳ QDP

                                ↳ QDP

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

A__SEL(s(from(x0)), cons(y1, y2)) → A__SEL(cons(x0, from(s(x0))), mark(y2))
A__SEL(s(sel1(y0, y1)), cons(y2, indx(x0, x1))) → A__SEL(a__sel1(mark(y0), mark(y1)), a__indx(mark(x0), x1))
A__SEL(s(dbl1(y0)), cons(y1, dbl(x0))) → A__SEL(a__dbl1(mark(y0)), a__dbl(mark(x0)))
A__SEL(s(dbl(y0)), cons(y1, dbl1(x0))) → A__SEL(a__dbl(mark(y0)), a__dbl1(mark(x0)))
A__SEL(s(from(x0)), cons(y1, y2)) → A__SEL(from(x0), mark(y2))
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(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(dbl1(y0)), cons(y1, cons(x0, x1))) → A__SEL(a__dbl1(mark(y0)), cons(x0, x1))
A__SEL(s(dbl(y0)), cons(y1, sel1(x0, x1))) → A__SEL(a__dbl(mark(y0)), a__sel1(mark(x0), mark(x1)))
A__SEL(s(sel1(y0, y1)), cons(y2, dbl1(x0))) → A__SEL(a__sel1(mark(y0), mark(y1)), a__dbl1(mark(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(sel1(y0, y1)), cons(y2, dbl(x0))) → A__SEL(a__sel1(mark(y0), mark(y1)), a__dbl(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(x0, x1)), cons(y1, y2)) → A__SEL(a__indx(mark(x0), x1), mark(y2))
A__SEL(s(sel1(y0, y1)), cons(y2, quote(x0))) → A__SEL(a__sel1(mark(y0), mark(y1)), a__quote(mark(x0)))
A__SEL(s(sel1(y0, y1)), cons(y2, sel(x0, x1))) → A__SEL(a__sel1(mark(y0), mark(y1)), a__sel(mark(x0), mark(x1)))
A__SEL(s(sel(y0, y1)), cons(y2, sel1(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), a__sel1(mark(x0), mark(x1)))
A__SEL(s(s(x0)), cons(y1, y2)) → A__SEL(s(x0), mark(y2))
A__SEL(s(sel1(y0, y1)), cons(y2, sel1(x0, x1))) → A__SEL(a__sel1(mark(y0), mark(y1)), a__sel1(mark(x0), mark(x1)))
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(sel1(y0, y1)), cons(y2, dbls(x0))) → A__SEL(a__sel1(mark(y0), mark(y1)), a__dbls(mark(x0)))
A__SEL(s(dbl1(y0)), cons(y1, sel(x0, x1))) → A__SEL(a__dbl1(mark(y0)), a__sel(mark(x0), mark(x1)))
A__SEL(s(sel(y0, y1)), cons(y2, dbl1(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__dbl1(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(dbl1(y0)), cons(y1, sel1(x0, x1))) → A__SEL(a__dbl1(mark(y0)), a__sel1(mark(x0), mark(x1)))
A__SEL(s(dbl1(y0)), cons(y1, dbls(x0))) → A__SEL(a__dbl1(mark(y0)), a__dbls(mark(x0)))
A__SEL(s(sel(y0, y1)), cons(y2, quote(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__quote(mark(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(dbl1(y0)), cons(y1, quote(x0))) → A__SEL(a__dbl1(mark(y0)), a__quote(mark(x0)))
A__SEL(s(sel1(y0, y1)), cons(y2, cons(x0, x1))) → A__SEL(a__sel1(mark(y0), mark(y1)), cons(x0, x1))
A__SEL(s(dbl(y0)), cons(y1, quote(x0))) → A__SEL(a__dbl(mark(y0)), a__quote(mark(x0)))
A__SEL(s(quote(x0)), cons(y1, y2)) → A__SEL(a__quote(mark(x0)), mark(y2))
A__SEL(s(sel1(y0, y1)), cons(y2, from(x0))) → A__SEL(a__sel1(mark(y0), mark(y1)), a__from(x0))
A__SEL(s(dbl1(y0)), cons(y1, from(x0))) → A__SEL(a__dbl1(mark(y0)), a__from(x0))
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(dbl1(y0)), cons(y1, dbl1(x0))) → A__SEL(a__dbl1(mark(y0)), a__dbl1(mark(x0)))
A__SEL(s(dbl1(y0)), cons(y1, indx(x0, x1))) → A__SEL(a__dbl1(mark(y0)), a__indx(mark(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)))
a__dbl1(0) → 01
a__dbl1(s(X)) → s1(s1(a__dbl1(mark(X))))
a__sel1(0, cons(X, Y)) → mark(X)
a__sel1(s(X), cons(Y, Z)) → a__sel1(mark(X), mark(Z))
a__quote(0) → 01
a__quote(s(X)) → s1(a__quote(mark(X)))
a__quote(dbl(X)) → a__dbl1(mark(X))
a__quote(sel(X, Y)) → a__sel1(mark(X), mark(Y))
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(dbl1(X)) → a__dbl1(mark(X))
mark(sel1(X1, X2)) → a__sel1(mark(X1), mark(X2))
mark(quote(X)) → a__quote(mark(X))
mark(0) → 0
mark(s(X)) → s(X)
mark(nil) → nil
mark(cons(X1, X2)) → cons(X1, X2)
mark(01) → 01
mark(s1(X)) → s1(mark(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__from(X) → from(X)
a__dbl1(X) → dbl1(X)
a__sel1(X1, X2) → sel1(X1, X2)
a__quote(X) → quote(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

                  ↳ DependencyGraphProof

                    ↳ AND

                      ↳ QDP

                      ↳ QDP

                        ↳ QDPOrderProof

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ AND

                                ↳ QDP

                                  ↳ QDPOrderProof

                                    ↳ QDP

                                      ↳ DependencyGraphProof

                                        ↳ AND

                                          ↳ QDP

                                            ↳ QDPOrderProof

                                              ↳ QDP

                                                ↳ DependencyGraphProof

                                                  ↳ AND

                                                    ↳ QDP

                                                    ↳ QDP

                                                      ↳ Narrowing

                                                        ↳ QDP

                                                          ↳ DependencyGraphProof

                                                            ↳ QDP

                                                              ↳ Narrowing

                                                                ↳ QDP

                                                                  ↳ DependencyGraphProof

                                                                    ↳ QDP

                                                                      ↳ Narrowing

                                                                        ↳ QDP

                                                                          ↳ DependencyGraphProof

                                                                            ↳ QDP

                                                                              ↳ Narrowing

                                                                                ↳ QDP

                                                                                  ↳ DependencyGraphProof

                                                                                    ↳ QDP

                                                                                      ↳ Narrowing

                                                                                        ↳ QDP

                                                                                          ↳ DependencyGraphProof

                                                                                            ↳ QDP

                                                                                              ↳ Narrowing

                                                                                                ↳ QDP

                                                                                                  ↳ DependencyGraphProof

                                                                                                    ↳ QDP

                                                                                                      ↳ Narrowing

                                          ↳ QDP

                                ↳ QDP

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

A__SEL(s(sel1(y0, y1)), cons(y2, indx(x0, x1))) → A__SEL(a__sel1(mark(y0), mark(y1)), a__indx(mark(x0), x1))
A__SEL(s(dbl(y0)), cons(y1, dbl1(x0))) → A__SEL(a__dbl(mark(y0)), a__dbl1(mark(x0)))
A__SEL(s(dbl1(y0)), cons(y1, dbl(x0))) → A__SEL(a__dbl1(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(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(dbl1(y0)), cons(y1, cons(x0, x1))) → A__SEL(a__dbl1(mark(y0)), cons(x0, x1))
A__SEL(s(dbl(y0)), cons(y1, sel1(x0, x1))) → A__SEL(a__dbl(mark(y0)), a__sel1(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(sel1(y0, y1)), cons(y2, dbl1(x0))) → A__SEL(a__sel1(mark(y0), mark(y1)), a__dbl1(mark(x0)))
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(sel1(y0, y1)), cons(y2, dbl(x0))) → A__SEL(a__sel1(mark(y0), mark(y1)), a__dbl(mark(x0)))
A__SEL(s(indx(x0, x1)), cons(y1, y2)) → A__SEL(a__indx(mark(x0), x1), mark(y2))
A__SEL(s(sel1(y0, y1)), cons(y2, quote(x0))) → A__SEL(a__sel1(mark(y0), mark(y1)), a__quote(mark(x0)))
A__SEL(s(sel(y0, y1)), cons(y2, sel1(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), a__sel1(mark(x0), mark(x1)))
A__SEL(s(sel1(y0, y1)), cons(y2, sel(x0, x1))) → A__SEL(a__sel1(mark(y0), mark(y1)), a__sel(mark(x0), mark(x1)))
A__SEL(s(s(x0)), cons(y1, y2)) → A__SEL(s(x0), mark(y2))
A__SEL(s(sel1(y0, y1)), cons(y2, sel1(x0, x1))) → A__SEL(a__sel1(mark(y0), mark(y1)), a__sel1(mark(x0), mark(x1)))
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(sel1(y0, y1)), cons(y2, dbls(x0))) → A__SEL(a__sel1(mark(y0), mark(y1)), a__dbls(mark(x0)))
A__SEL(s(dbl1(y0)), cons(y1, sel(x0, x1))) → A__SEL(a__dbl1(mark(y0)), a__sel(mark(x0), mark(x1)))
A__SEL(s(sel(y0, y1)), cons(y2, dbl1(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__dbl1(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(dbl1(y0)), cons(y1, sel1(x0, x1))) → A__SEL(a__dbl1(mark(y0)), a__sel1(mark(x0), mark(x1)))
A__SEL(s(sel(y0, y1)), cons(y2, quote(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__quote(mark(x0)))
A__SEL(s(dbl1(y0)), cons(y1, dbls(x0))) → A__SEL(a__dbl1(mark(y0)), a__dbls(mark(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(dbl1(y0)), cons(y1, quote(x0))) → A__SEL(a__dbl1(mark(y0)), a__quote(mark(x0)))
A__SEL(s(sel1(y0, y1)), cons(y2, cons(x0, x1))) → A__SEL(a__sel1(mark(y0), mark(y1)), cons(x0, x1))
A__SEL(s(dbl(y0)), cons(y1, quote(x0))) → A__SEL(a__dbl(mark(y0)), a__quote(mark(x0)))
A__SEL(s(quote(x0)), cons(y1, y2)) → A__SEL(a__quote(mark(x0)), mark(y2))
A__SEL(s(sel1(y0, y1)), cons(y2, from(x0))) → A__SEL(a__sel1(mark(y0), mark(y1)), a__from(x0))
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(dbl1(y0)), cons(y1, from(x0))) → A__SEL(a__dbl1(mark(y0)), a__from(x0))
A__SEL(s(dbl1(y0)), cons(y1, indx(x0, x1))) → A__SEL(a__dbl1(mark(y0)), a__indx(mark(x0), x1))
A__SEL(s(dbl1(y0)), cons(y1, dbl1(x0))) → A__SEL(a__dbl1(mark(y0)), a__dbl1(mark(x0)))
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)))
a__dbl1(0) → 01
a__dbl1(s(X)) → s1(s1(a__dbl1(mark(X))))
a__sel1(0, cons(X, Y)) → mark(X)
a__sel1(s(X), cons(Y, Z)) → a__sel1(mark(X), mark(Z))
a__quote(0) → 01
a__quote(s(X)) → s1(a__quote(mark(X)))
a__quote(dbl(X)) → a__dbl1(mark(X))
a__quote(sel(X, Y)) → a__sel1(mark(X), mark(Y))
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(dbl1(X)) → a__dbl1(mark(X))
mark(sel1(X1, X2)) → a__sel1(mark(X1), mark(X2))
mark(quote(X)) → a__quote(mark(X))
mark(0) → 0
mark(s(X)) → s(X)
mark(nil) → nil
mark(cons(X1, X2)) → cons(X1, X2)
mark(01) → 01
mark(s1(X)) → s1(mark(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__from(X) → from(X)
a__dbl1(X) → dbl1(X)
a__sel1(X1, X2) → sel1(X1, X2)
a__quote(X) → quote(X)

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

A__SEL(s(quote(y0)), cons(y1, dbls(x0))) → A__SEL(a__quote(mark(y0)), a__dbls(mark(x0)))
A__SEL(s(quote(y0)), cons(y1, cons(x0, x1))) → A__SEL(a__quote(mark(y0)), cons(x0, x1))
A__SEL(s(quote(y0)), cons(y1, indx(x0, x1))) → A__SEL(a__quote(mark(y0)), a__indx(mark(x0), x1))
A__SEL(s(quote(y0)), cons(y1, sel(x0, x1))) → A__SEL(a__quote(mark(y0)), a__sel(mark(x0), mark(x1)))
A__SEL(s(quote(y0)), cons(y1, from(x0))) → A__SEL(a__quote(mark(y0)), a__from(x0))
A__SEL(s(quote(y0)), cons(y1, 01)) → A__SEL(a__quote(mark(y0)), 01)
A__SEL(s(quote(y0)), cons(y1, s(x0))) → A__SEL(a__quote(mark(y0)), s(x0))
A__SEL(s(quote(y0)), cons(y1, sel1(x0, x1))) → A__SEL(a__quote(mark(y0)), a__sel1(mark(x0), mark(x1)))
A__SEL(s(quote(y0)), cons(y1, s1(x0))) → A__SEL(a__quote(mark(y0)), s1(mark(x0)))
A__SEL(s(quote(y0)), cons(y1, quote(x0))) → A__SEL(a__quote(mark(y0)), a__quote(mark(x0)))
A__SEL(s(quote(y0)), cons(y1, dbl1(x0))) → A__SEL(a__quote(mark(y0)), a__dbl1(mark(x0)))
A__SEL(s(quote(y0)), cons(y1, nil)) → A__SEL(a__quote(mark(y0)), nil)
A__SEL(s(quote(y0)), cons(y1, dbl(x0))) → A__SEL(a__quote(mark(y0)), a__dbl(mark(x0)))
A__SEL(s(quote(y0)), cons(y1, 0)) → A__SEL(a__quote(mark(y0)), 0)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ QDPOrderProof

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ AND

                      ↳ QDP

                      ↳ QDP

                        ↳ QDPOrderProof

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ AND

                                ↳ QDP

                                  ↳ QDPOrderProof

                                    ↳ QDP

                                      ↳ DependencyGraphProof

                                        ↳ AND

                                          ↳ QDP

                                            ↳ QDPOrderProof

                                              ↳ QDP

                                                ↳ DependencyGraphProof

                                                  ↳ AND

                                                    ↳ QDP

                                                    ↳ QDP

                                                      ↳ Narrowing

                                                        ↳ QDP

                                                          ↳ DependencyGraphProof

                                                            ↳ QDP

                                                              ↳ Narrowing

                                                                ↳ QDP

                                                                  ↳ DependencyGraphProof

                                                                    ↳ QDP

                                                                      ↳ Narrowing

                                                                        ↳ QDP

                                                                          ↳ DependencyGraphProof

                                                                            ↳ QDP

                                                                              ↳ Narrowing

                                                                                ↳ QDP

                                                                                  ↳ DependencyGraphProof

                                                                                    ↳ QDP

                                                                                      ↳ Narrowing

                                                                                        ↳ QDP

                                                                                          ↳ DependencyGraphProof

                                                                                            ↳ QDP

                                                                                              ↳ Narrowing

                                                                                                ↳ QDP

                                                                                                  ↳ DependencyGraphProof

                                                                                                    ↳ QDP

                                                                                                      ↳ Narrowing

                                                                                                        ↳ QDP

                                                                                                          ↳ DependencyGraphProof

                                          ↳ QDP

                                ↳ QDP

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

A__SEL(s(sel1(y0, y1)), cons(y2, indx(x0, x1))) → A__SEL(a__sel1(mark(y0), mark(y1)), a__indx(mark(x0), x1))
A__SEL(s(dbl(y0)), cons(y1, dbl1(x0))) → A__SEL(a__dbl(mark(y0)), a__dbl1(mark(x0)))
A__SEL(s(dbl1(y0)), cons(y1, dbl(x0))) → A__SEL(a__dbl1(mark(y0)), a__dbl(mark(x0)))
A__SEL(s(quote(y0)), cons(y1, indx(x0, x1))) → A__SEL(a__quote(mark(y0)), a__indx(mark(x0), x1))
A__SEL(s(dbl(y0)), cons(y1, from(x0))) → A__SEL(a__dbl(mark(y0)), a__from(x0))
A__SEL(s(quote(y0)), cons(y1, from(x0))) → A__SEL(a__quote(mark(y0)), a__from(x0))
A__SEL(s(quote(y0)), cons(y1, 01)) → A__SEL(a__quote(mark(y0)), 01)
A__SEL(s(quote(y0)), cons(y1, s(x0))) → A__SEL(a__quote(mark(y0)), s(x0))
A__SEL(s(dbl(y0)), cons(y1, sel1(x0, x1))) → A__SEL(a__dbl(mark(y0)), a__sel1(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(quote(y0)), cons(y1, quote(x0))) → A__SEL(a__quote(mark(y0)), a__quote(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(sel1(y0, y1)), cons(y2, dbl(x0))) → A__SEL(a__sel1(mark(y0), mark(y1)), a__dbl(mark(x0)))
A__SEL(s(sel(y0, y1)), cons(y2, sel1(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), a__sel1(mark(x0), mark(x1)))
A__SEL(s(sel1(y0, y1)), cons(y2, sel(x0, x1))) → A__SEL(a__sel1(mark(y0), mark(y1)), a__sel(mark(x0), mark(x1)))
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(sel1(y0, y1)), cons(y2, dbls(x0))) → A__SEL(a__sel1(mark(y0), mark(y1)), a__dbls(mark(x0)))
A__SEL(s(dbl1(y0)), cons(y1, sel(x0, x1))) → A__SEL(a__dbl1(mark(y0)), 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(quote(y0)), cons(y1, sel1(x0, x1))) → A__SEL(a__quote(mark(y0)), a__sel1(mark(x0), mark(x1)))
A__SEL(s(sel(y0, y1)), cons(y2, quote(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__quote(mark(x0)))
A__SEL(s(sel1(y0, y1)), cons(y2, cons(x0, x1))) → A__SEL(a__sel1(mark(y0), mark(y1)), cons(x0, x1))
A__SEL(s(quote(y0)), cons(y1, dbl(x0))) → A__SEL(a__quote(mark(y0)), a__dbl(mark(x0)))
A__SEL(s(dbl(y0)), cons(y1, quote(x0))) → A__SEL(a__dbl(mark(y0)), a__quote(mark(x0)))
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(quote(y0)), cons(y1, dbls(x0))) → A__SEL(a__quote(mark(y0)), a__dbls(mark(x0)))
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(dbl(y0)), cons(y1, sel(x0, x1))) → A__SEL(a__dbl(mark(y0)), a__sel(mark(x0), mark(x1)))
A__SEL(s(dbl1(y0)), cons(y1, cons(x0, x1))) → A__SEL(a__dbl1(mark(y0)), cons(x0, x1))
A__SEL(s(sel1(y0, y1)), cons(y2, dbl1(x0))) → A__SEL(a__sel1(mark(y0), mark(y1)), a__dbl1(mark(x0)))
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, 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(sel1(y0, y1)), cons(y2, quote(x0))) → A__SEL(a__sel1(mark(y0), mark(y1)), a__quote(mark(x0)))
A__SEL(s(quote(y0)), cons(y1, 0)) → A__SEL(a__quote(mark(y0)), 0)
A__SEL(s(s(x0)), cons(y1, y2)) → A__SEL(s(x0), mark(y2))
A__SEL(s(sel1(y0, y1)), cons(y2, sel1(x0, x1))) → A__SEL(a__sel1(mark(y0), mark(y1)), a__sel1(mark(x0), mark(x1)))
A__SEL(s(sel(y0, y1)), cons(y2, dbl1(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__dbl1(mark(x0)))
A__SEL(s(quote(y0)), cons(y1, cons(x0, x1))) → A__SEL(a__quote(mark(y0)), cons(x0, x1))
A__SEL(s(quote(y0)), cons(y1, sel(x0, x1))) → A__SEL(a__quote(mark(y0)), a__sel(mark(x0), mark(x1)))
A__SEL(s(dbl1(y0)), cons(y1, sel1(x0, x1))) → A__SEL(a__dbl1(mark(y0)), a__sel1(mark(x0), mark(x1)))
A__SEL(s(dbl1(y0)), cons(y1, dbls(x0))) → A__SEL(a__dbl1(mark(y0)), a__dbls(mark(x0)))
A__SEL(s(quote(y0)), cons(y1, s1(x0))) → A__SEL(a__quote(mark(y0)), s1(mark(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(quote(y0)), cons(y1, dbl1(x0))) → A__SEL(a__quote(mark(y0)), a__dbl1(mark(x0)))
A__SEL(s(dbl1(y0)), cons(y1, quote(x0))) → A__SEL(a__dbl1(mark(y0)), a__quote(mark(x0)))
A__SEL(s(quote(y0)), cons(y1, nil)) → A__SEL(a__quote(mark(y0)), nil)
A__SEL(s(sel1(y0, y1)), cons(y2, from(x0))) → A__SEL(a__sel1(mark(y0), mark(y1)), a__from(x0))
A__SEL(s(dbl1(y0)), cons(y1, from(x0))) → A__SEL(a__dbl1(mark(y0)), a__from(x0))
A__SEL(s(dbl1(y0)), cons(y1, indx(x0, x1))) → A__SEL(a__dbl1(mark(y0)), a__indx(mark(x0), x1))
A__SEL(s(dbl1(y0)), cons(y1, dbl1(x0))) → A__SEL(a__dbl1(mark(y0)), a__dbl1(mark(x0)))
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)))
a__dbl1(0) → 01
a__dbl1(s(X)) → s1(s1(a__dbl1(mark(X))))
a__sel1(0, cons(X, Y)) → mark(X)
a__sel1(s(X), cons(Y, Z)) → a__sel1(mark(X), mark(Z))
a__quote(0) → 01
a__quote(s(X)) → s1(a__quote(mark(X)))
a__quote(dbl(X)) → a__dbl1(mark(X))
a__quote(sel(X, Y)) → a__sel1(mark(X), mark(Y))
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(dbl1(X)) → a__dbl1(mark(X))
mark(sel1(X1, X2)) → a__sel1(mark(X1), mark(X2))
mark(quote(X)) → a__quote(mark(X))
mark(0) → 0
mark(s(X)) → s(X)
mark(nil) → nil
mark(cons(X1, X2)) → cons(X1, X2)
mark(01) → 01
mark(s1(X)) → s1(mark(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__from(X) → from(X)
a__dbl1(X) → dbl1(X)
a__sel1(X1, X2) → sel1(X1, X2)
a__quote(X) → quote(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 5 less nodes.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ QDPOrderProof

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ AND

                      ↳ QDP

                      ↳ QDP

                        ↳ QDPOrderProof

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ AND

                                ↳ QDP

                                  ↳ QDPOrderProof

                                    ↳ QDP

                                      ↳ DependencyGraphProof

                                        ↳ AND

                                          ↳ QDP

                                            ↳ QDPOrderProof

                                              ↳ QDP

                                                ↳ DependencyGraphProof

                                                  ↳ AND

                                                    ↳ QDP

                                                    ↳ QDP

                                                      ↳ Narrowing

                                                        ↳ QDP

                                                          ↳ DependencyGraphProof

                                                            ↳ QDP

                                                              ↳ Narrowing

                                                                ↳ QDP

                                                                  ↳ DependencyGraphProof

                                                                    ↳ QDP

                                                                      ↳ Narrowing

                                                                        ↳ QDP

                                                                          ↳ DependencyGraphProof

                                                                            ↳ QDP

                                                                              ↳ Narrowing

                                                                                ↳ QDP

                                                                                  ↳ DependencyGraphProof

                                                                                    ↳ QDP

                                                                                      ↳ Narrowing

                                                                                        ↳ QDP

                                                                                          ↳ DependencyGraphProof

                                                                                            ↳ QDP

                                                                                              ↳ Narrowing

                                                                                                ↳ QDP

                                                                                                  ↳ DependencyGraphProof

                                                                                                    ↳ QDP

                                                                                                      ↳ Narrowing

                                                                                                        ↳ QDP

                                                                                                          ↳ DependencyGraphProof

                                                                                                            ↳ QDP

                                                                                                              ↳ Narrowing

                                          ↳ QDP

                                ↳ QDP

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

A__SEL(s(quote(y0)), cons(y1, dbls(x0))) → A__SEL(a__quote(mark(y0)), a__dbls(mark(x0)))
A__SEL(s(sel1(y0, y1)), cons(y2, indx(x0, x1))) → A__SEL(a__sel1(mark(y0), mark(y1)), a__indx(mark(x0), x1))
A__SEL(s(dbl(y0)), cons(y1, dbl1(x0))) → A__SEL(a__dbl(mark(y0)), a__dbl1(mark(x0)))
A__SEL(s(dbl1(y0)), cons(y1, dbl(x0))) → A__SEL(a__dbl1(mark(y0)), a__dbl(mark(x0)))
A__SEL(s(quote(y0)), cons(y1, indx(x0, x1))) → A__SEL(a__quote(mark(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(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(quote(y0)), cons(y1, from(x0))) → A__SEL(a__quote(mark(y0)), a__from(x0))
A__SEL(s(dbl(y0)), cons(y1, from(x0))) → A__SEL(a__dbl(mark(y0)), a__from(x0))
A__SEL(s(dbl1(y0)), cons(y1, cons(x0, x1))) → A__SEL(a__dbl1(mark(y0)), cons(x0, x1))
A__SEL(s(dbl(y0)), cons(y1, sel1(x0, x1))) → A__SEL(a__dbl(mark(y0)), a__sel1(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(sel1(y0, y1)), cons(y2, dbl1(x0))) → A__SEL(a__sel1(mark(y0), mark(y1)), a__dbl1(mark(x0)))
A__SEL(s(quote(y0)), cons(y1, quote(x0))) → A__SEL(a__quote(mark(y0)), a__quote(mark(x0)))
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(sel1(y0, y1)), cons(y2, dbl(x0))) → A__SEL(a__sel1(mark(y0), mark(y1)), a__dbl(mark(x0)))
A__SEL(s(indx(x0, x1)), cons(y1, y2)) → A__SEL(a__indx(mark(x0), x1), mark(y2))
A__SEL(s(sel1(y0, y1)), cons(y2, quote(x0))) → A__SEL(a__sel1(mark(y0), mark(y1)), a__quote(mark(x0)))
A__SEL(s(sel(y0, y1)), cons(y2, sel1(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), a__sel1(mark(x0), mark(x1)))
A__SEL(s(sel1(y0, y1)), cons(y2, sel(x0, x1))) → A__SEL(a__sel1(mark(y0), mark(y1)), a__sel(mark(x0), mark(x1)))
A__SEL(s(s(x0)), cons(y1, y2)) → A__SEL(s(x0), mark(y2))
A__SEL(s(sel1(y0, y1)), cons(y2, sel1(x0, x1))) → A__SEL(a__sel1(mark(y0), mark(y1)), a__sel1(mark(x0), mark(x1)))
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(sel1(y0, y1)), cons(y2, dbls(x0))) → A__SEL(a__sel1(mark(y0), mark(y1)), a__dbls(mark(x0)))
A__SEL(s(dbl1(y0)), cons(y1, sel(x0, x1))) → A__SEL(a__dbl1(mark(y0)), a__sel(mark(x0), mark(x1)))
A__SEL(s(quote(y0)), cons(y1, cons(x0, x1))) → A__SEL(a__quote(mark(y0)), cons(x0, x1))
A__SEL(s(sel(y0, y1)), cons(y2, dbl1(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__dbl1(mark(x0)))
A__SEL(s(quote(y0)), cons(y1, sel(x0, x1))) → A__SEL(a__quote(mark(y0)), 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(quote(y0)), cons(y1, sel1(x0, x1))) → A__SEL(a__quote(mark(y0)), a__sel1(mark(x0), mark(x1)))
A__SEL(s(dbl1(y0)), cons(y1, sel1(x0, x1))) → A__SEL(a__dbl1(mark(y0)), a__sel1(mark(x0), mark(x1)))
A__SEL(s(sel(y0, y1)), cons(y2, quote(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__quote(mark(x0)))
A__SEL(s(dbl1(y0)), cons(y1, dbls(x0))) → A__SEL(a__dbl1(mark(y0)), a__dbls(mark(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(quote(y0)), cons(y1, dbl1(x0))) → A__SEL(a__quote(mark(y0)), a__dbl1(mark(x0)))
A__SEL(s(dbl1(y0)), cons(y1, quote(x0))) → A__SEL(a__dbl1(mark(y0)), a__quote(mark(x0)))
A__SEL(s(sel1(y0, y1)), cons(y2, cons(x0, x1))) → A__SEL(a__sel1(mark(y0), mark(y1)), cons(x0, x1))
A__SEL(s(quote(y0)), cons(y1, dbl(x0))) → A__SEL(a__quote(mark(y0)), a__dbl(mark(x0)))
A__SEL(s(dbl(y0)), cons(y1, quote(x0))) → A__SEL(a__dbl(mark(y0)), a__quote(mark(x0)))
A__SEL(s(sel1(y0, y1)), cons(y2, from(x0))) → A__SEL(a__sel1(mark(y0), mark(y1)), a__from(x0))
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(dbl1(y0)), cons(y1, from(x0))) → A__SEL(a__dbl1(mark(y0)), a__from(x0))
A__SEL(s(dbl1(y0)), cons(y1, indx(x0, x1))) → A__SEL(a__dbl1(mark(y0)), a__indx(mark(x0), x1))
A__SEL(s(dbl1(y0)), cons(y1, dbl1(x0))) → A__SEL(a__dbl1(mark(y0)), a__dbl1(mark(x0)))
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)))
a__dbl1(0) → 01
a__dbl1(s(X)) → s1(s1(a__dbl1(mark(X))))
a__sel1(0, cons(X, Y)) → mark(X)
a__sel1(s(X), cons(Y, Z)) → a__sel1(mark(X), mark(Z))
a__quote(0) → 01
a__quote(s(X)) → s1(a__quote(mark(X)))
a__quote(dbl(X)) → a__dbl1(mark(X))
a__quote(sel(X, Y)) → a__sel1(mark(X), mark(Y))
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(dbl1(X)) → a__dbl1(mark(X))
mark(sel1(X1, X2)) → a__sel1(mark(X1), mark(X2))
mark(quote(X)) → a__quote(mark(X))
mark(0) → 0
mark(s(X)) → s(X)
mark(nil) → nil
mark(cons(X1, X2)) → cons(X1, X2)
mark(01) → 01
mark(s1(X)) → s1(mark(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__from(X) → from(X)
a__dbl1(X) → dbl1(X)
a__sel1(X1, X2) → sel1(X1, X2)
a__quote(X) → quote(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, quote(x0))) → A__SEL(a__indx(mark(y0), y1), a__quote(mark(x0)))
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, sel1(x0, x1))) → A__SEL(a__indx(mark(y0), y1), a__sel1(mark(x0), mark(x1)))
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, 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, s1(x0))) → A__SEL(a__indx(mark(y0), y1), s1(mark(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, dbl1(x0))) → A__SEL(a__indx(mark(y0), y1), a__dbl1(mark(x0)))
A__SEL(s(indx(y0, y1)), cons(y2, 01)) → A__SEL(a__indx(mark(y0), y1), 01)
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(indx(y0, y1)), cons(y2, nil)) → A__SEL(a__indx(mark(y0), y1), nil)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ QDPOrderProof

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ AND

                      ↳ QDP

                      ↳ QDP

                        ↳ QDPOrderProof

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ AND

                                ↳ QDP

                                  ↳ QDPOrderProof

                                    ↳ QDP

                                      ↳ DependencyGraphProof

                                        ↳ AND

                                          ↳ QDP

                                            ↳ QDPOrderProof

                                              ↳ QDP

                                                ↳ DependencyGraphProof

                                                  ↳ AND

                                                    ↳ QDP

                                                    ↳ QDP

                                                      ↳ Narrowing

                                                        ↳ QDP

                                                          ↳ DependencyGraphProof

                                                            ↳ QDP

                                                              ↳ Narrowing

                                                                ↳ QDP

                                                                  ↳ DependencyGraphProof

                                                                    ↳ QDP

                                                                      ↳ Narrowing

                                                                        ↳ QDP

                                                                          ↳ DependencyGraphProof

                                                                            ↳ QDP

                                                                              ↳ Narrowing

                                                                                ↳ QDP

                                                                                  ↳ DependencyGraphProof

                                                                                    ↳ QDP

                                                                                      ↳ Narrowing

                                                                                        ↳ QDP

                                                                                          ↳ DependencyGraphProof

                                                                                            ↳ QDP

                                                                                              ↳ Narrowing

                                                                                                ↳ QDP

                                                                                                  ↳ DependencyGraphProof

                                                                                                    ↳ QDP

                                                                                                      ↳ Narrowing

                                                                                                        ↳ QDP

                                                                                                          ↳ DependencyGraphProof

                                                                                                            ↳ QDP

                                                                                                              ↳ Narrowing

                                                                                                                ↳ QDP

                                                                                                                  ↳ DependencyGraphProof

                                          ↳ QDP

                                ↳ QDP

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

A__SEL(s(sel1(y0, y1)), cons(y2, indx(x0, x1))) → A__SEL(a__sel1(mark(y0), mark(y1)), a__indx(mark(x0), x1))
A__SEL(s(dbl1(y0)), cons(y1, dbl(x0))) → A__SEL(a__dbl1(mark(y0)), a__dbl(mark(x0)))
A__SEL(s(dbl(y0)), cons(y1, dbl1(x0))) → A__SEL(a__dbl(mark(y0)), a__dbl1(mark(x0)))
A__SEL(s(quote(y0)), cons(y1, indx(x0, x1))) → A__SEL(a__quote(mark(y0)), a__indx(mark(x0), x1))
A__SEL(s(indx(y0, y1)), cons(y2, dbl(x0))) → A__SEL(a__indx(mark(y0), y1), a__dbl(mark(x0)))
A__SEL(s(dbl(y0)), cons(y1, from(x0))) → A__SEL(a__dbl(mark(y0)), a__from(x0))
A__SEL(s(quote(y0)), cons(y1, from(x0))) → A__SEL(a__quote(mark(y0)), a__from(x0))
A__SEL(s(dbl(y0)), cons(y1, sel1(x0, x1))) → A__SEL(a__dbl(mark(y0)), a__sel1(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(quote(y0)), cons(y1, quote(x0))) → A__SEL(a__quote(mark(y0)), a__quote(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(sel1(y0, y1)), cons(y2, dbl(x0))) → A__SEL(a__sel1(mark(y0), mark(y1)), a__dbl(mark(x0)))
A__SEL(s(indx(y0, y1)), cons(y2, 01)) → A__SEL(a__indx(mark(y0), y1), 01)
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(sel1(y0, y1)), cons(y2, sel(x0, x1))) → A__SEL(a__sel1(mark(y0), mark(y1)), a__sel(mark(x0), mark(x1)))
A__SEL(s(sel(y0, y1)), cons(y2, sel1(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), a__sel1(mark(x0), mark(x1)))
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(sel1(y0, y1)), cons(y2, dbls(x0))) → A__SEL(a__sel1(mark(y0), mark(y1)), a__dbls(mark(x0)))
A__SEL(s(dbl1(y0)), cons(y1, sel(x0, x1))) → A__SEL(a__dbl1(mark(y0)), 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(indx(y0, y1)), cons(y2, 0)) → A__SEL(a__indx(mark(y0), y1), 0)
A__SEL(s(quote(y0)), cons(y1, sel1(x0, x1))) → A__SEL(a__quote(mark(y0)), a__sel1(mark(x0), mark(x1)))
A__SEL(s(sel(y0, y1)), cons(y2, quote(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__quote(mark(x0)))
A__SEL(s(indx(y0, y1)), cons(y2, s(x0))) → A__SEL(a__indx(mark(y0), y1), s(x0))
A__SEL(s(sel1(y0, y1)), cons(y2, cons(x0, x1))) → A__SEL(a__sel1(mark(y0), mark(y1)), cons(x0, x1))
A__SEL(s(dbl(y0)), cons(y1, quote(x0))) → A__SEL(a__dbl(mark(y0)), a__quote(mark(x0)))
A__SEL(s(quote(y0)), cons(y1, dbl(x0))) → A__SEL(a__quote(mark(y0)), a__dbl(mark(x0)))
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(indx(y0, y1)), cons(y2, from(x0))) → A__SEL(a__indx(mark(y0), y1), a__from(x0))
A__SEL(s(quote(y0)), cons(y1, dbls(x0))) → A__SEL(a__quote(mark(y0)), a__dbls(mark(x0)))
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(dbl(y0)), cons(y1, sel(x0, x1))) → A__SEL(a__dbl(mark(y0)), a__sel(mark(x0), mark(x1)))
A__SEL(s(dbl1(y0)), cons(y1, cons(x0, x1))) → A__SEL(a__dbl1(mark(y0)), cons(x0, x1))
A__SEL(s(sel1(y0, y1)), cons(y2, dbl1(x0))) → A__SEL(a__sel1(mark(y0), mark(y1)), a__dbl1(mark(x0)))
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, dbls(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__dbls(mark(x0)))
A__SEL(s(sel1(y0, y1)), cons(y2, quote(x0))) → A__SEL(a__sel1(mark(y0), mark(y1)), a__quote(mark(x0)))
A__SEL(s(s(x0)), cons(y1, y2)) → A__SEL(s(x0), mark(y2))
A__SEL(s(sel1(y0, y1)), cons(y2, sel1(x0, x1))) → A__SEL(a__sel1(mark(y0), mark(y1)), a__sel1(mark(x0), mark(x1)))
A__SEL(s(indx(y0, y1)), cons(y2, quote(x0))) → A__SEL(a__indx(mark(y0), y1), a__quote(mark(x0)))
A__SEL(s(sel(y0, y1)), cons(y2, dbl1(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__dbl1(mark(x0)))
A__SEL(s(quote(y0)), cons(y1, cons(x0, x1))) → A__SEL(a__quote(mark(y0)), cons(x0, x1))
A__SEL(s(indx(y0, y1)), cons(y2, sel1(x0, x1))) → A__SEL(a__indx(mark(y0), y1), a__sel1(mark(x0), mark(x1)))
A__SEL(s(quote(y0)), cons(y1, sel(x0, x1))) → A__SEL(a__quote(mark(y0)), a__sel(mark(x0), mark(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(dbl1(y0)), cons(y1, sel1(x0, x1))) → A__SEL(a__dbl1(mark(y0)), a__sel1(mark(x0), mark(x1)))
A__SEL(s(dbl1(y0)), cons(y1, dbls(x0))) → A__SEL(a__dbl1(mark(y0)), a__dbls(mark(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(indx(y0, y1)), cons(y2, s1(x0))) → A__SEL(a__indx(mark(y0), y1), s1(mark(x0)))
A__SEL(s(dbl1(y0)), cons(y1, quote(x0))) → A__SEL(a__dbl1(mark(y0)), a__quote(mark(x0)))
A__SEL(s(quote(y0)), cons(y1, dbl1(x0))) → A__SEL(a__quote(mark(y0)), a__dbl1(mark(x0)))
A__SEL(s(indx(y0, y1)), cons(y2, dbl1(x0))) → A__SEL(a__indx(mark(y0), y1), a__dbl1(mark(x0)))
A__SEL(s(sel1(y0, y1)), cons(y2, from(x0))) → A__SEL(a__sel1(mark(y0), mark(y1)), a__from(x0))
A__SEL(s(dbl1(y0)), cons(y1, from(x0))) → A__SEL(a__dbl1(mark(y0)), a__from(x0))
A__SEL(s(indx(y0, y1)), cons(y2, nil)) → A__SEL(a__indx(mark(y0), y1), nil)
A__SEL(s(dbl1(y0)), cons(y1, dbl1(x0))) → A__SEL(a__dbl1(mark(y0)), a__dbl1(mark(x0)))
A__SEL(s(dbl1(y0)), cons(y1, indx(x0, x1))) → A__SEL(a__dbl1(mark(y0)), a__indx(mark(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)))
a__dbl1(0) → 01
a__dbl1(s(X)) → s1(s1(a__dbl1(mark(X))))
a__sel1(0, cons(X, Y)) → mark(X)
a__sel1(s(X), cons(Y, Z)) → a__sel1(mark(X), mark(Z))
a__quote(0) → 01
a__quote(s(X)) → s1(a__quote(mark(X)))
a__quote(dbl(X)) → a__dbl1(mark(X))
a__quote(sel(X, Y)) → a__sel1(mark(X), mark(Y))
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(dbl1(X)) → a__dbl1(mark(X))
mark(sel1(X1, X2)) → a__sel1(mark(X1), mark(X2))
mark(quote(X)) → a__quote(mark(X))
mark(0) → 0
mark(s(X)) → s(X)
mark(nil) → nil
mark(cons(X1, X2)) → cons(X1, X2)
mark(01) → 01
mark(s1(X)) → s1(mark(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__from(X) → from(X)
a__dbl1(X) → dbl1(X)
a__sel1(X1, X2) → sel1(X1, X2)
a__quote(X) → quote(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 5 less nodes.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ QDPOrderProof

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ AND

                      ↳ QDP

                      ↳ QDP

                        ↳ QDPOrderProof

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ AND

                                ↳ QDP

                                  ↳ QDPOrderProof

                                    ↳ QDP

                                      ↳ DependencyGraphProof

                                        ↳ AND

                                          ↳ QDP

                                            ↳ QDPOrderProof

                                              ↳ QDP

                                                ↳ DependencyGraphProof

                                                  ↳ AND

                                                    ↳ QDP

                                                    ↳ QDP

                                                      ↳ Narrowing

                                                        ↳ QDP

                                                          ↳ DependencyGraphProof

                                                            ↳ QDP

                                                              ↳ Narrowing

                                                                ↳ QDP

                                                                  ↳ DependencyGraphProof

                                                                    ↳ QDP

                                                                      ↳ Narrowing

                                                                        ↳ QDP

                                                                          ↳ DependencyGraphProof

                                                                            ↳ QDP

                                                                              ↳ Narrowing

                                                                                ↳ QDP

                                                                                  ↳ DependencyGraphProof

                                                                                    ↳ QDP

                                                                                      ↳ Narrowing

                                                                                        ↳ QDP

                                                                                          ↳ DependencyGraphProof

                                                                                            ↳ QDP

                                                                                              ↳ Narrowing

                                                                                                ↳ QDP

                                                                                                  ↳ DependencyGraphProof

                                                                                                    ↳ QDP

                                                                                                      ↳ Narrowing

                                                                                                        ↳ QDP

                                                                                                          ↳ DependencyGraphProof

                                                                                                            ↳ QDP

                                                                                                              ↳ Narrowing

                                                                                                                ↳ QDP

                                                                                                                  ↳ DependencyGraphProof

                                                                                                                    ↳ QDP

                                                                                                                      ↳ Narrowing

                                          ↳ QDP

                                ↳ QDP

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

A__SEL(s(sel1(y0, y1)), cons(y2, indx(x0, x1))) → A__SEL(a__sel1(mark(y0), mark(y1)), a__indx(mark(x0), x1))
A__SEL(s(dbl1(y0)), cons(y1, dbl(x0))) → A__SEL(a__dbl1(mark(y0)), a__dbl(mark(x0)))
A__SEL(s(dbl(y0)), cons(y1, dbl1(x0))) → A__SEL(a__dbl(mark(y0)), a__dbl1(mark(x0)))
A__SEL(s(quote(y0)), cons(y1, indx(x0, x1))) → A__SEL(a__quote(mark(y0)), a__indx(mark(x0), x1))
A__SEL(s(indx(y0, y1)), cons(y2, dbl(x0))) → A__SEL(a__indx(mark(y0), y1), a__dbl(mark(x0)))
A__SEL(s(dbl(y0)), cons(y1, from(x0))) → A__SEL(a__dbl(mark(y0)), a__from(x0))
A__SEL(s(quote(y0)), cons(y1, from(x0))) → A__SEL(a__quote(mark(y0)), a__from(x0))
A__SEL(s(dbl(y0)), cons(y1, sel1(x0, x1))) → A__SEL(a__dbl(mark(y0)), a__sel1(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(quote(y0)), cons(y1, quote(x0))) → A__SEL(a__quote(mark(y0)), a__quote(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(sel1(y0, y1)), cons(y2, dbl(x0))) → A__SEL(a__sel1(mark(y0), mark(y1)), a__dbl(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(sel1(y0, y1)), cons(y2, sel(x0, x1))) → A__SEL(a__sel1(mark(y0), mark(y1)), a__sel(mark(x0), mark(x1)))
A__SEL(s(sel(y0, y1)), cons(y2, sel1(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), a__sel1(mark(x0), mark(x1)))
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(sel1(y0, y1)), cons(y2, dbls(x0))) → A__SEL(a__sel1(mark(y0), mark(y1)), a__dbls(mark(x0)))
A__SEL(s(dbl1(y0)), cons(y1, sel(x0, x1))) → A__SEL(a__dbl1(mark(y0)), 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(quote(y0)), cons(y1, sel1(x0, x1))) → A__SEL(a__quote(mark(y0)), a__sel1(mark(x0), mark(x1)))
A__SEL(s(sel(y0, y1)), cons(y2, quote(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__quote(mark(x0)))
A__SEL(s(sel1(y0, y1)), cons(y2, cons(x0, x1))) → A__SEL(a__sel1(mark(y0), mark(y1)), cons(x0, x1))
A__SEL(s(dbl(y0)), cons(y1, quote(x0))) → A__SEL(a__dbl(mark(y0)), a__quote(mark(x0)))
A__SEL(s(quote(y0)), cons(y1, dbl(x0))) → A__SEL(a__quote(mark(y0)), a__dbl(mark(x0)))
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(quote(y0)), cons(y1, dbls(x0))) → A__SEL(a__quote(mark(y0)), a__dbls(mark(x0)))
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(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(dbl1(y0)), cons(y1, cons(x0, x1))) → A__SEL(a__dbl1(mark(y0)), cons(x0, x1))
A__SEL(s(sel1(y0, y1)), cons(y2, dbl1(x0))) → A__SEL(a__sel1(mark(y0), mark(y1)), a__dbl1(mark(x0)))
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, dbls(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__dbls(mark(x0)))
A__SEL(s(sel1(y0, y1)), cons(y2, quote(x0))) → A__SEL(a__sel1(mark(y0), mark(y1)), a__quote(mark(x0)))
A__SEL(s(s(x0)), cons(y1, y2)) → A__SEL(s(x0), mark(y2))
A__SEL(s(sel1(y0, y1)), cons(y2, sel1(x0, x1))) → A__SEL(a__sel1(mark(y0), mark(y1)), a__sel1(mark(x0), mark(x1)))
A__SEL(s(indx(y0, y1)), cons(y2, quote(x0))) → A__SEL(a__indx(mark(y0), y1), a__quote(mark(x0)))
A__SEL(s(sel(y0, y1)), cons(y2, dbl1(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__dbl1(mark(x0)))
A__SEL(s(quote(y0)), cons(y1, cons(x0, x1))) → A__SEL(a__quote(mark(y0)), cons(x0, x1))
A__SEL(s(indx(y0, y1)), cons(y2, sel1(x0, x1))) → A__SEL(a__indx(mark(y0), y1), a__sel1(mark(x0), mark(x1)))
A__SEL(s(quote(y0)), cons(y1, sel(x0, x1))) → A__SEL(a__quote(mark(y0)), a__sel(mark(x0), mark(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(dbl1(y0)), cons(y1, sel1(x0, x1))) → A__SEL(a__dbl1(mark(y0)), a__sel1(mark(x0), mark(x1)))
A__SEL(s(dbl1(y0)), cons(y1, dbls(x0))) → A__SEL(a__dbl1(mark(y0)), a__dbls(mark(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(dbl1(y0)), cons(y1, quote(x0))) → A__SEL(a__dbl1(mark(y0)), a__quote(mark(x0)))
A__SEL(s(quote(y0)), cons(y1, dbl1(x0))) → A__SEL(a__quote(mark(y0)), a__dbl1(mark(x0)))
A__SEL(s(indx(y0, y1)), cons(y2, dbl1(x0))) → A__SEL(a__indx(mark(y0), y1), a__dbl1(mark(x0)))
A__SEL(s(sel1(y0, y1)), cons(y2, from(x0))) → A__SEL(a__sel1(mark(y0), mark(y1)), a__from(x0))
A__SEL(s(dbl1(y0)), cons(y1, from(x0))) → A__SEL(a__dbl1(mark(y0)), a__from(x0))
A__SEL(s(dbl1(y0)), cons(y1, dbl1(x0))) → A__SEL(a__dbl1(mark(y0)), a__dbl1(mark(x0)))
A__SEL(s(dbl1(y0)), cons(y1, indx(x0, x1))) → A__SEL(a__dbl1(mark(y0)), a__indx(mark(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)))
a__dbl1(0) → 01
a__dbl1(s(X)) → s1(s1(a__dbl1(mark(X))))
a__sel1(0, cons(X, Y)) → mark(X)
a__sel1(s(X), cons(Y, Z)) → a__sel1(mark(X), mark(Z))
a__quote(0) → 01
a__quote(s(X)) → s1(a__quote(mark(X)))
a__quote(dbl(X)) → a__dbl1(mark(X))
a__quote(sel(X, Y)) → a__sel1(mark(X), mark(Y))
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(dbl1(X)) → a__dbl1(mark(X))
mark(sel1(X1, X2)) → a__sel1(mark(X1), mark(X2))
mark(quote(X)) → a__quote(mark(X))
mark(0) → 0
mark(s(X)) → s(X)
mark(nil) → nil
mark(cons(X1, X2)) → cons(X1, X2)
mark(01) → 01
mark(s1(X)) → s1(mark(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__from(X) → from(X)
a__dbl1(X) → dbl1(X)
a__sel1(X1, X2) → sel1(X1, X2)
a__quote(X) → quote(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, sel1(x0, x1))) → A__SEL(s(y0), a__sel1(mark(x0), mark(x1)))
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, s(x0))) → A__SEL(s(y0), s(x0))
A__SEL(s(s(y0)), cons(y1, s1(x0))) → A__SEL(s(y0), s1(mark(x0)))
A__SEL(s(s(y0)), cons(y1, 01)) → A__SEL(s(y0), 01)
A__SEL(s(s(y0)), cons(y1, 0)) → A__SEL(s(y0), 0)
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, quote(x0))) → A__SEL(s(y0), a__quote(mark(x0)))
A__SEL(s(s(y0)), cons(y1, nil)) → A__SEL(s(y0), nil)
A__SEL(s(s(y0)), cons(y1, dbl1(x0))) → A__SEL(s(y0), a__dbl1(mark(x0)))
A__SEL(s(s(y0)), cons(y1, dbl(x0))) → A__SEL(s(y0), a__dbl(mark(x0)))
A__SEL(s(s(y0)), cons(y1, cons(x0, x1))) → A__SEL(s(y0), cons(x0, x1))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ QDPOrderProof

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ AND

                      ↳ QDP

                      ↳ QDP

                        ↳ QDPOrderProof

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ AND

                                ↳ QDP

                                  ↳ QDPOrderProof

                                    ↳ QDP

                                      ↳ DependencyGraphProof

                                        ↳ AND

                                          ↳ QDP

                                            ↳ QDPOrderProof

                                              ↳ QDP

                                                ↳ DependencyGraphProof

                                                  ↳ AND

                                                    ↳ QDP

                                                    ↳ QDP

                                                      ↳ Narrowing

                                                        ↳ QDP

                                                          ↳ DependencyGraphProof

                                                            ↳ QDP

                                                              ↳ Narrowing

                                                                ↳ QDP

                                                                  ↳ DependencyGraphProof

                                                                    ↳ QDP

                                                                      ↳ Narrowing

                                                                        ↳ QDP

                                                                          ↳ DependencyGraphProof

                                                                            ↳ QDP

                                                                              ↳ Narrowing

                                                                                ↳ QDP

                                                                                  ↳ DependencyGraphProof

                                                                                    ↳ QDP

                                                                                      ↳ Narrowing

                                                                                        ↳ QDP

                                                                                          ↳ DependencyGraphProof

                                                                                            ↳ QDP

                                                                                              ↳ Narrowing

                                                                                                ↳ QDP

                                                                                                  ↳ DependencyGraphProof

                                                                                                    ↳ QDP

                                                                                                      ↳ Narrowing

                                                                                                        ↳ QDP

                                                                                                          ↳ DependencyGraphProof

                                                                                                            ↳ QDP

                                                                                                              ↳ Narrowing

                                                                                                                ↳ QDP

                                                                                                                  ↳ DependencyGraphProof

                                                                                                                    ↳ QDP

                                                                                                                      ↳ Narrowing

                                                                                                                        ↳ QDP

                                                                                                                          ↳ DependencyGraphProof

                                          ↳ QDP

                                ↳ QDP

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

A__SEL(s(sel1(y0, y1)), cons(y2, indx(x0, x1))) → A__SEL(a__sel1(mark(y0), mark(y1)), a__indx(mark(x0), x1))
A__SEL(s(s(y0)), cons(y1, sel1(x0, x1))) → A__SEL(s(y0), a__sel1(mark(x0), mark(x1)))
A__SEL(s(dbl(y0)), cons(y1, dbl1(x0))) → A__SEL(a__dbl(mark(y0)), a__dbl1(mark(x0)))
A__SEL(s(dbl1(y0)), cons(y1, dbl(x0))) → A__SEL(a__dbl1(mark(y0)), a__dbl(mark(x0)))
A__SEL(s(s(y0)), cons(y1, s(x0))) → A__SEL(s(y0), s(x0))
A__SEL(s(quote(y0)), cons(y1, indx(x0, x1))) → A__SEL(a__quote(mark(y0)), a__indx(mark(x0), x1))
A__SEL(s(indx(y0, y1)), cons(y2, dbl(x0))) → A__SEL(a__indx(mark(y0), y1), a__dbl(mark(x0)))
A__SEL(s(quote(y0)), cons(y1, from(x0))) → A__SEL(a__quote(mark(y0)), a__from(x0))
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, sel1(x0, x1))) → A__SEL(a__dbl(mark(y0)), a__sel1(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(quote(y0)), cons(y1, quote(x0))) → A__SEL(a__quote(mark(y0)), a__quote(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(sel1(y0, y1)), cons(y2, dbl(x0))) → A__SEL(a__sel1(mark(y0), mark(y1)), a__dbl(mark(x0)))
A__SEL(s(s(y0)), cons(y1, 01)) → A__SEL(s(y0), 01)
A__SEL(s(s(y0)), cons(y1, 0)) → A__SEL(s(y0), 0)
A__SEL(s(s(y0)), cons(y1, quote(x0))) → A__SEL(s(y0), a__quote(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, sel1(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), a__sel1(mark(x0), mark(x1)))
A__SEL(s(sel1(y0, y1)), cons(y2, sel(x0, x1))) → A__SEL(a__sel1(mark(y0), mark(y1)), a__sel(mark(x0), mark(x1)))
A__SEL(s(s(y0)), cons(y1, dbl1(x0))) → A__SEL(s(y0), a__dbl1(mark(x0)))
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(sel1(y0, y1)), cons(y2, dbls(x0))) → A__SEL(a__sel1(mark(y0), mark(y1)), a__dbls(mark(x0)))
A__SEL(s(dbl1(y0)), cons(y1, sel(x0, x1))) → A__SEL(a__dbl1(mark(y0)), 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(quote(y0)), cons(y1, sel1(x0, x1))) → A__SEL(a__quote(mark(y0)), a__sel1(mark(x0), mark(x1)))
A__SEL(s(sel(y0, y1)), cons(y2, quote(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__quote(mark(x0)))
A__SEL(s(sel1(y0, y1)), cons(y2, cons(x0, x1))) → A__SEL(a__sel1(mark(y0), mark(y1)), cons(x0, x1))
A__SEL(s(quote(y0)), cons(y1, dbl(x0))) → A__SEL(a__quote(mark(y0)), a__dbl(mark(x0)))
A__SEL(s(dbl(y0)), cons(y1, quote(x0))) → A__SEL(a__dbl(mark(y0)), a__quote(mark(x0)))
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(indx(y0, y1)), cons(y2, from(x0))) → A__SEL(a__indx(mark(y0), y1), a__from(x0))
A__SEL(s(quote(y0)), cons(y1, dbls(x0))) → A__SEL(a__quote(mark(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(dbl(y0)), cons(y1, dbls(x0))) → A__SEL(a__dbl(mark(y0)), a__dbls(mark(x0)))
A__SEL(s(s(y0)), cons(y1, s1(x0))) → A__SEL(s(y0), s1(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(dbl1(y0)), cons(y1, cons(x0, x1))) → A__SEL(a__dbl1(mark(y0)), cons(x0, x1))
A__SEL(s(sel1(y0, y1)), cons(y2, dbl1(x0))) → A__SEL(a__sel1(mark(y0), mark(y1)), a__dbl1(mark(x0)))
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, dbls(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__dbls(mark(x0)))
A__SEL(s(sel1(y0, y1)), cons(y2, quote(x0))) → A__SEL(a__sel1(mark(y0), mark(y1)), a__quote(mark(x0)))
A__SEL(s(sel1(y0, y1)), cons(y2, sel1(x0, x1))) → A__SEL(a__sel1(mark(y0), mark(y1)), a__sel1(mark(x0), mark(x1)))
A__SEL(s(s(y0)), cons(y1, from(x0))) → A__SEL(s(y0), a__from(x0))
A__SEL(s(indx(y0, y1)), cons(y2, quote(x0))) → A__SEL(a__indx(mark(y0), y1), a__quote(mark(x0)))
A__SEL(s(s(y0)), cons(y1, dbls(x0))) → A__SEL(s(y0), a__dbls(mark(x0)))
A__SEL(s(quote(y0)), cons(y1, cons(x0, x1))) → A__SEL(a__quote(mark(y0)), cons(x0, x1))
A__SEL(s(sel(y0, y1)), cons(y2, dbl1(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__dbl1(mark(x0)))
A__SEL(s(indx(y0, y1)), cons(y2, sel1(x0, x1))) → A__SEL(a__indx(mark(y0), y1), a__sel1(mark(x0), mark(x1)))
A__SEL(s(quote(y0)), cons(y1, sel(x0, x1))) → A__SEL(a__quote(mark(y0)), a__sel(mark(x0), mark(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(dbl1(y0)), cons(y1, sel1(x0, x1))) → A__SEL(a__dbl1(mark(y0)), a__sel1(mark(x0), mark(x1)))
A__SEL(s(dbl1(y0)), cons(y1, dbls(x0))) → A__SEL(a__dbl1(mark(y0)), a__dbls(mark(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(quote(y0)), cons(y1, dbl1(x0))) → A__SEL(a__quote(mark(y0)), a__dbl1(mark(x0)))
A__SEL(s(dbl1(y0)), cons(y1, quote(x0))) → A__SEL(a__dbl1(mark(y0)), a__quote(mark(x0)))
A__SEL(s(indx(y0, y1)), cons(y2, dbl1(x0))) → A__SEL(a__indx(mark(y0), y1), a__dbl1(mark(x0)))
A__SEL(s(sel1(y0, y1)), cons(y2, from(x0))) → A__SEL(a__sel1(mark(y0), mark(y1)), a__from(x0))
A__SEL(s(s(y0)), cons(y1, sel(x0, x1))) → A__SEL(s(y0), a__sel(mark(x0), mark(x1)))
A__SEL(s(dbl1(y0)), cons(y1, from(x0))) → A__SEL(a__dbl1(mark(y0)), a__from(x0))
A__SEL(s(s(y0)), cons(y1, nil)) → A__SEL(s(y0), nil)
A__SEL(s(dbl1(y0)), cons(y1, indx(x0, x1))) → A__SEL(a__dbl1(mark(y0)), a__indx(mark(x0), x1))
A__SEL(s(dbl1(y0)), cons(y1, dbl1(x0))) → A__SEL(a__dbl1(mark(y0)), a__dbl1(mark(x0)))
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)))
a__dbl1(0) → 01
a__dbl1(s(X)) → s1(s1(a__dbl1(mark(X))))
a__sel1(0, cons(X, Y)) → mark(X)
a__sel1(s(X), cons(Y, Z)) → a__sel1(mark(X), mark(Z))
a__quote(0) → 01
a__quote(s(X)) → s1(a__quote(mark(X)))
a__quote(dbl(X)) → a__dbl1(mark(X))
a__quote(sel(X, Y)) → a__sel1(mark(X), mark(Y))
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(dbl1(X)) → a__dbl1(mark(X))
mark(sel1(X1, X2)) → a__sel1(mark(X1), mark(X2))
mark(quote(X)) → a__quote(mark(X))
mark(0) → 0
mark(s(X)) → s(X)
mark(nil) → nil
mark(cons(X1, X2)) → cons(X1, X2)
mark(01) → 01
mark(s1(X)) → s1(mark(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__from(X) → from(X)
a__dbl1(X) → dbl1(X)
a__sel1(X1, X2) → sel1(X1, X2)
a__quote(X) → quote(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 5 less nodes.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ QDPOrderProof

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ AND

                      ↳ QDP

                      ↳ QDP

                        ↳ QDPOrderProof

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ AND

                                ↳ QDP

                                  ↳ QDPOrderProof

                                    ↳ QDP

                                      ↳ DependencyGraphProof

                                        ↳ AND

                                          ↳ QDP

                                            ↳ QDPOrderProof

                                              ↳ QDP

                                                ↳ DependencyGraphProof

                                                  ↳ AND

                                                    ↳ QDP

                                                    ↳ QDP

                                                      ↳ Narrowing

                                                        ↳ QDP

                                                          ↳ DependencyGraphProof

                                                            ↳ QDP

                                                              ↳ Narrowing

                                                                ↳ QDP

                                                                  ↳ DependencyGraphProof

                                                                    ↳ QDP

                                                                      ↳ Narrowing

                                                                        ↳ QDP

                                                                          ↳ DependencyGraphProof

                                                                            ↳ QDP

                                                                              ↳ Narrowing

                                                                                ↳ QDP

                                                                                  ↳ DependencyGraphProof

                                                                                    ↳ QDP

                                                                                      ↳ Narrowing

                                                                                        ↳ QDP

                                                                                          ↳ DependencyGraphProof

                                                                                            ↳ QDP

                                                                                              ↳ Narrowing

                                                                                                ↳ QDP

                                                                                                  ↳ DependencyGraphProof

                                                                                                    ↳ QDP

                                                                                                      ↳ Narrowing

                                                                                                        ↳ QDP

                                                                                                          ↳ DependencyGraphProof

                                                                                                            ↳ QDP

                                                                                                              ↳ Narrowing

                                                                                                                ↳ QDP

                                                                                                                  ↳ DependencyGraphProof

                                                                                                                    ↳ QDP

                                                                                                                      ↳ Narrowing

                                                                                                                        ↳ QDP

                                                                                                                          ↳ DependencyGraphProof

                                                                                                                            ↳ QDP

                                                                                                                              ↳ Narrowing

                                          ↳ QDP

                                ↳ QDP

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

A__SEL(s(sel1(y0, y1)), cons(y2, indx(x0, x1))) → A__SEL(a__sel1(mark(y0), mark(y1)), a__indx(mark(x0), x1))
A__SEL(s(s(y0)), cons(y1, sel1(x0, x1))) → A__SEL(s(y0), a__sel1(mark(x0), mark(x1)))
A__SEL(s(dbl1(y0)), cons(y1, dbl(x0))) → A__SEL(a__dbl1(mark(y0)), a__dbl(mark(x0)))
A__SEL(s(dbl(y0)), cons(y1, dbl1(x0))) → A__SEL(a__dbl(mark(y0)), a__dbl1(mark(x0)))
A__SEL(s(quote(y0)), cons(y1, indx(x0, x1))) → A__SEL(a__quote(mark(y0)), a__indx(mark(x0), x1))
A__SEL(s(indx(y0, y1)), cons(y2, dbl(x0))) → A__SEL(a__indx(mark(y0), y1), a__dbl(mark(x0)))
A__SEL(s(dbl(y0)), cons(y1, from(x0))) → A__SEL(a__dbl(mark(y0)), a__from(x0))
A__SEL(s(quote(y0)), cons(y1, from(x0))) → A__SEL(a__quote(mark(y0)), a__from(x0))
A__SEL(s(dbl(y0)), cons(y1, sel1(x0, x1))) → A__SEL(a__dbl(mark(y0)), a__sel1(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(quote(y0)), cons(y1, quote(x0))) → A__SEL(a__quote(mark(y0)), a__quote(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(sel1(y0, y1)), cons(y2, dbl(x0))) → A__SEL(a__sel1(mark(y0), mark(y1)), a__dbl(mark(x0)))
A__SEL(s(s(y0)), cons(y1, quote(x0))) → A__SEL(s(y0), a__quote(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(sel1(y0, y1)), cons(y2, sel(x0, x1))) → A__SEL(a__sel1(mark(y0), mark(y1)), a__sel(mark(x0), mark(x1)))
A__SEL(s(sel(y0, y1)), cons(y2, sel1(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), a__sel1(mark(x0), mark(x1)))
A__SEL(s(s(y0)), cons(y1, dbl1(x0))) → A__SEL(s(y0), a__dbl1(mark(x0)))
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(sel1(y0, y1)), cons(y2, dbls(x0))) → A__SEL(a__sel1(mark(y0), mark(y1)), a__dbls(mark(x0)))
A__SEL(s(dbl1(y0)), cons(y1, sel(x0, x1))) → A__SEL(a__dbl1(mark(y0)), 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(quote(y0)), cons(y1, sel1(x0, x1))) → A__SEL(a__quote(mark(y0)), a__sel1(mark(x0), mark(x1)))
A__SEL(s(sel(y0, y1)), cons(y2, quote(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__quote(mark(x0)))
A__SEL(s(sel1(y0, y1)), cons(y2, cons(x0, x1))) → A__SEL(a__sel1(mark(y0), mark(y1)), cons(x0, x1))
A__SEL(s(dbl(y0)), cons(y1, quote(x0))) → A__SEL(a__dbl(mark(y0)), a__quote(mark(x0)))
A__SEL(s(quote(y0)), cons(y1, dbl(x0))) → A__SEL(a__quote(mark(y0)), a__dbl(mark(x0)))
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(quote(y0)), cons(y1, dbls(x0))) → A__SEL(a__quote(mark(y0)), a__dbls(mark(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(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(dbl1(y0)), cons(y1, cons(x0, x1))) → A__SEL(a__dbl1(mark(y0)), cons(x0, x1))
A__SEL(s(sel1(y0, y1)), cons(y2, dbl1(x0))) → A__SEL(a__sel1(mark(y0), mark(y1)), a__dbl1(mark(x0)))
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, dbls(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__dbls(mark(x0)))
A__SEL(s(sel1(y0, y1)), cons(y2, quote(x0))) → A__SEL(a__sel1(mark(y0), mark(y1)), a__quote(mark(x0)))
A__SEL(s(sel1(y0, y1)), cons(y2, sel1(x0, x1))) → A__SEL(a__sel1(mark(y0), mark(y1)), a__sel1(mark(x0), mark(x1)))
A__SEL(s(s(y0)), cons(y1, from(x0))) → A__SEL(s(y0), a__from(x0))
A__SEL(s(indx(y0, y1)), cons(y2, quote(x0))) → A__SEL(a__indx(mark(y0), y1), a__quote(mark(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, dbl1(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__dbl1(mark(x0)))
A__SEL(s(quote(y0)), cons(y1, cons(x0, x1))) → A__SEL(a__quote(mark(y0)), cons(x0, x1))
A__SEL(s(indx(y0, y1)), cons(y2, sel1(x0, x1))) → A__SEL(a__indx(mark(y0), y1), a__sel1(mark(x0), mark(x1)))
A__SEL(s(quote(y0)), cons(y1, sel(x0, x1))) → A__SEL(a__quote(mark(y0)), a__sel(mark(x0), mark(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(dbl1(y0)), cons(y1, sel1(x0, x1))) → A__SEL(a__dbl1(mark(y0)), a__sel1(mark(x0), mark(x1)))
A__SEL(s(dbl1(y0)), cons(y1, dbls(x0))) → A__SEL(a__dbl1(mark(y0)), a__dbls(mark(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(dbl1(y0)), cons(y1, quote(x0))) → A__SEL(a__dbl1(mark(y0)), a__quote(mark(x0)))
A__SEL(s(quote(y0)), cons(y1, dbl1(x0))) → A__SEL(a__quote(mark(y0)), a__dbl1(mark(x0)))
A__SEL(s(indx(y0, y1)), cons(y2, dbl1(x0))) → A__SEL(a__indx(mark(y0), y1), a__dbl1(mark(x0)))
A__SEL(s(sel1(y0, y1)), cons(y2, from(x0))) → A__SEL(a__sel1(mark(y0), mark(y1)), a__from(x0))
A__SEL(s(s(y0)), cons(y1, sel(x0, x1))) → A__SEL(s(y0), a__sel(mark(x0), mark(x1)))
A__SEL(s(dbl1(y0)), cons(y1, from(x0))) → A__SEL(a__dbl1(mark(y0)), a__from(x0))
A__SEL(s(dbl1(y0)), cons(y1, dbl1(x0))) → A__SEL(a__dbl1(mark(y0)), a__dbl1(mark(x0)))
A__SEL(s(dbl1(y0)), cons(y1, indx(x0, x1))) → A__SEL(a__dbl1(mark(y0)), a__indx(mark(x0), x1))
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)))
a__dbl1(0) → 01
a__dbl1(s(X)) → s1(s1(a__dbl1(mark(X))))
a__sel1(0, cons(X, Y)) → mark(X)
a__sel1(s(X), cons(Y, Z)) → a__sel1(mark(X), mark(Z))
a__quote(0) → 01
a__quote(s(X)) → s1(a__quote(mark(X)))
a__quote(dbl(X)) → a__dbl1(mark(X))
a__quote(sel(X, Y)) → a__sel1(mark(X), mark(Y))
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(dbl1(X)) → a__dbl1(mark(X))
mark(sel1(X1, X2)) → a__sel1(mark(X1), mark(X2))
mark(quote(X)) → a__quote(mark(X))
mark(0) → 0
mark(s(X)) → s(X)
mark(nil) → nil
mark(cons(X1, X2)) → cons(X1, X2)
mark(01) → 01
mark(s1(X)) → s1(mark(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__from(X) → from(X)
a__dbl1(X) → dbl1(X)
a__sel1(X1, X2) → sel1(X1, X2)
a__quote(X) → quote(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, quote(x0))) → A__SEL(a__dbls(mark(y0)), a__quote(mark(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, sel1(x0, x1))) → A__SEL(a__dbls(mark(y0)), a__sel1(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(dbls(y0)), cons(y1, dbl1(x0))) → A__SEL(a__dbls(mark(y0)), a__dbl1(mark(x0)))
A__SEL(s(dbls(y0)), cons(y1, 01)) → A__SEL(a__dbls(mark(y0)), 01)
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, s1(x0))) → A__SEL(a__dbls(mark(y0)), s1(mark(x0)))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ QDPOrderProof

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ AND

                      ↳ QDP

                      ↳ QDP

                        ↳ QDPOrderProof

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ AND

                                ↳ QDP

                                  ↳ QDPOrderProof

                                    ↳ QDP

                                      ↳ DependencyGraphProof

                                        ↳ AND

                                          ↳ QDP

                                            ↳ QDPOrderProof

                                              ↳ QDP

                                                ↳ DependencyGraphProof

                                                  ↳ AND

                                                    ↳ QDP

                                                    ↳ QDP

                                                      ↳ Narrowing

                                                        ↳ QDP

                                                          ↳ DependencyGraphProof

                                                            ↳ QDP

                                                              ↳ Narrowing

                                                                ↳ QDP

                                                                  ↳ DependencyGraphProof

                                                                    ↳ QDP

                                                                      ↳ Narrowing

                                                                        ↳ QDP

                                                                          ↳ DependencyGraphProof

                                                                            ↳ QDP

                                                                              ↳ Narrowing

                                                                                ↳ QDP

                                                                                  ↳ DependencyGraphProof

                                                                                    ↳ QDP

                                                                                      ↳ Narrowing

                                                                                        ↳ QDP

                                                                                          ↳ DependencyGraphProof

                                                                                            ↳ QDP

                                                                                              ↳ Narrowing

                                                                                                ↳ QDP

                                                                                                  ↳ DependencyGraphProof

                                                                                                    ↳ QDP

                                                                                                      ↳ Narrowing

                                                                                                        ↳ QDP

                                                                                                          ↳ DependencyGraphProof

                                                                                                            ↳ QDP

                                                                                                              ↳ Narrowing

                                                                                                                ↳ QDP

                                                                                                                  ↳ DependencyGraphProof

                                                                                                                    ↳ QDP

                                                                                                                      ↳ Narrowing

                                                                                                                        ↳ QDP

                                                                                                                          ↳ DependencyGraphProof

                                                                                                                            ↳ QDP

                                                                                                                              ↳ Narrowing

                                                                                                                                ↳ QDP

                                                                                                                                  ↳ DependencyGraphProof

                                          ↳ QDP

                                ↳ QDP

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

A__SEL(s(sel1(y0, y1)), cons(y2, indx(x0, x1))) → A__SEL(a__sel1(mark(y0), mark(y1)), a__indx(mark(x0), x1))
A__SEL(s(s(y0)), cons(y1, sel1(x0, x1))) → A__SEL(s(y0), a__sel1(mark(x0), mark(x1)))
A__SEL(s(dbl(y0)), cons(y1, dbl1(x0))) → A__SEL(a__dbl(mark(y0)), a__dbl1(mark(x0)))
A__SEL(s(dbl1(y0)), cons(y1, dbl(x0))) → A__SEL(a__dbl1(mark(y0)), a__dbl(mark(x0)))
A__SEL(s(quote(y0)), cons(y1, indx(x0, x1))) → A__SEL(a__quote(mark(y0)), a__indx(mark(x0), x1))
A__SEL(s(indx(y0, y1)), cons(y2, dbl(x0))) → A__SEL(a__indx(mark(y0), y1), a__dbl(mark(x0)))
A__SEL(s(quote(y0)), cons(y1, from(x0))) → A__SEL(a__quote(mark(y0)), a__from(x0))
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, sel1(x0, x1))) → A__SEL(a__dbl(mark(y0)), a__sel1(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(quote(y0)), cons(y1, quote(x0))) → A__SEL(a__quote(mark(y0)), a__quote(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, 01)) → A__SEL(a__dbls(mark(y0)), 01)
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(sel1(y0, y1)), cons(y2, dbl(x0))) → A__SEL(a__sel1(mark(y0), mark(y1)), a__dbl(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(s(y0)), cons(y1, quote(x0))) → A__SEL(s(y0), a__quote(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, sel1(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), a__sel1(mark(x0), mark(x1)))
A__SEL(s(sel1(y0, y1)), cons(y2, sel(x0, x1))) → A__SEL(a__sel1(mark(y0), mark(y1)), a__sel(mark(x0), mark(x1)))
A__SEL(s(s(y0)), cons(y1, dbl1(x0))) → A__SEL(s(y0), a__dbl1(mark(x0)))
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(sel1(y0, y1)), cons(y2, dbls(x0))) → A__SEL(a__sel1(mark(y0), mark(y1)), a__dbls(mark(x0)))
A__SEL(s(dbl1(y0)), cons(y1, sel(x0, x1))) → A__SEL(a__dbl1(mark(y0)), a__sel(mark(x0), mark(x1)))
A__SEL(s(dbls(y0)), cons(y1, sel1(x0, x1))) → A__SEL(a__dbls(mark(y0)), a__sel1(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(dbls(y0)), cons(y1, nil)) → A__SEL(a__dbls(mark(y0)), nil)
A__SEL(s(quote(y0)), cons(y1, sel1(x0, x1))) → A__SEL(a__quote(mark(y0)), a__sel1(mark(x0), mark(x1)))
A__SEL(s(sel(y0, y1)), cons(y2, quote(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__quote(mark(x0)))
A__SEL(s(sel1(y0, y1)), cons(y2, cons(x0, x1))) → A__SEL(a__sel1(mark(y0), mark(y1)), cons(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(quote(y0)), cons(y1, dbl(x0))) → A__SEL(a__quote(mark(y0)), a__dbl(mark(x0)))
A__SEL(s(dbl(y0)), cons(y1, quote(x0))) → A__SEL(a__dbl(mark(y0)), a__quote(mark(x0)))
A__SEL(s(dbls(y0)), cons(y1, s1(x0))) → A__SEL(a__dbls(mark(y0)), s1(mark(x0)))
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(y0)), cons(y1, indx(x0, x1))) → A__SEL(a__dbls(mark(y0)), a__indx(mark(x0), x1))
A__SEL(s(dbls(y0)), cons(y1, quote(x0))) → A__SEL(a__dbls(mark(y0)), a__quote(mark(x0)))
A__SEL(s(indx(y0, y1)), cons(y2, from(x0))) → A__SEL(a__indx(mark(y0), y1), a__from(x0))
A__SEL(s(quote(y0)), cons(y1, dbls(x0))) → A__SEL(a__quote(mark(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(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(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(dbl1(y0)), cons(y1, cons(x0, x1))) → A__SEL(a__dbl1(mark(y0)), cons(x0, x1))
A__SEL(s(sel1(y0, y1)), cons(y2, dbl1(x0))) → A__SEL(a__sel1(mark(y0), mark(y1)), a__dbl1(mark(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, from(x0))) → A__SEL(a__dbls(mark(y0)), a__from(x0))
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(sel1(y0, y1)), cons(y2, quote(x0))) → A__SEL(a__sel1(mark(y0), mark(y1)), a__quote(mark(x0)))
A__SEL(s(sel1(y0, y1)), cons(y2, sel1(x0, x1))) → A__SEL(a__sel1(mark(y0), mark(y1)), a__sel1(mark(x0), mark(x1)))
A__SEL(s(s(y0)), cons(y1, from(x0))) → A__SEL(s(y0), a__from(x0))
A__SEL(s(indx(y0, y1)), cons(y2, quote(x0))) → A__SEL(a__indx(mark(y0), y1), a__quote(mark(x0)))
A__SEL(s(s(y0)), cons(y1, dbls(x0))) → A__SEL(s(y0), a__dbls(mark(x0)))
A__SEL(s(quote(y0)), cons(y1, cons(x0, x1))) → A__SEL(a__quote(mark(y0)), cons(x0, x1))
A__SEL(s(sel(y0, y1)), cons(y2, dbl1(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__dbl1(mark(x0)))
A__SEL(s(indx(y0, y1)), cons(y2, sel1(x0, x1))) → A__SEL(a__indx(mark(y0), y1), a__sel1(mark(x0), mark(x1)))
A__SEL(s(quote(y0)), cons(y1, sel(x0, x1))) → A__SEL(a__quote(mark(y0)), a__sel(mark(x0), mark(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(dbl1(y0)), cons(y1, sel1(x0, x1))) → A__SEL(a__dbl1(mark(y0)), a__sel1(mark(x0), mark(x1)))
A__SEL(s(dbls(y0)), cons(y1, dbl1(x0))) → A__SEL(a__dbls(mark(y0)), a__dbl1(mark(x0)))
A__SEL(s(dbls(y0)), cons(y1, cons(x0, x1))) → A__SEL(a__dbls(mark(y0)), cons(x0, x1))
A__SEL(s(dbl1(y0)), cons(y1, dbls(x0))) → A__SEL(a__dbl1(mark(y0)), a__dbls(mark(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(quote(y0)), cons(y1, dbl1(x0))) → A__SEL(a__quote(mark(y0)), a__dbl1(mark(x0)))
A__SEL(s(dbl1(y0)), cons(y1, quote(x0))) → A__SEL(a__dbl1(mark(y0)), a__quote(mark(x0)))
A__SEL(s(indx(y0, y1)), cons(y2, dbl1(x0))) → A__SEL(a__indx(mark(y0), y1), a__dbl1(mark(x0)))
A__SEL(s(sel1(y0, y1)), cons(y2, from(x0))) → A__SEL(a__sel1(mark(y0), mark(y1)), a__from(x0))
A__SEL(s(s(y0)), cons(y1, sel(x0, x1))) → A__SEL(s(y0), a__sel(mark(x0), mark(x1)))
A__SEL(s(dbl1(y0)), cons(y1, from(x0))) → A__SEL(a__dbl1(mark(y0)), a__from(x0))
A__SEL(s(dbl1(y0)), cons(y1, indx(x0, x1))) → A__SEL(a__dbl1(mark(y0)), a__indx(mark(x0), x1))
A__SEL(s(dbl1(y0)), cons(y1, dbl1(x0))) → A__SEL(a__dbl1(mark(y0)), a__dbl1(mark(x0)))
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)))
a__dbl1(0) → 01
a__dbl1(s(X)) → s1(s1(a__dbl1(mark(X))))
a__sel1(0, cons(X, Y)) → mark(X)
a__sel1(s(X), cons(Y, Z)) → a__sel1(mark(X), mark(Z))
a__quote(0) → 01
a__quote(s(X)) → s1(a__quote(mark(X)))
a__quote(dbl(X)) → a__dbl1(mark(X))
a__quote(sel(X, Y)) → a__sel1(mark(X), mark(Y))
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(dbl1(X)) → a__dbl1(mark(X))
mark(sel1(X1, X2)) → a__sel1(mark(X1), mark(X2))
mark(quote(X)) → a__quote(mark(X))
mark(0) → 0
mark(s(X)) → s(X)
mark(nil) → nil
mark(cons(X1, X2)) → cons(X1, X2)
mark(01) → 01
mark(s1(X)) → s1(mark(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__from(X) → from(X)
a__dbl1(X) → dbl1(X)
a__sel1(X1, X2) → sel1(X1, X2)
a__quote(X) → quote(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 5 less nodes.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ QDPOrderProof

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ AND

                      ↳ QDP

                      ↳ QDP

                        ↳ QDPOrderProof

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ AND

                                ↳ QDP

                                  ↳ QDPOrderProof

                                    ↳ QDP

                                      ↳ DependencyGraphProof

                                        ↳ AND

                                          ↳ QDP

                                            ↳ QDPOrderProof

                                              ↳ QDP

                                                ↳ DependencyGraphProof

                                                  ↳ AND

                                                    ↳ QDP

                                                    ↳ QDP

                                                      ↳ Narrowing

                                                        ↳ QDP

                                                          ↳ DependencyGraphProof

                                                            ↳ QDP

                                                              ↳ Narrowing

                                                                ↳ QDP

                                                                  ↳ DependencyGraphProof

                                                                    ↳ QDP

                                                                      ↳ Narrowing

                                                                        ↳ QDP

                                                                          ↳ DependencyGraphProof

                                                                            ↳ QDP

                                                                              ↳ Narrowing

                                                                                ↳ QDP

                                                                                  ↳ DependencyGraphProof

                                                                                    ↳ QDP

                                                                                      ↳ Narrowing

                                                                                        ↳ QDP

                                                                                          ↳ DependencyGraphProof

                                                                                            ↳ QDP

                                                                                              ↳ Narrowing

                                                                                                ↳ QDP

                                                                                                  ↳ DependencyGraphProof

                                                                                                    ↳ QDP

                                                                                                      ↳ Narrowing

                                                                                                        ↳ QDP

                                                                                                          ↳ DependencyGraphProof

                                                                                                            ↳ QDP

                                                                                                              ↳ Narrowing

                                                                                                                ↳ QDP

                                                                                                                  ↳ DependencyGraphProof

                                                                                                                    ↳ QDP

                                                                                                                      ↳ Narrowing

                                                                                                                        ↳ QDP

                                                                                                                          ↳ DependencyGraphProof

                                                                                                                            ↳ QDP

                                                                                                                              ↳ Narrowing

                                                                                                                                ↳ QDP

                                                                                                                                  ↳ DependencyGraphProof

                                                                                                                                    ↳ QDP

                                                                                                                                      ↳ Narrowing

                                          ↳ QDP

                                ↳ QDP

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

A__SEL(s(sel1(y0, y1)), cons(y2, indx(x0, x1))) → A__SEL(a__sel1(mark(y0), mark(y1)), a__indx(mark(x0), x1))
A__SEL(s(s(y0)), cons(y1, sel1(x0, x1))) → A__SEL(s(y0), a__sel1(mark(x0), mark(x1)))
A__SEL(s(dbl1(y0)), cons(y1, dbl(x0))) → A__SEL(a__dbl1(mark(y0)), a__dbl(mark(x0)))
A__SEL(s(dbl(y0)), cons(y1, dbl1(x0))) → A__SEL(a__dbl(mark(y0)), a__dbl1(mark(x0)))
A__SEL(s(quote(y0)), cons(y1, indx(x0, x1))) → A__SEL(a__quote(mark(y0)), a__indx(mark(x0), x1))
A__SEL(s(indx(y0, y1)), cons(y2, dbl(x0))) → A__SEL(a__indx(mark(y0), y1), a__dbl(mark(x0)))
A__SEL(s(dbl(y0)), cons(y1, from(x0))) → A__SEL(a__dbl(mark(y0)), a__from(x0))
A__SEL(s(quote(y0)), cons(y1, from(x0))) → A__SEL(a__quote(mark(y0)), a__from(x0))
A__SEL(s(dbl(y0)), cons(y1, sel1(x0, x1))) → A__SEL(a__dbl(mark(y0)), a__sel1(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(quote(y0)), cons(y1, quote(x0))) → A__SEL(a__quote(mark(y0)), a__quote(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(sel1(y0, y1)), cons(y2, dbl(x0))) → A__SEL(a__sel1(mark(y0), mark(y1)), a__dbl(mark(x0)))
A__SEL(s(s(y0)), cons(y1, quote(x0))) → A__SEL(s(y0), a__quote(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(sel1(y0, y1)), cons(y2, sel(x0, x1))) → A__SEL(a__sel1(mark(y0), mark(y1)), a__sel(mark(x0), mark(x1)))
A__SEL(s(sel(y0, y1)), cons(y2, sel1(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), a__sel1(mark(x0), mark(x1)))
A__SEL(s(s(y0)), cons(y1, dbl1(x0))) → A__SEL(s(y0), a__dbl1(mark(x0)))
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(sel1(y0, y1)), cons(y2, dbls(x0))) → A__SEL(a__sel1(mark(y0), mark(y1)), a__dbls(mark(x0)))
A__SEL(s(dbl1(y0)), cons(y1, sel(x0, x1))) → A__SEL(a__dbl1(mark(y0)), a__sel(mark(x0), mark(x1)))
A__SEL(s(dbls(y0)), cons(y1, sel1(x0, x1))) → A__SEL(a__dbls(mark(y0)), a__sel1(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(quote(y0)), cons(y1, sel1(x0, x1))) → A__SEL(a__quote(mark(y0)), a__sel1(mark(x0), mark(x1)))
A__SEL(s(sel(y0, y1)), cons(y2, quote(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__quote(mark(x0)))
A__SEL(s(sel1(y0, y1)), cons(y2, cons(x0, x1))) → A__SEL(a__sel1(mark(y0), mark(y1)), cons(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(dbl(y0)), cons(y1, quote(x0))) → A__SEL(a__dbl(mark(y0)), a__quote(mark(x0)))
A__SEL(s(quote(y0)), cons(y1, dbl(x0))) → A__SEL(a__quote(mark(y0)), a__dbl(mark(x0)))
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(y0)), cons(y1, indx(x0, x1))) → A__SEL(a__dbls(mark(y0)), a__indx(mark(x0), x1))
A__SEL(s(dbls(y0)), cons(y1, quote(x0))) → A__SEL(a__dbls(mark(y0)), a__quote(mark(x0)))
A__SEL(s(quote(y0)), cons(y1, dbls(x0))) → A__SEL(a__quote(mark(y0)), a__dbls(mark(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(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(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(dbl1(y0)), cons(y1, cons(x0, x1))) → A__SEL(a__dbl1(mark(y0)), cons(x0, x1))
A__SEL(s(sel1(y0, y1)), cons(y2, dbl1(x0))) → A__SEL(a__sel1(mark(y0), mark(y1)), a__dbl1(mark(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, 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(sel1(y0, y1)), cons(y2, quote(x0))) → A__SEL(a__sel1(mark(y0), mark(y1)), a__quote(mark(x0)))
A__SEL(s(sel1(y0, y1)), cons(y2, sel1(x0, x1))) → A__SEL(a__sel1(mark(y0), mark(y1)), a__sel1(mark(x0), mark(x1)))
A__SEL(s(s(y0)), cons(y1, from(x0))) → A__SEL(s(y0), a__from(x0))
A__SEL(s(indx(y0, y1)), cons(y2, quote(x0))) → A__SEL(a__indx(mark(y0), y1), a__quote(mark(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, dbl1(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__dbl1(mark(x0)))
A__SEL(s(quote(y0)), cons(y1, cons(x0, x1))) → A__SEL(a__quote(mark(y0)), cons(x0, x1))
A__SEL(s(indx(y0, y1)), cons(y2, sel1(x0, x1))) → A__SEL(a__indx(mark(y0), y1), a__sel1(mark(x0), mark(x1)))
A__SEL(s(quote(y0)), cons(y1, sel(x0, x1))) → A__SEL(a__quote(mark(y0)), a__sel(mark(x0), mark(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(dbl1(y0)), cons(y1, sel1(x0, x1))) → A__SEL(a__dbl1(mark(y0)), a__sel1(mark(x0), mark(x1)))
A__SEL(s(dbls(y0)), cons(y1, dbl1(x0))) → A__SEL(a__dbls(mark(y0)), a__dbl1(mark(x0)))
A__SEL(s(dbls(y0)), cons(y1, cons(x0, x1))) → A__SEL(a__dbls(mark(y0)), cons(x0, x1))
A__SEL(s(dbl1(y0)), cons(y1, dbls(x0))) → A__SEL(a__dbl1(mark(y0)), a__dbls(mark(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(dbl1(y0)), cons(y1, quote(x0))) → A__SEL(a__dbl1(mark(y0)), a__quote(mark(x0)))
A__SEL(s(quote(y0)), cons(y1, dbl1(x0))) → A__SEL(a__quote(mark(y0)), a__dbl1(mark(x0)))
A__SEL(s(indx(y0, y1)), cons(y2, dbl1(x0))) → A__SEL(a__indx(mark(y0), y1), a__dbl1(mark(x0)))
A__SEL(s(sel1(y0, y1)), cons(y2, from(x0))) → A__SEL(a__sel1(mark(y0), mark(y1)), a__from(x0))
A__SEL(s(s(y0)), cons(y1, sel(x0, x1))) → A__SEL(s(y0), a__sel(mark(x0), mark(x1)))
A__SEL(s(dbl1(y0)), cons(y1, from(x0))) → A__SEL(a__dbl1(mark(y0)), a__from(x0))
A__SEL(s(dbl1(y0)), cons(y1, dbl1(x0))) → A__SEL(a__dbl1(mark(y0)), a__dbl1(mark(x0)))
A__SEL(s(dbl1(y0)), cons(y1, indx(x0, x1))) → A__SEL(a__dbl1(mark(y0)), a__indx(mark(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)))
a__dbl1(0) → 01
a__dbl1(s(X)) → s1(s1(a__dbl1(mark(X))))
a__sel1(0, cons(X, Y)) → mark(X)
a__sel1(s(X), cons(Y, Z)) → a__sel1(mark(X), mark(Z))
a__quote(0) → 01
a__quote(s(X)) → s1(a__quote(mark(X)))
a__quote(dbl(X)) → a__dbl1(mark(X))
a__quote(sel(X, Y)) → a__sel1(mark(X), mark(Y))
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(dbl1(X)) → a__dbl1(mark(X))
mark(sel1(X1, X2)) → a__sel1(mark(X1), mark(X2))
mark(quote(X)) → a__quote(mark(X))
mark(0) → 0
mark(s(X)) → s(X)
mark(nil) → nil
mark(cons(X1, X2)) → cons(X1, X2)
mark(01) → 01
mark(s1(X)) → s1(mark(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__from(X) → from(X)
a__dbl1(X) → dbl1(X)
a__sel1(X1, X2) → sel1(X1, X2)
a__quote(X) → quote(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)), from(x0))
A__SEL(s(dbl(y0)), cons(y1, from(x0))) → A__SEL(a__dbl(mark(y0)), cons(x0, from(s(x0))))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ QDPOrderProof

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ AND

                      ↳ QDP

                      ↳ QDP

                        ↳ QDPOrderProof

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ AND

                                ↳ QDP

                                  ↳ QDPOrderProof

                                    ↳ QDP

                                      ↳ DependencyGraphProof

                                        ↳ AND

                                          ↳ QDP

                                            ↳ QDPOrderProof

                                              ↳ QDP

                                                ↳ DependencyGraphProof

                                                  ↳ AND

                                                    ↳ QDP

                                                    ↳ QDP

                                                      ↳ Narrowing

                                                        ↳ QDP

                                                          ↳ DependencyGraphProof

                                                            ↳ QDP

                                                              ↳ Narrowing

                                                                ↳ QDP

                                                                  ↳ DependencyGraphProof

                                                                    ↳ QDP

                                                                      ↳ Narrowing

                                                                        ↳ QDP

                                                                          ↳ DependencyGraphProof

                                                                            ↳ QDP

                                                                              ↳ Narrowing

                                                                                ↳ QDP

                                                                                  ↳ DependencyGraphProof

                                                                                    ↳ QDP

                                                                                      ↳ Narrowing

                                                                                        ↳ QDP

                                                                                          ↳ DependencyGraphProof

                                                                                            ↳ QDP

                                                                                              ↳ Narrowing

                                                                                                ↳ QDP

                                                                                                  ↳ DependencyGraphProof

                                                                                                    ↳ QDP

                                                                                                      ↳ Narrowing

                                                                                                        ↳ QDP

                                                                                                          ↳ DependencyGraphProof

                                                                                                            ↳ QDP

                                                                                                              ↳ Narrowing

                                                                                                                ↳ QDP

                                                                                                                  ↳ DependencyGraphProof

                                                                                                                    ↳ QDP

                                                                                                                      ↳ Narrowing

                                                                                                                        ↳ QDP

                                                                                                                          ↳ DependencyGraphProof

                                                                                                                            ↳ QDP

                                                                                                                              ↳ Narrowing

                                                                                                                                ↳ QDP

                                                                                                                                  ↳ DependencyGraphProof

                                                                                                                                    ↳ QDP

                                                                                                                                      ↳ Narrowing

                                                                                                                                        ↳ QDP

                                                                                                                                          ↳ DependencyGraphProof

                                          ↳ QDP

                                ↳ QDP

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

A__SEL(s(sel1(y0, y1)), cons(y2, indx(x0, x1))) → A__SEL(a__sel1(mark(y0), mark(y1)), a__indx(mark(x0), x1))
A__SEL(s(s(y0)), cons(y1, sel1(x0, x1))) → A__SEL(s(y0), a__sel1(mark(x0), mark(x1)))
A__SEL(s(dbl(y0)), cons(y1, dbl1(x0))) → A__SEL(a__dbl(mark(y0)), a__dbl1(mark(x0)))
A__SEL(s(dbl1(y0)), cons(y1, dbl(x0))) → A__SEL(a__dbl1(mark(y0)), a__dbl(mark(x0)))
A__SEL(s(quote(y0)), cons(y1, indx(x0, x1))) → A__SEL(a__quote(mark(y0)), a__indx(mark(x0), x1))
A__SEL(s(indx(y0, y1)), cons(y2, dbl(x0))) → A__SEL(a__indx(mark(y0), y1), a__dbl(mark(x0)))
A__SEL(s(quote(y0)), cons(y1, from(x0))) → A__SEL(a__quote(mark(y0)), a__from(x0))
A__SEL(s(dbl(y0)), cons(y1, sel1(x0, x1))) → A__SEL(a__dbl(mark(y0)), a__sel1(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(quote(y0)), cons(y1, quote(x0))) → A__SEL(a__quote(mark(y0)), a__quote(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(dbls(y0)), cons(y1, dbls(x0))) → A__SEL(a__dbls(mark(y0)), a__dbls(mark(x0)))
A__SEL(s(sel1(y0, y1)), cons(y2, dbl(x0))) → A__SEL(a__sel1(mark(y0), mark(y1)), a__dbl(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(s(y0)), cons(y1, quote(x0))) → A__SEL(s(y0), a__quote(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, sel1(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), a__sel1(mark(x0), mark(x1)))
A__SEL(s(sel1(y0, y1)), cons(y2, sel(x0, x1))) → A__SEL(a__sel1(mark(y0), mark(y1)), a__sel(mark(x0), mark(x1)))
A__SEL(s(s(y0)), cons(y1, dbl1(x0))) → A__SEL(s(y0), a__dbl1(mark(x0)))
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(sel1(y0, y1)), cons(y2, dbls(x0))) → A__SEL(a__sel1(mark(y0), mark(y1)), a__dbls(mark(x0)))
A__SEL(s(dbl1(y0)), cons(y1, sel(x0, x1))) → A__SEL(a__dbl1(mark(y0)), 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(dbls(y0)), cons(y1, sel1(x0, x1))) → A__SEL(a__dbls(mark(y0)), a__sel1(mark(x0), mark(x1)))
A__SEL(s(quote(y0)), cons(y1, sel1(x0, x1))) → A__SEL(a__quote(mark(y0)), a__sel1(mark(x0), mark(x1)))
A__SEL(s(sel(y0, y1)), cons(y2, quote(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__quote(mark(x0)))
A__SEL(s(sel1(y0, y1)), cons(y2, cons(x0, x1))) → A__SEL(a__sel1(mark(y0), mark(y1)), cons(x0, x1))
A__SEL(s(quote(y0)), cons(y1, dbl(x0))) → A__SEL(a__quote(mark(y0)), a__dbl(mark(x0)))
A__SEL(s(dbl(y0)), cons(y1, quote(x0))) → A__SEL(a__dbl(mark(y0)), a__quote(mark(x0)))
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(sel(y0, y1)), cons(y2, cons(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), cons(x0, x1))
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(quote(y0)), cons(y1, dbls(x0))) → A__SEL(a__quote(mark(y0)), a__dbls(mark(x0)))
A__SEL(s(dbls(y0)), cons(y1, quote(x0))) → A__SEL(a__dbls(mark(y0)), a__quote(mark(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(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(dbl1(y0)), cons(y1, cons(x0, x1))) → A__SEL(a__dbl1(mark(y0)), cons(x0, x1))
A__SEL(s(sel1(y0, y1)), cons(y2, dbl1(x0))) → A__SEL(a__sel1(mark(y0), mark(y1)), a__dbl1(mark(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, 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(sel1(y0, y1)), cons(y2, quote(x0))) → A__SEL(a__sel1(mark(y0), mark(y1)), a__quote(mark(x0)))
A__SEL(s(sel1(y0, y1)), cons(y2, sel1(x0, x1))) → A__SEL(a__sel1(mark(y0), mark(y1)), a__sel1(mark(x0), mark(x1)))
A__SEL(s(s(y0)), cons(y1, from(x0))) → A__SEL(s(y0), a__from(x0))
A__SEL(s(indx(y0, y1)), cons(y2, quote(x0))) → A__SEL(a__indx(mark(y0), y1), a__quote(mark(x0)))
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(quote(y0)), cons(y1, cons(x0, x1))) → A__SEL(a__quote(mark(y0)), cons(x0, x1))
A__SEL(s(sel(y0, y1)), cons(y2, dbl1(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__dbl1(mark(x0)))
A__SEL(s(indx(y0, y1)), cons(y2, sel1(x0, x1))) → A__SEL(a__indx(mark(y0), y1), a__sel1(mark(x0), mark(x1)))
A__SEL(s(quote(y0)), cons(y1, sel(x0, x1))) → A__SEL(a__quote(mark(y0)), a__sel(mark(x0), mark(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(dbl1(y0)), cons(y1, sel1(x0, x1))) → A__SEL(a__dbl1(mark(y0)), a__sel1(mark(x0), mark(x1)))
A__SEL(s(dbl1(y0)), cons(y1, dbls(x0))) → A__SEL(a__dbl1(mark(y0)), a__dbls(mark(x0)))
A__SEL(s(dbls(y0)), cons(y1, cons(x0, x1))) → A__SEL(a__dbls(mark(y0)), cons(x0, x1))
A__SEL(s(dbls(y0)), cons(y1, dbl1(x0))) → A__SEL(a__dbls(mark(y0)), a__dbl1(mark(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(quote(y0)), cons(y1, dbl1(x0))) → A__SEL(a__quote(mark(y0)), a__dbl1(mark(x0)))
A__SEL(s(dbl1(y0)), cons(y1, quote(x0))) → A__SEL(a__dbl1(mark(y0)), a__quote(mark(x0)))
A__SEL(s(indx(y0, y1)), cons(y2, dbl1(x0))) → A__SEL(a__indx(mark(y0), y1), a__dbl1(mark(x0)))
A__SEL(s(sel1(y0, y1)), cons(y2, from(x0))) → A__SEL(a__sel1(mark(y0), mark(y1)), a__from(x0))
A__SEL(s(s(y0)), cons(y1, sel(x0, x1))) → A__SEL(s(y0), a__sel(mark(x0), mark(x1)))
A__SEL(s(dbl1(y0)), cons(y1, from(x0))) → A__SEL(a__dbl1(mark(y0)), a__from(x0))
A__SEL(s(dbl1(y0)), cons(y1, indx(x0, x1))) → A__SEL(a__dbl1(mark(y0)), a__indx(mark(x0), x1))
A__SEL(s(dbl1(y0)), cons(y1, dbl1(x0))) → A__SEL(a__dbl1(mark(y0)), a__dbl1(mark(x0)))
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)))
a__dbl1(0) → 01
a__dbl1(s(X)) → s1(s1(a__dbl1(mark(X))))
a__sel1(0, cons(X, Y)) → mark(X)
a__sel1(s(X), cons(Y, Z)) → a__sel1(mark(X), mark(Z))
a__quote(0) → 01
a__quote(s(X)) → s1(a__quote(mark(X)))
a__quote(dbl(X)) → a__dbl1(mark(X))
a__quote(sel(X, Y)) → a__sel1(mark(X), mark(Y))
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(dbl1(X)) → a__dbl1(mark(X))
mark(sel1(X1, X2)) → a__sel1(mark(X1), mark(X2))
mark(quote(X)) → a__quote(mark(X))
mark(0) → 0
mark(s(X)) → s(X)
mark(nil) → nil
mark(cons(X1, X2)) → cons(X1, X2)
mark(01) → 01
mark(s1(X)) → s1(mark(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__from(X) → from(X)
a__dbl1(X) → dbl1(X)
a__sel1(X1, X2) → sel1(X1, X2)
a__quote(X) → quote(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

                  ↳ DependencyGraphProof

                    ↳ AND

                      ↳ QDP

                      ↳ QDP

                        ↳ QDPOrderProof

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ AND

                                ↳ QDP

                                  ↳ QDPOrderProof

                                    ↳ QDP

                                      ↳ DependencyGraphProof

                                        ↳ AND

                                          ↳ QDP

                                            ↳ QDPOrderProof

                                              ↳ QDP

                                                ↳ DependencyGraphProof

                                                  ↳ AND

                                                    ↳ QDP

                                                    ↳ QDP

                                                      ↳ Narrowing

                                                        ↳ QDP

                                                          ↳ DependencyGraphProof

                                                            ↳ QDP

                                                              ↳ Narrowing

                                                                ↳ QDP

                                                                  ↳ DependencyGraphProof

                                                                    ↳ QDP

                                                                      ↳ Narrowing

                                                                        ↳ QDP

                                                                          ↳ DependencyGraphProof

                                                                            ↳ QDP

                                                                              ↳ Narrowing

                                                                                ↳ QDP

                                                                                  ↳ DependencyGraphProof

                                                                                    ↳ QDP

                                                                                      ↳ Narrowing

                                                                                        ↳ QDP

                                                                                          ↳ DependencyGraphProof

                                                                                            ↳ QDP

                                                                                              ↳ Narrowing

                                                                                                ↳ QDP

                                                                                                  ↳ DependencyGraphProof

                                                                                                    ↳ QDP

                                                                                                      ↳ Narrowing

                                                                                                        ↳ QDP

                                                                                                          ↳ DependencyGraphProof

                                                                                                            ↳ QDP

                                                                                                              ↳ Narrowing

                                                                                                                ↳ QDP

                                                                                                                  ↳ DependencyGraphProof

                                                                                                                    ↳ QDP

                                                                                                                      ↳ Narrowing

                                                                                                                        ↳ QDP

                                                                                                                          ↳ DependencyGraphProof

                                                                                                                            ↳ QDP

                                                                                                                              ↳ Narrowing

                                                                                                                                ↳ QDP

                                                                                                                                  ↳ DependencyGraphProof

                                                                                                                                    ↳ QDP

                                                                                                                                      ↳ Narrowing

                                                                                                                                        ↳ QDP

                                                                                                                                          ↳ DependencyGraphProof

                                                                                                                                            ↳ QDP

                                                                                                                                              ↳ QDPOrderProof

                                          ↳ QDP

                                ↳ QDP

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

A__SEL(s(sel1(y0, y1)), cons(y2, indx(x0, x1))) → A__SEL(a__sel1(mark(y0), mark(y1)), a__indx(mark(x0), x1))
A__SEL(s(s(y0)), cons(y1, sel1(x0, x1))) → A__SEL(s(y0), a__sel1(mark(x0), mark(x1)))
A__SEL(s(dbl1(y0)), cons(y1, dbl(x0))) → A__SEL(a__dbl1(mark(y0)), a__dbl(mark(x0)))
A__SEL(s(dbl(y0)), cons(y1, dbl1(x0))) → A__SEL(a__dbl(mark(y0)), a__dbl1(mark(x0)))
A__SEL(s(quote(y0)), cons(y1, indx(x0, x1))) → A__SEL(a__quote(mark(y0)), a__indx(mark(x0), x1))
A__SEL(s(indx(y0, y1)), cons(y2, dbl(x0))) → A__SEL(a__indx(mark(y0), y1), a__dbl(mark(x0)))
A__SEL(s(quote(y0)), cons(y1, from(x0))) → A__SEL(a__quote(mark(y0)), a__from(x0))
A__SEL(s(dbl(y0)), cons(y1, sel1(x0, x1))) → A__SEL(a__dbl(mark(y0)), a__sel1(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(quote(y0)), cons(y1, quote(x0))) → A__SEL(a__quote(mark(y0)), a__quote(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(sel1(y0, y1)), cons(y2, dbl(x0))) → A__SEL(a__sel1(mark(y0), mark(y1)), a__dbl(mark(x0)))
A__SEL(s(s(y0)), cons(y1, quote(x0))) → A__SEL(s(y0), a__quote(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(sel1(y0, y1)), cons(y2, sel(x0, x1))) → A__SEL(a__sel1(mark(y0), mark(y1)), a__sel(mark(x0), mark(x1)))
A__SEL(s(sel(y0, y1)), cons(y2, sel1(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), a__sel1(mark(x0), mark(x1)))
A__SEL(s(s(y0)), cons(y1, dbl1(x0))) → A__SEL(s(y0), a__dbl1(mark(x0)))
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(sel1(y0, y1)), cons(y2, dbls(x0))) → A__SEL(a__sel1(mark(y0), mark(y1)), a__dbls(mark(x0)))
A__SEL(s(dbl1(y0)), cons(y1, sel(x0, x1))) → A__SEL(a__dbl1(mark(y0)), a__sel(mark(x0), mark(x1)))
A__SEL(s(dbls(y0)), cons(y1, sel1(x0, x1))) → A__SEL(a__dbls(mark(y0)), a__sel1(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(quote(y0)), cons(y1, sel1(x0, x1))) → A__SEL(a__quote(mark(y0)), a__sel1(mark(x0), mark(x1)))
A__SEL(s(sel(y0, y1)), cons(y2, quote(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__quote(mark(x0)))
A__SEL(s(sel1(y0, y1)), cons(y2, cons(x0, x1))) → A__SEL(a__sel1(mark(y0), mark(y1)), cons(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(dbl(y0)), cons(y1, quote(x0))) → A__SEL(a__dbl(mark(y0)), a__quote(mark(x0)))
A__SEL(s(quote(y0)), cons(y1, dbl(x0))) → A__SEL(a__quote(mark(y0)), a__dbl(mark(x0)))
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(y0)), cons(y1, indx(x0, x1))) → A__SEL(a__dbls(mark(y0)), a__indx(mark(x0), x1))
A__SEL(s(dbls(y0)), cons(y1, quote(x0))) → A__SEL(a__dbls(mark(y0)), a__quote(mark(x0)))
A__SEL(s(quote(y0)), cons(y1, dbls(x0))) → A__SEL(a__quote(mark(y0)), a__dbls(mark(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(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(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(dbl1(y0)), cons(y1, cons(x0, x1))) → A__SEL(a__dbl1(mark(y0)), cons(x0, x1))
A__SEL(s(sel1(y0, y1)), cons(y2, dbl1(x0))) → A__SEL(a__sel1(mark(y0), mark(y1)), a__dbl1(mark(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, 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(sel1(y0, y1)), cons(y2, quote(x0))) → A__SEL(a__sel1(mark(y0), mark(y1)), a__quote(mark(x0)))
A__SEL(s(sel1(y0, y1)), cons(y2, sel1(x0, x1))) → A__SEL(a__sel1(mark(y0), mark(y1)), a__sel1(mark(x0), mark(x1)))
A__SEL(s(s(y0)), cons(y1, from(x0))) → A__SEL(s(y0), a__from(x0))
A__SEL(s(indx(y0, y1)), cons(y2, quote(x0))) → A__SEL(a__indx(mark(y0), y1), a__quote(mark(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, dbl1(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__dbl1(mark(x0)))
A__SEL(s(quote(y0)), cons(y1, cons(x0, x1))) → A__SEL(a__quote(mark(y0)), cons(x0, x1))
A__SEL(s(indx(y0, y1)), cons(y2, sel1(x0, x1))) → A__SEL(a__indx(mark(y0), y1), a__sel1(mark(x0), mark(x1)))
A__SEL(s(quote(y0)), cons(y1, sel(x0, x1))) → A__SEL(a__quote(mark(y0)), a__sel(mark(x0), mark(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(dbl1(y0)), cons(y1, sel1(x0, x1))) → A__SEL(a__dbl1(mark(y0)), a__sel1(mark(x0), mark(x1)))
A__SEL(s(dbls(y0)), cons(y1, dbl1(x0))) → A__SEL(a__dbls(mark(y0)), a__dbl1(mark(x0)))
A__SEL(s(dbls(y0)), cons(y1, cons(x0, x1))) → A__SEL(a__dbls(mark(y0)), cons(x0, x1))
A__SEL(s(dbl1(y0)), cons(y1, dbls(x0))) → A__SEL(a__dbl1(mark(y0)), a__dbls(mark(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(dbl1(y0)), cons(y1, quote(x0))) → A__SEL(a__dbl1(mark(y0)), a__quote(mark(x0)))
A__SEL(s(quote(y0)), cons(y1, dbl1(x0))) → A__SEL(a__quote(mark(y0)), a__dbl1(mark(x0)))
A__SEL(s(indx(y0, y1)), cons(y2, dbl1(x0))) → A__SEL(a__indx(mark(y0), y1), a__dbl1(mark(x0)))
A__SEL(s(sel1(y0, y1)), cons(y2, from(x0))) → A__SEL(a__sel1(mark(y0), mark(y1)), a__from(x0))
A__SEL(s(s(y0)), cons(y1, sel(x0, x1))) → A__SEL(s(y0), a__sel(mark(x0), mark(x1)))
A__SEL(s(dbl1(y0)), cons(y1, from(x0))) → A__SEL(a__dbl1(mark(y0)), a__from(x0))
A__SEL(s(dbl1(y0)), cons(y1, dbl1(x0))) → A__SEL(a__dbl1(mark(y0)), a__dbl1(mark(x0)))
A__SEL(s(dbl1(y0)), cons(y1, indx(x0, x1))) → A__SEL(a__dbl1(mark(y0)), a__indx(mark(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)))
a__dbl1(0) → 01
a__dbl1(s(X)) → s1(s1(a__dbl1(mark(X))))
a__sel1(0, cons(X, Y)) → mark(X)
a__sel1(s(X), cons(Y, Z)) → a__sel1(mark(X), mark(Z))
a__quote(0) → 01
a__quote(s(X)) → s1(a__quote(mark(X)))
a__quote(dbl(X)) → a__dbl1(mark(X))
a__quote(sel(X, Y)) → a__sel1(mark(X), mark(Y))
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(dbl1(X)) → a__dbl1(mark(X))
mark(sel1(X1, X2)) → a__sel1(mark(X1), mark(X2))
mark(quote(X)) → a__quote(mark(X))
mark(0) → 0
mark(s(X)) → s(X)
mark(nil) → nil
mark(cons(X1, X2)) → cons(X1, X2)
mark(01) → 01
mark(s1(X)) → s1(mark(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__from(X) → from(X)
a__dbl1(X) → dbl1(X)
a__sel1(X1, X2) → sel1(X1, X2)
a__quote(X) → quote(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(dbl1(y0)), cons(y1, dbl(x0))) → A__SEL(a__dbl1(mark(y0)), a__dbl(mark(x0)))
A__SEL(s(dbl(y0)), cons(y1, dbl1(x0))) → A__SEL(a__dbl(mark(y0)), a__dbl1(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(sel1(y0, y1)), cons(y2, dbl(x0))) → A__SEL(a__sel1(mark(y0), mark(y1)), a__dbl(mark(x0)))
A__SEL(s(s(y0)), cons(y1, dbl1(x0))) → A__SEL(s(y0), a__dbl1(mark(x0)))
A__SEL(s(s(y0)), cons(y1, dbl(x0))) → A__SEL(s(y0), a__dbl(mark(x0)))
A__SEL(s(quote(y0)), cons(y1, dbl(x0))) → A__SEL(a__quote(mark(y0)), a__dbl(mark(x0)))
A__SEL(s(dbls(y0)), cons(y1, dbl(x0))) → A__SEL(a__dbls(mark(y0)), 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(sel1(y0, y1)), cons(y2, dbl1(x0))) → A__SEL(a__sel1(mark(y0), mark(y1)), a__dbl1(mark(x0)))
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, dbl1(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__dbl1(mark(x0)))
A__SEL(s(dbls(y0)), cons(y1, dbl1(x0))) → A__SEL(a__dbls(mark(y0)), a__dbl1(mark(x0)))
A__SEL(s(quote(y0)), cons(y1, dbl1(x0))) → A__SEL(a__quote(mark(y0)), a__dbl1(mark(x0)))
A__SEL(s(indx(y0, y1)), cons(y2, dbl1(x0))) → A__SEL(a__indx(mark(y0), y1), a__dbl1(mark(x0)))
A__SEL(s(dbl1(y0)), cons(y1, dbl1(x0))) → A__SEL(a__dbl1(mark(y0)), a__dbl1(mark(x0)))

The remaining pairs can at least be oriented weakly.

A__SEL(s(sel1(y0, y1)), cons(y2, indx(x0, x1))) → A__SEL(a__sel1(mark(y0), mark(y1)), a__indx(mark(x0), x1))
A__SEL(s(s(y0)), cons(y1, sel1(x0, x1))) → A__SEL(s(y0), a__sel1(mark(x0), mark(x1)))
A__SEL(s(quote(y0)), cons(y1, indx(x0, x1))) → A__SEL(a__quote(mark(y0)), a__indx(mark(x0), x1))
A__SEL(s(quote(y0)), cons(y1, from(x0))) → A__SEL(a__quote(mark(y0)), a__from(x0))
A__SEL(s(dbl(y0)), cons(y1, sel1(x0, x1))) → A__SEL(a__dbl(mark(y0)), a__sel1(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(quote(y0)), cons(y1, quote(x0))) → A__SEL(a__quote(mark(y0)), a__quote(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(s(y0)), cons(y1, quote(x0))) → A__SEL(s(y0), a__quote(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(sel1(y0, y1)), cons(y2, sel(x0, x1))) → A__SEL(a__sel1(mark(y0), mark(y1)), a__sel(mark(x0), mark(x1)))
A__SEL(s(sel(y0, y1)), cons(y2, sel1(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), a__sel1(mark(x0), mark(x1)))
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(sel1(y0, y1)), cons(y2, dbls(x0))) → A__SEL(a__sel1(mark(y0), mark(y1)), a__dbls(mark(x0)))
A__SEL(s(dbl1(y0)), cons(y1, sel(x0, x1))) → A__SEL(a__dbl1(mark(y0)), a__sel(mark(x0), mark(x1)))
A__SEL(s(dbls(y0)), cons(y1, sel1(x0, x1))) → A__SEL(a__dbls(mark(y0)), a__sel1(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(quote(y0)), cons(y1, sel1(x0, x1))) → A__SEL(a__quote(mark(y0)), a__sel1(mark(x0), mark(x1)))
A__SEL(s(sel(y0, y1)), cons(y2, quote(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__quote(mark(x0)))
A__SEL(s(sel1(y0, y1)), cons(y2, cons(x0, x1))) → A__SEL(a__sel1(mark(y0), mark(y1)), cons(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(dbl(y0)), cons(y1, quote(x0))) → A__SEL(a__dbl(mark(y0)), a__quote(mark(x0)))
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(y0)), cons(y1, indx(x0, x1))) → A__SEL(a__dbls(mark(y0)), a__indx(mark(x0), x1))
A__SEL(s(dbls(y0)), cons(y1, quote(x0))) → A__SEL(a__dbls(mark(y0)), a__quote(mark(x0)))
A__SEL(s(quote(y0)), cons(y1, dbls(x0))) → A__SEL(a__quote(mark(y0)), a__dbls(mark(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(dbl(y0)), cons(y1, sel(x0, x1))) → A__SEL(a__dbl(mark(y0)), a__sel(mark(x0), mark(x1)))
A__SEL(s(dbl1(y0)), cons(y1, cons(x0, x1))) → A__SEL(a__dbl1(mark(y0)), cons(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(sel1(y0, y1)), cons(y2, quote(x0))) → A__SEL(a__sel1(mark(y0), mark(y1)), a__quote(mark(x0)))
A__SEL(s(sel1(y0, y1)), cons(y2, sel1(x0, x1))) → A__SEL(a__sel1(mark(y0), mark(y1)), a__sel1(mark(x0), mark(x1)))
A__SEL(s(s(y0)), cons(y1, from(x0))) → A__SEL(s(y0), a__from(x0))
A__SEL(s(indx(y0, y1)), cons(y2, quote(x0))) → A__SEL(a__indx(mark(y0), y1), a__quote(mark(x0)))
A__SEL(s(s(y0)), cons(y1, dbls(x0))) → A__SEL(s(y0), a__dbls(mark(x0)))
A__SEL(s(quote(y0)), cons(y1, cons(x0, x1))) → A__SEL(a__quote(mark(y0)), cons(x0, x1))
A__SEL(s(indx(y0, y1)), cons(y2, sel1(x0, x1))) → A__SEL(a__indx(mark(y0), y1), a__sel1(mark(x0), mark(x1)))
A__SEL(s(quote(y0)), cons(y1, sel(x0, x1))) → A__SEL(a__quote(mark(y0)), a__sel(mark(x0), mark(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(dbl1(y0)), cons(y1, sel1(x0, x1))) → A__SEL(a__dbl1(mark(y0)), a__sel1(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(dbl1(y0)), cons(y1, dbls(x0))) → A__SEL(a__dbl1(mark(y0)), a__dbls(mark(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(dbl1(y0)), cons(y1, quote(x0))) → A__SEL(a__dbl1(mark(y0)), a__quote(mark(x0)))
A__SEL(s(sel1(y0, y1)), cons(y2, from(x0))) → A__SEL(a__sel1(mark(y0), mark(y1)), a__from(x0))
A__SEL(s(s(y0)), cons(y1, sel(x0, x1))) → A__SEL(s(y0), a__sel(mark(x0), mark(x1)))
A__SEL(s(dbl1(y0)), cons(y1, from(x0))) → A__SEL(a__dbl1(mark(y0)), a__from(x0))
A__SEL(s(dbl1(y0)), cons(y1, indx(x0, x1))) → A__SEL(a__dbl1(mark(y0)), a__indx(mark(x0), x1))
A__SEL(s(s(y0)), cons(y1, cons(x0, x1))) → A__SEL(s(y0), cons(x0, x1))

Used ordering: Polynomial interpretation [25]:

POL(0) = 0
POL(01) = 0
POL(A__SEL(x₁, x₂)) = x₂
POL(a__dbl(x₁)) = 0
POL(a__dbl1(x₁)) = 0
POL(a__dbls(x₁)) = x₁
POL(a__from(x₁)) = 1
POL(a__indx(x₁, x₂)) = 1
POL(a__quote(x₁)) = 1
POL(a__sel(x₁, x₂)) = 1
POL(a__sel1(x₁, x₂)) = x₂
POL(cons(x₁, x₂)) = 1
POL(dbl(x₁)) = 0
POL(dbl1(x₁)) = 0
POL(dbls(x₁)) = 0
POL(from(x₁)) = 0
POL(indx(x₁, x₂)) = 0
POL(mark(x₁)) = 1
POL(nil) = 0
POL(quote(x₁)) = 0
POL(s(x₁)) = 0
POL(s1(x₁)) = 0
POL(sel(x₁, x₂)) = 0
POL(sel1(x₁, x₂)) = 0

The following usable rules [17] were oriented:

a__dbls(cons(X, Y)) → cons(dbl(X), dbls(Y))
a__dbls(nil) → nil
a__dbl(s(X)) → s(s(dbl(X)))
a__dbl(0) → 0
a__dbl1(s(X)) → s1(s1(a__dbl1(mark(X))))
a__quote(0) → 01
a__indx(cons(X, Y), Z) → cons(sel(X, Z), indx(Y, Z))
a__indx(nil, X) → nil
a__dbl1(0) → 01
a__from(X) → cons(X, from(s(X)))
mark(indx(X1, X2)) → a__indx(mark(X1), X2)
mark(from(X)) → a__from(X)
mark(dbls(X)) → a__dbls(mark(X))
mark(sel1(X1, X2)) → a__sel1(mark(X1), mark(X2))
a__sel(s(X), cons(Y, Z)) → a__sel(mark(X), mark(Z))
a__sel1(0, cons(X, Y)) → mark(X)
a__sel(0, cons(X, Y)) → mark(X)
a__sel1(s(X), cons(Y, Z)) → a__sel1(mark(X), mark(Z))
mark(quote(X)) → a__quote(mark(X))
a__quote(sel(X, Y)) → a__sel1(mark(X), mark(Y))
mark(sel(X1, X2)) → a__sel(mark(X1), mark(X2))
mark(dbl(X)) → a__dbl(mark(X))
a__quote(s(X)) → s1(a__quote(mark(X)))
a__quote(dbl(X)) → a__dbl1(mark(X))
mark(cons(X1, X2)) → cons(X1, X2)
mark(01) → 01
mark(s(X)) → s(X)
mark(nil) → nil
mark(0) → 0
mark(dbl1(X)) → a__dbl1(mark(X))
a__sel1(X1, X2) → sel1(X1, X2)
a__dbl1(X) → dbl1(X)
a__from(X) → from(X)
a__indx(X1, X2) → indx(X1, X2)
a__sel(X1, X2) → sel(X1, X2)
a__dbls(X) → dbls(X)
a__dbl(X) → dbl(X)
mark(s1(X)) → s1(mark(X))
a__quote(X) → quote(X)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ QDPOrderProof

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ AND

                      ↳ QDP

                      ↳ QDP

                        ↳ QDPOrderProof

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ AND

                                ↳ QDP

                                  ↳ QDPOrderProof

                                    ↳ QDP

                                      ↳ DependencyGraphProof

                                        ↳ AND

                                          ↳ QDP

                                            ↳ QDPOrderProof

                                              ↳ QDP

                                                ↳ DependencyGraphProof

                                                  ↳ AND

                                                    ↳ QDP

                                                    ↳ QDP

                                                      ↳ Narrowing

                                                        ↳ QDP

                                                          ↳ DependencyGraphProof

                                                            ↳ QDP

                                                              ↳ Narrowing

                                                                ↳ QDP

                                                                  ↳ DependencyGraphProof

                                                                    ↳ QDP

                                                                      ↳ Narrowing

                                                                        ↳ QDP

                                                                          ↳ DependencyGraphProof

                                                                            ↳ QDP

                                                                              ↳ Narrowing

                                                                                ↳ QDP

                                                                                  ↳ DependencyGraphProof

                                                                                    ↳ QDP

                                                                                      ↳ Narrowing

                                                                                        ↳ QDP

                                                                                          ↳ DependencyGraphProof

                                                                                            ↳ QDP

                                                                                              ↳ Narrowing

                                                                                                ↳ QDP

                                                                                                  ↳ DependencyGraphProof

                                                                                                    ↳ QDP

                                                                                                      ↳ Narrowing

                                                                                                        ↳ QDP

                                                                                                          ↳ DependencyGraphProof

                                                                                                            ↳ QDP

                                                                                                              ↳ Narrowing

                                                                                                                ↳ QDP

                                                                                                                  ↳ DependencyGraphProof

                                                                                                                    ↳ QDP

                                                                                                                      ↳ Narrowing

                                                                                                                        ↳ QDP

                                                                                                                          ↳ DependencyGraphProof

                                                                                                                            ↳ QDP

                                                                                                                              ↳ Narrowing

                                                                                                                                ↳ QDP

                                                                                                                                  ↳ DependencyGraphProof

                                                                                                                                    ↳ QDP

                                                                                                                                      ↳ Narrowing

                                                                                                                                        ↳ QDP

                                                                                                                                          ↳ DependencyGraphProof

                                                                                                                                            ↳ QDP

                                                                                                                                              ↳ QDPOrderProof

                                                                                                                                                ↳ QDP

                                                                                                                                                  ↳ QDPOrderProof

                                          ↳ QDP

                                ↳ QDP

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

A__SEL(s(sel1(y0, y1)), cons(y2, indx(x0, x1))) → A__SEL(a__sel1(mark(y0), mark(y1)), a__indx(mark(x0), x1))
A__SEL(s(s(y0)), cons(y1, sel1(x0, x1))) → A__SEL(s(y0), a__sel1(mark(x0), mark(x1)))
A__SEL(s(quote(y0)), cons(y1, indx(x0, x1))) → A__SEL(a__quote(mark(y0)), a__indx(mark(x0), x1))
A__SEL(s(quote(y0)), cons(y1, from(x0))) → A__SEL(a__quote(mark(y0)), a__from(x0))
A__SEL(s(dbl(y0)), cons(y1, sel1(x0, x1))) → A__SEL(a__dbl(mark(y0)), a__sel1(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(quote(y0)), cons(y1, quote(x0))) → A__SEL(a__quote(mark(y0)), a__quote(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(s(y0)), cons(y1, quote(x0))) → A__SEL(s(y0), a__quote(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, sel1(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), a__sel1(mark(x0), mark(x1)))
A__SEL(s(sel1(y0, y1)), cons(y2, sel(x0, x1))) → A__SEL(a__sel1(mark(y0), mark(y1)), a__sel(mark(x0), mark(x1)))
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(sel1(y0, y1)), cons(y2, dbls(x0))) → A__SEL(a__sel1(mark(y0), mark(y1)), a__dbls(mark(x0)))
A__SEL(s(dbl1(y0)), cons(y1, sel(x0, x1))) → A__SEL(a__dbl1(mark(y0)), 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(dbls(y0)), cons(y1, sel1(x0, x1))) → A__SEL(a__dbls(mark(y0)), a__sel1(mark(x0), mark(x1)))
A__SEL(s(quote(y0)), cons(y1, sel1(x0, x1))) → A__SEL(a__quote(mark(y0)), a__sel1(mark(x0), mark(x1)))
A__SEL(s(sel(y0, y1)), cons(y2, quote(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__quote(mark(x0)))
A__SEL(s(sel1(y0, y1)), cons(y2, cons(x0, x1))) → A__SEL(a__sel1(mark(y0), mark(y1)), cons(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(dbl(y0)), cons(y1, quote(x0))) → A__SEL(a__dbl(mark(y0)), a__quote(mark(x0)))
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(y0)), cons(y1, indx(x0, x1))) → A__SEL(a__dbls(mark(y0)), a__indx(mark(x0), x1))
A__SEL(s(dbls(y0)), cons(y1, quote(x0))) → A__SEL(a__dbls(mark(y0)), a__quote(mark(x0)))
A__SEL(s(quote(y0)), cons(y1, dbls(x0))) → A__SEL(a__quote(mark(y0)), a__dbls(mark(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(dbl(y0)), cons(y1, sel(x0, x1))) → A__SEL(a__dbl(mark(y0)), a__sel(mark(x0), mark(x1)))
A__SEL(s(dbl1(y0)), cons(y1, cons(x0, x1))) → A__SEL(a__dbl1(mark(y0)), cons(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(sel1(y0, y1)), cons(y2, quote(x0))) → A__SEL(a__sel1(mark(y0), mark(y1)), a__quote(mark(x0)))
A__SEL(s(sel1(y0, y1)), cons(y2, sel1(x0, x1))) → A__SEL(a__sel1(mark(y0), mark(y1)), a__sel1(mark(x0), mark(x1)))
A__SEL(s(s(y0)), cons(y1, from(x0))) → A__SEL(s(y0), a__from(x0))
A__SEL(s(indx(y0, y1)), cons(y2, quote(x0))) → A__SEL(a__indx(mark(y0), y1), a__quote(mark(x0)))
A__SEL(s(s(y0)), cons(y1, dbls(x0))) → A__SEL(s(y0), a__dbls(mark(x0)))
A__SEL(s(quote(y0)), cons(y1, cons(x0, x1))) → A__SEL(a__quote(mark(y0)), cons(x0, x1))
A__SEL(s(indx(y0, y1)), cons(y2, sel1(x0, x1))) → A__SEL(a__indx(mark(y0), y1), a__sel1(mark(x0), mark(x1)))
A__SEL(s(quote(y0)), cons(y1, sel(x0, x1))) → A__SEL(a__quote(mark(y0)), a__sel(mark(x0), mark(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(dbl1(y0)), cons(y1, sel1(x0, x1))) → A__SEL(a__dbl1(mark(y0)), a__sel1(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(dbl1(y0)), cons(y1, dbls(x0))) → A__SEL(a__dbl1(mark(y0)), a__dbls(mark(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(dbl1(y0)), cons(y1, quote(x0))) → A__SEL(a__dbl1(mark(y0)), a__quote(mark(x0)))
A__SEL(s(sel1(y0, y1)), cons(y2, from(x0))) → A__SEL(a__sel1(mark(y0), mark(y1)), a__from(x0))
A__SEL(s(s(y0)), cons(y1, sel(x0, x1))) → A__SEL(s(y0), a__sel(mark(x0), mark(x1)))
A__SEL(s(dbl1(y0)), cons(y1, from(x0))) → A__SEL(a__dbl1(mark(y0)), a__from(x0))
A__SEL(s(dbl1(y0)), cons(y1, indx(x0, x1))) → A__SEL(a__dbl1(mark(y0)), a__indx(mark(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)))
a__dbl1(0) → 01
a__dbl1(s(X)) → s1(s1(a__dbl1(mark(X))))
a__sel1(0, cons(X, Y)) → mark(X)
a__sel1(s(X), cons(Y, Z)) → a__sel1(mark(X), mark(Z))
a__quote(0) → 01
a__quote(s(X)) → s1(a__quote(mark(X)))
a__quote(dbl(X)) → a__dbl1(mark(X))
a__quote(sel(X, Y)) → a__sel1(mark(X), mark(Y))
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(dbl1(X)) → a__dbl1(mark(X))
mark(sel1(X1, X2)) → a__sel1(mark(X1), mark(X2))
mark(quote(X)) → a__quote(mark(X))
mark(0) → 0
mark(s(X)) → s(X)
mark(nil) → nil
mark(cons(X1, X2)) → cons(X1, X2)
mark(01) → 01
mark(s1(X)) → s1(mark(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__from(X) → from(X)
a__dbl1(X) → dbl1(X)
a__sel1(X1, X2) → sel1(X1, X2)
a__quote(X) → quote(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(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(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(dbl1(y0)), cons(y1, sel(x0, x1))) → A__SEL(a__dbl1(mark(y0)), a__sel(mark(x0), mark(x1)))
A__SEL(s(dbls(y0)), cons(y1, sel1(x0, x1))) → A__SEL(a__dbls(mark(y0)), a__sel1(mark(x0), mark(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, indx(x0, x1))) → A__SEL(a__dbls(mark(y0)), a__indx(mark(x0), x1))
A__SEL(s(dbls(y0)), cons(y1, quote(x0))) → A__SEL(a__dbls(mark(y0)), a__quote(mark(x0)))
A__SEL(s(indx(y0, y1)), cons(y2, from(x0))) → A__SEL(a__indx(mark(y0), y1), a__from(x0))
A__SEL(s(dbl1(y0)), cons(y1, cons(x0, x1))) → A__SEL(a__dbl1(mark(y0)), cons(x0, x1))
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, quote(x0))) → A__SEL(a__indx(mark(y0), y1), a__quote(mark(x0)))
A__SEL(s(indx(y0, y1)), cons(y2, sel1(x0, x1))) → A__SEL(a__indx(mark(y0), y1), a__sel1(mark(x0), mark(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(dbl1(y0)), cons(y1, sel1(x0, x1))) → A__SEL(a__dbl1(mark(y0)), a__sel1(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(dbl1(y0)), cons(y1, dbls(x0))) → A__SEL(a__dbl1(mark(y0)), a__dbls(mark(x0)))
A__SEL(s(dbl1(y0)), cons(y1, quote(x0))) → A__SEL(a__dbl1(mark(y0)), a__quote(mark(x0)))
A__SEL(s(dbl1(y0)), cons(y1, from(x0))) → A__SEL(a__dbl1(mark(y0)), a__from(x0))
A__SEL(s(dbl1(y0)), cons(y1, indx(x0, x1))) → A__SEL(a__dbl1(mark(y0)), a__indx(mark(x0), x1))

The remaining pairs can at least be oriented weakly.

A__SEL(s(sel1(y0, y1)), cons(y2, indx(x0, x1))) → A__SEL(a__sel1(mark(y0), mark(y1)), a__indx(mark(x0), x1))
A__SEL(s(s(y0)), cons(y1, sel1(x0, x1))) → A__SEL(s(y0), a__sel1(mark(x0), mark(x1)))
A__SEL(s(quote(y0)), cons(y1, indx(x0, x1))) → A__SEL(a__quote(mark(y0)), a__indx(mark(x0), x1))
A__SEL(s(quote(y0)), cons(y1, from(x0))) → A__SEL(a__quote(mark(y0)), a__from(x0))
A__SEL(s(dbl(y0)), cons(y1, sel1(x0, x1))) → A__SEL(a__dbl(mark(y0)), a__sel1(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(quote(y0)), cons(y1, quote(x0))) → A__SEL(a__quote(mark(y0)), a__quote(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(s(y0)), cons(y1, quote(x0))) → A__SEL(s(y0), a__quote(mark(x0)))
A__SEL(s(sel(y0, y1)), cons(y2, sel1(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), a__sel1(mark(x0), mark(x1)))
A__SEL(s(sel1(y0, y1)), cons(y2, sel(x0, x1))) → A__SEL(a__sel1(mark(y0), mark(y1)), a__sel(mark(x0), mark(x1)))
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(sel1(y0, y1)), cons(y2, dbls(x0))) → A__SEL(a__sel1(mark(y0), mark(y1)), 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(quote(y0)), cons(y1, sel1(x0, x1))) → A__SEL(a__quote(mark(y0)), a__sel1(mark(x0), mark(x1)))
A__SEL(s(sel(y0, y1)), cons(y2, quote(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__quote(mark(x0)))
A__SEL(s(sel1(y0, y1)), cons(y2, cons(x0, x1))) → A__SEL(a__sel1(mark(y0), mark(y1)), cons(x0, x1))
A__SEL(s(dbl(y0)), cons(y1, quote(x0))) → A__SEL(a__dbl(mark(y0)), a__quote(mark(x0)))
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(quote(y0)), cons(y1, dbls(x0))) → A__SEL(a__quote(mark(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(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(sel(y0, y1)), cons(y2, dbls(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__dbls(mark(x0)))
A__SEL(s(sel1(y0, y1)), cons(y2, quote(x0))) → A__SEL(a__sel1(mark(y0), mark(y1)), a__quote(mark(x0)))
A__SEL(s(sel1(y0, y1)), cons(y2, sel1(x0, x1))) → A__SEL(a__sel1(mark(y0), mark(y1)), a__sel1(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(quote(y0)), cons(y1, cons(x0, x1))) → A__SEL(a__quote(mark(y0)), cons(x0, x1))
A__SEL(s(quote(y0)), cons(y1, sel(x0, x1))) → A__SEL(a__quote(mark(y0)), 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(sel1(y0, y1)), cons(y2, from(x0))) → A__SEL(a__sel1(mark(y0), mark(y1)), a__from(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, cons(x0, x1))) → A__SEL(s(y0), cons(x0, x1))

Used ordering: Polynomial interpretation [25]:

POL(0) = 1
POL(01) = 0
POL(A__SEL(x₁, x₂)) = x₁
POL(a__dbl(x₁)) = 1
POL(a__dbl1(x₁)) = 0
POL(a__dbls(x₁)) = 0
POL(a__from(x₁)) = 0
POL(a__indx(x₁, x₂)) = 0
POL(a__quote(x₁)) = 1
POL(a__sel(x₁, x₂)) = 1
POL(a__sel1(x₁, x₂)) = 1
POL(cons(x₁, x₂)) = 0
POL(dbl(x₁)) = 0
POL(dbl1(x₁)) = 0
POL(dbls(x₁)) = 0
POL(from(x₁)) = 0
POL(indx(x₁, x₂)) = 0
POL(mark(x₁)) = 1
POL(nil) = 0
POL(quote(x₁)) = 0
POL(s(x₁)) = 1
POL(s1(x₁)) = 0
POL(sel(x₁, x₂)) = 0
POL(sel1(x₁, x₂)) = 0

The following usable rules [17] were oriented:

a__dbls(cons(X, Y)) → cons(dbl(X), dbls(Y))
a__dbls(nil) → nil
a__dbl(s(X)) → s(s(dbl(X)))
a__dbl(0) → 0
a__dbl1(s(X)) → s1(s1(a__dbl1(mark(X))))
a__quote(0) → 01
a__indx(cons(X, Y), Z) → cons(sel(X, Z), indx(Y, Z))
a__indx(nil, X) → nil
a__dbl1(0) → 01
a__from(X) → cons(X, from(s(X)))
mark(indx(X1, X2)) → a__indx(mark(X1), X2)
mark(from(X)) → a__from(X)
mark(dbls(X)) → a__dbls(mark(X))
mark(sel1(X1, X2)) → a__sel1(mark(X1), mark(X2))
a__sel(s(X), cons(Y, Z)) → a__sel(mark(X), mark(Z))
a__sel1(0, cons(X, Y)) → mark(X)
a__sel(0, cons(X, Y)) → mark(X)
a__sel1(s(X), cons(Y, Z)) → a__sel1(mark(X), mark(Z))
mark(quote(X)) → a__quote(mark(X))
a__quote(sel(X, Y)) → a__sel1(mark(X), mark(Y))
mark(sel(X1, X2)) → a__sel(mark(X1), mark(X2))
mark(dbl(X)) → a__dbl(mark(X))
a__quote(s(X)) → s1(a__quote(mark(X)))
a__quote(dbl(X)) → a__dbl1(mark(X))
mark(cons(X1, X2)) → cons(X1, X2)
mark(01) → 01
mark(s(X)) → s(X)
mark(nil) → nil
mark(0) → 0
mark(dbl1(X)) → a__dbl1(mark(X))
a__sel1(X1, X2) → sel1(X1, X2)
a__dbl1(X) → dbl1(X)
a__from(X) → from(X)
a__indx(X1, X2) → indx(X1, X2)
a__sel(X1, X2) → sel(X1, X2)
a__dbls(X) → dbls(X)
a__dbl(X) → dbl(X)
mark(s1(X)) → s1(mark(X))
a__quote(X) → quote(X)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ QDPOrderProof

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ AND

                      ↳ QDP

                      ↳ QDP

                        ↳ QDPOrderProof

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ AND

                                ↳ QDP

                                  ↳ QDPOrderProof

                                    ↳ QDP

                                      ↳ DependencyGraphProof

                                        ↳ AND

                                          ↳ QDP

                                            ↳ QDPOrderProof

                                              ↳ QDP

                                                ↳ DependencyGraphProof

                                                  ↳ AND

                                                    ↳ QDP

                                                    ↳ QDP

                                                      ↳ Narrowing

                                                        ↳ QDP

                                                          ↳ DependencyGraphProof

                                                            ↳ QDP

                                                              ↳ Narrowing

                                                                ↳ QDP

                                                                  ↳ DependencyGraphProof

                                                                    ↳ QDP

                                                                      ↳ Narrowing

                                                                        ↳ QDP

                                                                          ↳ DependencyGraphProof

                                                                            ↳ QDP

                                                                              ↳ Narrowing

                                                                                ↳ QDP

                                                                                  ↳ DependencyGraphProof

                                                                                    ↳ QDP

                                                                                      ↳ Narrowing

                                                                                        ↳ QDP

                                                                                          ↳ DependencyGraphProof

                                                                                            ↳ QDP

                                                                                              ↳ Narrowing

                                                                                                ↳ QDP

                                                                                                  ↳ DependencyGraphProof

                                                                                                    ↳ QDP

                                                                                                      ↳ Narrowing

                                                                                                        ↳ QDP

                                                                                                          ↳ DependencyGraphProof

                                                                                                            ↳ QDP

                                                                                                              ↳ Narrowing

                                                                                                                ↳ QDP

                                                                                                                  ↳ DependencyGraphProof

                                                                                                                    ↳ QDP

                                                                                                                      ↳ Narrowing

                                                                                                                        ↳ QDP

                                                                                                                          ↳ DependencyGraphProof

                                                                                                                            ↳ QDP

                                                                                                                              ↳ Narrowing

                                                                                                                                ↳ QDP

                                                                                                                                  ↳ DependencyGraphProof

                                                                                                                                    ↳ QDP

                                                                                                                                      ↳ Narrowing

                                                                                                                                        ↳ QDP

                                                                                                                                          ↳ DependencyGraphProof

                                                                                                                                            ↳ QDP

                                                                                                                                              ↳ QDPOrderProof

                                                                                                                                                ↳ QDP

                                                                                                                                                  ↳ QDPOrderProof

                                                                                                                                                    ↳ QDP

                                                                                                                                                      ↳ Narrowing

                                          ↳ QDP

                                ↳ QDP

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

A__SEL(s(sel1(y0, y1)), cons(y2, indx(x0, x1))) → A__SEL(a__sel1(mark(y0), mark(y1)), a__indx(mark(x0), x1))
A__SEL(s(quote(y0)), cons(y1, dbls(x0))) → A__SEL(a__quote(mark(y0)), a__dbls(mark(x0)))
A__SEL(s(s(y0)), cons(y1, sel1(x0, x1))) → A__SEL(s(y0), a__sel1(mark(x0), mark(x1)))
A__SEL(s(s(y0)), cons(y1, indx(x0, x1))) → A__SEL(s(y0), a__indx(mark(x0), x1))
A__SEL(s(quote(y0)), cons(y1, indx(x0, x1))) → A__SEL(a__quote(mark(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(dbl(y0)), cons(y1, sel(x0, x1))) → A__SEL(a__dbl(mark(y0)), a__sel(mark(x0), mark(x1)))
A__SEL(s(quote(y0)), cons(y1, from(x0))) → A__SEL(a__quote(mark(y0)), a__from(x0))
A__SEL(s(dbl(y0)), cons(y1, sel1(x0, x1))) → A__SEL(a__dbl(mark(y0)), a__sel1(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(quote(y0)), cons(y1, quote(x0))) → A__SEL(a__quote(mark(y0)), a__quote(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(sel1(y0, y1)), cons(y2, quote(x0))) → A__SEL(a__sel1(mark(y0), mark(y1)), a__quote(mark(x0)))
A__SEL(s(s(y0)), cons(y1, quote(x0))) → A__SEL(s(y0), a__quote(mark(x0)))
A__SEL(s(sel1(y0, y1)), cons(y2, sel(x0, x1))) → A__SEL(a__sel1(mark(y0), mark(y1)), a__sel(mark(x0), mark(x1)))
A__SEL(s(sel(y0, y1)), cons(y2, sel1(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), a__sel1(mark(x0), mark(x1)))
A__SEL(s(sel1(y0, y1)), cons(y2, sel1(x0, x1))) → A__SEL(a__sel1(mark(y0), mark(y1)), a__sel1(mark(x0), mark(x1)))
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(sel1(y0, y1)), cons(y2, dbls(x0))) → A__SEL(a__sel1(mark(y0), mark(y1)), a__dbls(mark(x0)))
A__SEL(s(quote(y0)), cons(y1, cons(x0, x1))) → A__SEL(a__quote(mark(y0)), cons(x0, x1))
A__SEL(s(quote(y0)), cons(y1, sel(x0, x1))) → A__SEL(a__quote(mark(y0)), 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(quote(y0)), cons(y1, sel1(x0, x1))) → A__SEL(a__quote(mark(y0)), a__sel1(mark(x0), mark(x1)))
A__SEL(s(sel(y0, y1)), cons(y2, quote(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__quote(mark(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(sel1(y0, y1)), cons(y2, cons(x0, x1))) → A__SEL(a__sel1(mark(y0), mark(y1)), cons(x0, x1))
A__SEL(s(dbl(y0)), cons(y1, quote(x0))) → A__SEL(a__dbl(mark(y0)), a__quote(mark(x0)))
A__SEL(s(sel1(y0, y1)), cons(y2, from(x0))) → A__SEL(a__sel1(mark(y0), mark(y1)), a__from(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)))
a__dbl1(0) → 01
a__dbl1(s(X)) → s1(s1(a__dbl1(mark(X))))
a__sel1(0, cons(X, Y)) → mark(X)
a__sel1(s(X), cons(Y, Z)) → a__sel1(mark(X), mark(Z))
a__quote(0) → 01
a__quote(s(X)) → s1(a__quote(mark(X)))
a__quote(dbl(X)) → a__dbl1(mark(X))
a__quote(sel(X, Y)) → a__sel1(mark(X), mark(Y))
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(dbl1(X)) → a__dbl1(mark(X))
mark(sel1(X1, X2)) → a__sel1(mark(X1), mark(X2))
mark(quote(X)) → a__quote(mark(X))
mark(0) → 0
mark(s(X)) → s(X)
mark(nil) → nil
mark(cons(X1, X2)) → cons(X1, X2)
mark(01) → 01
mark(s1(X)) → s1(mark(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__from(X) → from(X)
a__dbl1(X) → dbl1(X)
a__sel1(X1, X2) → sel1(X1, X2)
a__quote(X) → quote(X)

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

A__SEL(s(quote(y0)), cons(y1, from(x0))) → A__SEL(a__quote(mark(y0)), from(x0))
A__SEL(s(quote(y0)), cons(y1, from(x0))) → A__SEL(a__quote(mark(y0)), cons(x0, from(s(x0))))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ QDPOrderProof

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ AND

                      ↳ QDP

                      ↳ QDP

                        ↳ QDPOrderProof

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ AND

                                ↳ QDP

                                  ↳ QDPOrderProof

                                    ↳ QDP

                                      ↳ DependencyGraphProof

                                        ↳ AND

                                          ↳ QDP

                                            ↳ QDPOrderProof

                                              ↳ QDP

                                                ↳ DependencyGraphProof

                                                  ↳ AND

                                                    ↳ QDP

                                                    ↳ QDP

                                                      ↳ Narrowing

                                                        ↳ QDP

                                                          ↳ DependencyGraphProof

                                                            ↳ QDP

                                                              ↳ Narrowing

                                                                ↳ QDP

                                                                  ↳ DependencyGraphProof

                                                                    ↳ QDP

                                                                      ↳ Narrowing

                                                                        ↳ QDP

                                                                          ↳ DependencyGraphProof

                                                                            ↳ QDP

                                                                              ↳ Narrowing

                                                                                ↳ QDP

                                                                                  ↳ DependencyGraphProof

                                                                                    ↳ QDP

                                                                                      ↳ Narrowing

                                                                                        ↳ QDP

                                                                                          ↳ DependencyGraphProof

                                                                                            ↳ QDP

                                                                                              ↳ Narrowing

                                                                                                ↳ QDP

                                                                                                  ↳ DependencyGraphProof

                                                                                                    ↳ QDP

                                                                                                      ↳ Narrowing

                                                                                                        ↳ QDP

                                                                                                          ↳ DependencyGraphProof

                                                                                                            ↳ QDP

                                                                                                              ↳ Narrowing

                                                                                                                ↳ QDP

                                                                                                                  ↳ DependencyGraphProof

                                                                                                                    ↳ QDP

                                                                                                                      ↳ Narrowing

                                                                                                                        ↳ QDP

                                                                                                                          ↳ DependencyGraphProof

                                                                                                                            ↳ QDP

                                                                                                                              ↳ Narrowing

                                                                                                                                ↳ QDP

                                                                                                                                  ↳ DependencyGraphProof

                                                                                                                                    ↳ QDP

                                                                                                                                      ↳ Narrowing

                                                                                                                                        ↳ QDP

                                                                                                                                          ↳ DependencyGraphProof

                                                                                                                                            ↳ QDP

                                                                                                                                              ↳ QDPOrderProof

                                                                                                                                                ↳ QDP

                                                                                                                                                  ↳ QDPOrderProof

                                                                                                                                                    ↳ QDP

                                                                                                                                                      ↳ Narrowing

                                                                                                                                                        ↳ QDP

                                                                                                                                                          ↳ DependencyGraphProof

                                          ↳ QDP

                                ↳ QDP

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

A__SEL(s(quote(y0)), cons(y1, dbls(x0))) → A__SEL(a__quote(mark(y0)), a__dbls(mark(x0)))
A__SEL(s(sel1(y0, y1)), cons(y2, indx(x0, x1))) → A__SEL(a__sel1(mark(y0), mark(y1)), a__indx(mark(x0), x1))
A__SEL(s(s(y0)), cons(y1, sel1(x0, x1))) → A__SEL(s(y0), a__sel1(mark(x0), mark(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(quote(y0)), cons(y1, indx(x0, x1))) → A__SEL(a__quote(mark(y0)), a__indx(mark(x0), 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(quote(y0)), cons(y1, from(x0))) → A__SEL(a__quote(mark(y0)), from(x0))
A__SEL(s(dbl(y0)), cons(y1, sel1(x0, x1))) → A__SEL(a__dbl(mark(y0)), a__sel1(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(quote(y0)), cons(y1, quote(x0))) → A__SEL(a__quote(mark(y0)), a__quote(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(s(y0)), cons(y1, quote(x0))) → A__SEL(s(y0), a__quote(mark(x0)))
A__SEL(s(sel1(y0, y1)), cons(y2, quote(x0))) → A__SEL(a__sel1(mark(y0), mark(y1)), a__quote(mark(x0)))
A__SEL(s(sel(y0, y1)), cons(y2, sel1(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), a__sel1(mark(x0), mark(x1)))
A__SEL(s(sel1(y0, y1)), cons(y2, sel(x0, x1))) → A__SEL(a__sel1(mark(y0), mark(y1)), a__sel(mark(x0), mark(x1)))
A__SEL(s(sel1(y0, y1)), cons(y2, sel1(x0, x1))) → A__SEL(a__sel1(mark(y0), mark(y1)), a__sel1(mark(x0), mark(x1)))
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(sel1(y0, y1)), cons(y2, dbls(x0))) → A__SEL(a__sel1(mark(y0), mark(y1)), a__dbls(mark(x0)))
A__SEL(s(quote(y0)), cons(y1, cons(x0, x1))) → A__SEL(a__quote(mark(y0)), cons(x0, 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(quote(y0)), cons(y1, sel(x0, x1))) → A__SEL(a__quote(mark(y0)), a__sel(mark(x0), mark(x1)))
A__SEL(s(quote(y0)), cons(y1, sel1(x0, x1))) → A__SEL(a__quote(mark(y0)), a__sel1(mark(x0), mark(x1)))
A__SEL(s(sel(y0, y1)), cons(y2, quote(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__quote(mark(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(sel1(y0, y1)), cons(y2, cons(x0, x1))) → A__SEL(a__sel1(mark(y0), mark(y1)), cons(x0, x1))
A__SEL(s(dbl(y0)), cons(y1, quote(x0))) → A__SEL(a__dbl(mark(y0)), a__quote(mark(x0)))
A__SEL(s(sel1(y0, y1)), cons(y2, from(x0))) → A__SEL(a__sel1(mark(y0), mark(y1)), a__from(x0))
A__SEL(s(quote(y0)), cons(y1, from(x0))) → A__SEL(a__quote(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)))
a__dbl1(0) → 01
a__dbl1(s(X)) → s1(s1(a__dbl1(mark(X))))
a__sel1(0, cons(X, Y)) → mark(X)
a__sel1(s(X), cons(Y, Z)) → a__sel1(mark(X), mark(Z))
a__quote(0) → 01
a__quote(s(X)) → s1(a__quote(mark(X)))
a__quote(dbl(X)) → a__dbl1(mark(X))
a__quote(sel(X, Y)) → a__sel1(mark(X), mark(Y))
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(dbl1(X)) → a__dbl1(mark(X))
mark(sel1(X1, X2)) → a__sel1(mark(X1), mark(X2))
mark(quote(X)) → a__quote(mark(X))
mark(0) → 0
mark(s(X)) → s(X)
mark(nil) → nil
mark(cons(X1, X2)) → cons(X1, X2)
mark(01) → 01
mark(s1(X)) → s1(mark(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__from(X) → from(X)
a__dbl1(X) → dbl1(X)
a__sel1(X1, X2) → sel1(X1, X2)
a__quote(X) → quote(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

                  ↳ DependencyGraphProof

                    ↳ AND

                      ↳ QDP

                      ↳ QDP

                        ↳ QDPOrderProof

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ AND

                                ↳ QDP

                                  ↳ QDPOrderProof

                                    ↳ QDP

                                      ↳ DependencyGraphProof

                                        ↳ AND

                                          ↳ QDP

                                            ↳ QDPOrderProof

                                              ↳ QDP

                                                ↳ DependencyGraphProof

                                                  ↳ AND

                                                    ↳ QDP

                                                    ↳ QDP

                                                      ↳ Narrowing

                                                        ↳ QDP

                                                          ↳ DependencyGraphProof

                                                            ↳ QDP

                                                              ↳ Narrowing

                                                                ↳ QDP

                                                                  ↳ DependencyGraphProof

                                                                    ↳ QDP

                                                                      ↳ Narrowing

                                                                        ↳ QDP

                                                                          ↳ DependencyGraphProof

                                                                            ↳ QDP

                                                                              ↳ Narrowing

                                                                                ↳ QDP

                                                                                  ↳ DependencyGraphProof

                                                                                    ↳ QDP

                                                                                      ↳ Narrowing

                                                                                        ↳ QDP

                                                                                          ↳ DependencyGraphProof

                                                                                            ↳ QDP

                                                                                              ↳ Narrowing

                                                                                                ↳ QDP

                                                                                                  ↳ DependencyGraphProof

                                                                                                    ↳ QDP

                                                                                                      ↳ Narrowing

                                                                                                        ↳ QDP

                                                                                                          ↳ DependencyGraphProof

                                                                                                            ↳ QDP

                                                                                                              ↳ Narrowing

                                                                                                                ↳ QDP

                                                                                                                  ↳ DependencyGraphProof

                                                                                                                    ↳ QDP

                                                                                                                      ↳ Narrowing

                                                                                                                        ↳ QDP

                                                                                                                          ↳ DependencyGraphProof

                                                                                                                            ↳ QDP

                                                                                                                              ↳ Narrowing

                                                                                                                                ↳ QDP

                                                                                                                                  ↳ DependencyGraphProof

                                                                                                                                    ↳ QDP

                                                                                                                                      ↳ Narrowing

                                                                                                                                        ↳ QDP

                                                                                                                                          ↳ DependencyGraphProof

                                                                                                                                            ↳ QDP

                                                                                                                                              ↳ QDPOrderProof

                                                                                                                                                ↳ QDP

                                                                                                                                                  ↳ QDPOrderProof

                                                                                                                                                    ↳ QDP

                                                                                                                                                      ↳ Narrowing

                                                                                                                                                        ↳ QDP

                                                                                                                                                          ↳ DependencyGraphProof

                                                                                                                                                            ↳ QDP

                                                                                                                                                              ↳ QDPOrderProof

                                          ↳ QDP

                                ↳ QDP

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

A__SEL(s(sel1(y0, y1)), cons(y2, indx(x0, x1))) → A__SEL(a__sel1(mark(y0), mark(y1)), a__indx(mark(x0), x1))
A__SEL(s(quote(y0)), cons(y1, dbls(x0))) → A__SEL(a__quote(mark(y0)), a__dbls(mark(x0)))
A__SEL(s(s(y0)), cons(y1, sel1(x0, x1))) → A__SEL(s(y0), a__sel1(mark(x0), mark(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(quote(y0)), cons(y1, indx(x0, x1))) → A__SEL(a__quote(mark(y0)), a__indx(mark(x0), 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, sel1(x0, x1))) → A__SEL(a__dbl(mark(y0)), a__sel1(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(quote(y0)), cons(y1, quote(x0))) → A__SEL(a__quote(mark(y0)), a__quote(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(s(y0)), cons(y1, quote(x0))) → A__SEL(s(y0), a__quote(mark(x0)))
A__SEL(s(sel1(y0, y1)), cons(y2, quote(x0))) → A__SEL(a__sel1(mark(y0), mark(y1)), a__quote(mark(x0)))
A__SEL(s(sel1(y0, y1)), cons(y2, sel(x0, x1))) → A__SEL(a__sel1(mark(y0), mark(y1)), a__sel(mark(x0), mark(x1)))
A__SEL(s(sel(y0, y1)), cons(y2, sel1(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), a__sel1(mark(x0), mark(x1)))
A__SEL(s(sel1(y0, y1)), cons(y2, sel1(x0, x1))) → A__SEL(a__sel1(mark(y0), mark(y1)), a__sel1(mark(x0), mark(x1)))
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(sel1(y0, y1)), cons(y2, dbls(x0))) → A__SEL(a__sel1(mark(y0), mark(y1)), a__dbls(mark(x0)))
A__SEL(s(quote(y0)), cons(y1, cons(x0, x1))) → A__SEL(a__quote(mark(y0)), cons(x0, 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(quote(y0)), cons(y1, sel(x0, x1))) → A__SEL(a__quote(mark(y0)), a__sel(mark(x0), mark(x1)))
A__SEL(s(quote(y0)), cons(y1, sel1(x0, x1))) → A__SEL(a__quote(mark(y0)), a__sel1(mark(x0), mark(x1)))
A__SEL(s(sel(y0, y1)), cons(y2, quote(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__quote(mark(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(sel1(y0, y1)), cons(y2, cons(x0, x1))) → A__SEL(a__sel1(mark(y0), mark(y1)), cons(x0, x1))
A__SEL(s(dbl(y0)), cons(y1, quote(x0))) → A__SEL(a__dbl(mark(y0)), a__quote(mark(x0)))
A__SEL(s(sel1(y0, y1)), cons(y2, from(x0))) → A__SEL(a__sel1(mark(y0), mark(y1)), a__from(x0))
A__SEL(s(quote(y0)), cons(y1, from(x0))) → A__SEL(a__quote(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)))
a__dbl1(0) → 01
a__dbl1(s(X)) → s1(s1(a__dbl1(mark(X))))
a__sel1(0, cons(X, Y)) → mark(X)
a__sel1(s(X), cons(Y, Z)) → a__sel1(mark(X), mark(Z))
a__quote(0) → 01
a__quote(s(X)) → s1(a__quote(mark(X)))
a__quote(dbl(X)) → a__dbl1(mark(X))
a__quote(sel(X, Y)) → a__sel1(mark(X), mark(Y))
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(dbl1(X)) → a__dbl1(mark(X))
mark(sel1(X1, X2)) → a__sel1(mark(X1), mark(X2))
mark(quote(X)) → a__quote(mark(X))
mark(0) → 0
mark(s(X)) → s(X)
mark(nil) → nil
mark(cons(X1, X2)) → cons(X1, X2)
mark(01) → 01
mark(s1(X)) → s1(mark(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__from(X) → from(X)
a__dbl1(X) → dbl1(X)
a__sel1(X1, X2) → sel1(X1, X2)
a__quote(X) → quote(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(dbl(y0)), cons(y1, cons(x0, x1))) → A__SEL(a__dbl(mark(y0)), cons(x0, x1))
A__SEL(s(quote(y0)), cons(y1, cons(x0, x1))) → A__SEL(a__quote(mark(y0)), cons(x0, x1))
A__SEL(s(sel1(y0, y1)), cons(y2, cons(x0, x1))) → A__SEL(a__sel1(mark(y0), mark(y1)), cons(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(y0)), cons(y1, cons(x0, x1))) → A__SEL(s(y0), cons(x0, x1))

The remaining pairs can at least be oriented weakly.

A__SEL(s(sel1(y0, y1)), cons(y2, indx(x0, x1))) → A__SEL(a__sel1(mark(y0), mark(y1)), a__indx(mark(x0), x1))
A__SEL(s(quote(y0)), cons(y1, dbls(x0))) → A__SEL(a__quote(mark(y0)), a__dbls(mark(x0)))
A__SEL(s(s(y0)), cons(y1, sel1(x0, x1))) → A__SEL(s(y0), a__sel1(mark(x0), mark(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(quote(y0)), cons(y1, indx(x0, x1))) → A__SEL(a__quote(mark(y0)), a__indx(mark(x0), 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, sel1(x0, x1))) → A__SEL(a__dbl(mark(y0)), a__sel1(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(quote(y0)), cons(y1, quote(x0))) → A__SEL(a__quote(mark(y0)), a__quote(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(s(y0)), cons(y1, quote(x0))) → A__SEL(s(y0), a__quote(mark(x0)))
A__SEL(s(sel1(y0, y1)), cons(y2, quote(x0))) → A__SEL(a__sel1(mark(y0), mark(y1)), a__quote(mark(x0)))
A__SEL(s(sel1(y0, y1)), cons(y2, sel(x0, x1))) → A__SEL(a__sel1(mark(y0), mark(y1)), a__sel(mark(x0), mark(x1)))
A__SEL(s(sel(y0, y1)), cons(y2, sel1(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), a__sel1(mark(x0), mark(x1)))
A__SEL(s(sel1(y0, y1)), cons(y2, sel1(x0, x1))) → A__SEL(a__sel1(mark(y0), mark(y1)), a__sel1(mark(x0), mark(x1)))
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(sel1(y0, y1)), cons(y2, dbls(x0))) → A__SEL(a__sel1(mark(y0), mark(y1)), 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(quote(y0)), cons(y1, sel(x0, x1))) → A__SEL(a__quote(mark(y0)), a__sel(mark(x0), mark(x1)))
A__SEL(s(quote(y0)), cons(y1, sel1(x0, x1))) → A__SEL(a__quote(mark(y0)), a__sel1(mark(x0), mark(x1)))
A__SEL(s(sel(y0, y1)), cons(y2, quote(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__quote(mark(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(dbl(y0)), cons(y1, quote(x0))) → A__SEL(a__dbl(mark(y0)), a__quote(mark(x0)))
A__SEL(s(sel1(y0, y1)), cons(y2, from(x0))) → A__SEL(a__sel1(mark(y0), mark(y1)), a__from(x0))
A__SEL(s(quote(y0)), cons(y1, from(x0))) → A__SEL(a__quote(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)))

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

POL(0) = 0
POL(01) = 1
POL(A__SEL(x₁, x₂)) = x₂
POL(a__dbl(x₁)) = 0
POL(a__dbl1(x₁)) = 1
POL(a__dbls(x₁)) = x₁
POL(a__from(x₁)) = 1 + x₁
POL(a__indx(x₁, x₂)) = 1 + x₂
POL(a__quote(x₁)) = 1 + x₁
POL(a__sel(x₁, x₂)) = x₂
POL(a__sel1(x₁, x₂)) = x₂
POL(cons(x₁, x₂)) = 1 + max(x₁, x₂)
POL(dbl(x₁)) = 0
POL(dbl1(x₁)) = 1
POL(dbls(x₁)) = x₁
POL(from(x₁)) = x₁
POL(indx(x₁, x₂)) = x₂
POL(mark(x₁)) = 1 + x₁
POL(nil) = 0
POL(quote(x₁)) = 1 + x₁
POL(s(x₁)) = 0
POL(s1(x₁)) = 1
POL(sel(x₁, x₂)) = x₂
POL(sel1(x₁, x₂)) = x₂

The following usable rules [17] were oriented:

a__dbls(cons(X, Y)) → cons(dbl(X), dbls(Y))
a__dbls(nil) → nil
a__dbl(s(X)) → s(s(dbl(X)))
a__dbl(0) → 0
a__dbl1(s(X)) → s1(s1(a__dbl1(mark(X))))
a__quote(0) → 01
a__indx(cons(X, Y), Z) → cons(sel(X, Z), indx(Y, Z))
a__indx(nil, X) → nil
a__dbl1(0) → 01
a__from(X) → cons(X, from(s(X)))
mark(indx(X1, X2)) → a__indx(mark(X1), X2)
mark(from(X)) → a__from(X)
mark(dbls(X)) → a__dbls(mark(X))
mark(sel1(X1, X2)) → a__sel1(mark(X1), mark(X2))
a__sel(s(X), cons(Y, Z)) → a__sel(mark(X), mark(Z))
a__sel1(0, cons(X, Y)) → mark(X)
a__sel(0, cons(X, Y)) → mark(X)
a__sel1(s(X), cons(Y, Z)) → a__sel1(mark(X), mark(Z))
mark(quote(X)) → a__quote(mark(X))
a__quote(sel(X, Y)) → a__sel1(mark(X), mark(Y))
mark(sel(X1, X2)) → a__sel(mark(X1), mark(X2))
mark(dbl(X)) → a__dbl(mark(X))
a__quote(s(X)) → s1(a__quote(mark(X)))
a__quote(dbl(X)) → a__dbl1(mark(X))
mark(cons(X1, X2)) → cons(X1, X2)
mark(01) → 01
mark(s(X)) → s(X)
mark(nil) → nil
mark(0) → 0
mark(dbl1(X)) → a__dbl1(mark(X))
a__sel1(X1, X2) → sel1(X1, X2)
a__dbl1(X) → dbl1(X)
a__from(X) → from(X)
a__indx(X1, X2) → indx(X1, X2)
a__sel(X1, X2) → sel(X1, X2)
a__dbls(X) → dbls(X)
a__dbl(X) → dbl(X)
mark(s1(X)) → s1(mark(X))
a__quote(X) → quote(X)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ QDPOrderProof

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ AND

                      ↳ QDP

                      ↳ QDP

                        ↳ QDPOrderProof

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ AND

                                ↳ QDP

                                  ↳ QDPOrderProof

                                    ↳ QDP

                                      ↳ DependencyGraphProof

                                        ↳ AND

                                          ↳ QDP

                                            ↳ QDPOrderProof

                                              ↳ QDP

                                                ↳ DependencyGraphProof

                                                  ↳ AND

                                                    ↳ QDP

                                                    ↳ QDP

                                                      ↳ Narrowing

                                                        ↳ QDP

                                                          ↳ DependencyGraphProof

                                                            ↳ QDP

                                                              ↳ Narrowing

                                                                ↳ QDP

                                                                  ↳ DependencyGraphProof

                                                                    ↳ QDP

                                                                      ↳ Narrowing

                                                                        ↳ QDP

                                                                          ↳ DependencyGraphProof

                                                                            ↳ QDP

                                                                              ↳ Narrowing

                                                                                ↳ QDP

                                                                                  ↳ DependencyGraphProof

                                                                                    ↳ QDP

                                                                                      ↳ Narrowing

                                                                                        ↳ QDP

                                                                                          ↳ DependencyGraphProof

                                                                                            ↳ QDP

                                                                                              ↳ Narrowing

                                                                                                ↳ QDP

                                                                                                  ↳ DependencyGraphProof

                                                                                                    ↳ QDP

                                                                                                      ↳ Narrowing

                                                                                                        ↳ QDP

                                                                                                          ↳ DependencyGraphProof

                                                                                                            ↳ QDP

                                                                                                              ↳ Narrowing

                                                                                                                ↳ QDP

                                                                                                                  ↳ DependencyGraphProof

                                                                                                                    ↳ QDP

                                                                                                                      ↳ Narrowing

                                                                                                                        ↳ QDP

                                                                                                                          ↳ DependencyGraphProof

                                                                                                                            ↳ QDP

                                                                                                                              ↳ Narrowing

                                                                                                                                ↳ QDP

                                                                                                                                  ↳ DependencyGraphProof

                                                                                                                                    ↳ QDP

                                                                                                                                      ↳ Narrowing

                                                                                                                                        ↳ QDP

                                                                                                                                          ↳ DependencyGraphProof

                                                                                                                                            ↳ QDP

                                                                                                                                              ↳ QDPOrderProof

                                                                                                                                                ↳ QDP

                                                                                                                                                  ↳ QDPOrderProof

                                                                                                                                                    ↳ QDP

                                                                                                                                                      ↳ Narrowing

                                                                                                                                                        ↳ QDP

                                                                                                                                                          ↳ DependencyGraphProof

                                                                                                                                                            ↳ QDP

                                                                                                                                                              ↳ QDPOrderProof

                                                                                                                                                                ↳ QDP

                                          ↳ QDP

                                ↳ QDP

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

A__SEL(s(quote(y0)), cons(y1, dbls(x0))) → A__SEL(a__quote(mark(y0)), a__dbls(mark(x0)))
A__SEL(s(sel1(y0, y1)), cons(y2, indx(x0, x1))) → A__SEL(a__sel1(mark(y0), mark(y1)), a__indx(mark(x0), x1))
A__SEL(s(s(y0)), cons(y1, sel1(x0, x1))) → A__SEL(s(y0), a__sel1(mark(x0), mark(x1)))
A__SEL(s(s(y0)), cons(y1, indx(x0, x1))) → A__SEL(s(y0), a__indx(mark(x0), x1))
A__SEL(s(quote(y0)), cons(y1, indx(x0, x1))) → A__SEL(a__quote(mark(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(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, sel1(x0, x1))) → A__SEL(a__dbl(mark(y0)), a__sel1(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(quote(y0)), cons(y1, quote(x0))) → A__SEL(a__quote(mark(y0)), a__quote(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(s(y0)), cons(y1, quote(x0))) → A__SEL(s(y0), a__quote(mark(x0)))
A__SEL(s(sel1(y0, y1)), cons(y2, quote(x0))) → A__SEL(a__sel1(mark(y0), mark(y1)), a__quote(mark(x0)))
A__SEL(s(sel1(y0, y1)), cons(y2, sel(x0, x1))) → A__SEL(a__sel1(mark(y0), mark(y1)), a__sel(mark(x0), mark(x1)))
A__SEL(s(sel(y0, y1)), cons(y2, sel1(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), a__sel1(mark(x0), mark(x1)))
A__SEL(s(sel1(y0, y1)), cons(y2, sel1(x0, x1))) → A__SEL(a__sel1(mark(y0), mark(y1)), a__sel1(mark(x0), mark(x1)))
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(sel1(y0, y1)), cons(y2, dbls(x0))) → A__SEL(a__sel1(mark(y0), mark(y1)), 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(quote(y0)), cons(y1, sel(x0, x1))) → A__SEL(a__quote(mark(y0)), a__sel(mark(x0), mark(x1)))
A__SEL(s(quote(y0)), cons(y1, sel1(x0, x1))) → A__SEL(a__quote(mark(y0)), a__sel1(mark(x0), mark(x1)))
A__SEL(s(sel(y0, y1)), cons(y2, quote(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__quote(mark(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(dbl(y0)), cons(y1, quote(x0))) → A__SEL(a__dbl(mark(y0)), a__quote(mark(x0)))
A__SEL(s(sel1(y0, y1)), cons(y2, from(x0))) → A__SEL(a__sel1(mark(y0), mark(y1)), a__from(x0))
A__SEL(s(s(y0)), cons(y1, sel(x0, x1))) → A__SEL(s(y0), a__sel(mark(x0), mark(x1)))
A__SEL(s(quote(y0)), cons(y1, from(x0))) → A__SEL(a__quote(mark(y0)), 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)))
a__dbl1(0) → 01
a__dbl1(s(X)) → s1(s1(a__dbl1(mark(X))))
a__sel1(0, cons(X, Y)) → mark(X)
a__sel1(s(X), cons(Y, Z)) → a__sel1(mark(X), mark(Z))
a__quote(0) → 01
a__quote(s(X)) → s1(a__quote(mark(X)))
a__quote(dbl(X)) → a__dbl1(mark(X))
a__quote(sel(X, Y)) → a__sel1(mark(X), mark(Y))
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(dbl1(X)) → a__dbl1(mark(X))
mark(sel1(X1, X2)) → a__sel1(mark(X1), mark(X2))
mark(quote(X)) → a__quote(mark(X))
mark(0) → 0
mark(s(X)) → s(X)
mark(nil) → nil
mark(cons(X1, X2)) → cons(X1, X2)
mark(01) → 01
mark(s1(X)) → s1(mark(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__from(X) → from(X)
a__dbl1(X) → dbl1(X)
a__sel1(X1, X2) → sel1(X1, X2)
a__quote(X) → quote(X)

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ QDPOrderProof

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ AND

                      ↳ QDP

                      ↳ QDP

                        ↳ QDPOrderProof

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ AND

                                ↳ QDP

                                  ↳ QDPOrderProof

                                    ↳ QDP

                                      ↳ DependencyGraphProof

                                        ↳ AND

                                          ↳ QDP

                                          ↳ QDP

                                ↳ QDP

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

A__DBL1(s(X)) → A__DBL1(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)))
a__dbl1(0) → 01
a__dbl1(s(X)) → s1(s1(a__dbl1(mark(X))))
a__sel1(0, cons(X, Y)) → mark(X)
a__sel1(s(X), cons(Y, Z)) → a__sel1(mark(X), mark(Z))
a__quote(0) → 01
a__quote(s(X)) → s1(a__quote(mark(X)))
a__quote(dbl(X)) → a__dbl1(mark(X))
a__quote(sel(X, Y)) → a__sel1(mark(X), mark(Y))
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(dbl1(X)) → a__dbl1(mark(X))
mark(sel1(X1, X2)) → a__sel1(mark(X1), mark(X2))
mark(quote(X)) → a__quote(mark(X))
mark(0) → 0
mark(s(X)) → s(X)
mark(nil) → nil
mark(cons(X1, X2)) → cons(X1, X2)
mark(01) → 01
mark(s1(X)) → s1(mark(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__from(X) → from(X)
a__dbl1(X) → dbl1(X)
a__sel1(X1, X2) → sel1(X1, X2)
a__quote(X) → quote(X)

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ QDPOrderProof

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ AND

                      ↳ QDP

                      ↳ QDP

                        ↳ QDPOrderProof

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ AND

                                ↳ QDP

                                ↳ QDP

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

A__SEL1(s(X), cons(Y, Z)) → A__SEL1(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)))
a__dbl1(0) → 01
a__dbl1(s(X)) → s1(s1(a__dbl1(mark(X))))
a__sel1(0, cons(X, Y)) → mark(X)
a__sel1(s(X), cons(Y, Z)) → a__sel1(mark(X), mark(Z))
a__quote(0) → 01
a__quote(s(X)) → s1(a__quote(mark(X)))
a__quote(dbl(X)) → a__dbl1(mark(X))
a__quote(sel(X, Y)) → a__sel1(mark(X), mark(Y))
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(dbl1(X)) → a__dbl1(mark(X))
mark(sel1(X1, X2)) → a__sel1(mark(X1), mark(X2))
mark(quote(X)) → a__quote(mark(X))
mark(0) → 0
mark(s(X)) → s(X)
mark(nil) → nil
mark(cons(X1, X2)) → cons(X1, X2)
mark(01) → 01
mark(s1(X)) → s1(mark(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__from(X) → from(X)
a__dbl1(X) → dbl1(X)
a__sel1(X1, X2) → sel1(X1, X2)
a__quote(X) → quote(X)

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