Termination w.r.t. Q proof of /tmp/tmpnVTa9c/Ex1_Luc02b

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__from(X) → cons(mark(X), from(s(X)))
a__first(0, Z) → nil
a__first(s(X), cons(Y, Z)) → cons(mark(Y), first(X, Z))
a__sel(0, cons(X, Z)) → mark(X)
a__sel(s(X), cons(Y, Z)) → a__sel(mark(X), mark(Z))
mark(from(X)) → a__from(mark(X))
mark(first(X1, X2)) → a__first(mark(X1), mark(X2))
mark(sel(X1, X2)) → a__sel(mark(X1), mark(X2))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(s(X)) → s(mark(X))
mark(0) → 0
mark(nil) → nil
a__from(X) → from(X)
a__first(X1, X2) → first(X1, X2)
a__sel(X1, X2) → sel(X1, X2)

Q is empty.

↳ QTRS

  ↳ DependencyPairsProof

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

a__from(X) → cons(mark(X), from(s(X)))
a__first(0, Z) → nil
a__first(s(X), cons(Y, Z)) → cons(mark(Y), first(X, Z))
a__sel(0, cons(X, Z)) → mark(X)
a__sel(s(X), cons(Y, Z)) → a__sel(mark(X), mark(Z))
mark(from(X)) → a__from(mark(X))
mark(first(X1, X2)) → a__first(mark(X1), mark(X2))
mark(sel(X1, X2)) → a__sel(mark(X1), mark(X2))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(s(X)) → s(mark(X))
mark(0) → 0
mark(nil) → nil
a__from(X) → from(X)
a__first(X1, X2) → first(X1, X2)
a__sel(X1, X2) → sel(X1, X2)

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(first(X1, X2)) → MARK(X1)
MARK(from(X)) → MARK(X)
MARK(sel(X1, X2)) → MARK(X1)
MARK(sel(X1, X2)) → A__SEL(mark(X1), mark(X2))
MARK(from(X)) → A__FROM(mark(X))
MARK(s(X)) → MARK(X)
A__SEL(s(X), cons(Y, Z)) → A__SEL(mark(X), mark(Z))
MARK(cons(X1, X2)) → MARK(X1)
MARK(first(X1, X2)) → MARK(X2)
A__FROM(X) → MARK(X)
MARK(first(X1, X2)) → A__FIRST(mark(X1), mark(X2))
A__SEL(0, cons(X, Z)) → MARK(X)
A__FIRST(s(X), cons(Y, Z)) → MARK(Y)
A__SEL(s(X), cons(Y, Z)) → MARK(Z)
MARK(sel(X1, X2)) → MARK(X2)
A__SEL(s(X), cons(Y, Z)) → MARK(X)

The TRS R consists of the following rules:

a__from(X) → cons(mark(X), from(s(X)))
a__first(0, Z) → nil
a__first(s(X), cons(Y, Z)) → cons(mark(Y), first(X, Z))
a__sel(0, cons(X, Z)) → mark(X)
a__sel(s(X), cons(Y, Z)) → a__sel(mark(X), mark(Z))
mark(from(X)) → a__from(mark(X))
mark(first(X1, X2)) → a__first(mark(X1), mark(X2))
mark(sel(X1, X2)) → a__sel(mark(X1), mark(X2))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(s(X)) → s(mark(X))
mark(0) → 0
mark(nil) → nil
a__from(X) → from(X)
a__first(X1, X2) → first(X1, X2)
a__sel(X1, X2) → sel(X1, X2)

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ Narrowing

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

MARK(first(X1, X2)) → MARK(X1)
MARK(from(X)) → MARK(X)
MARK(sel(X1, X2)) → MARK(X1)
MARK(sel(X1, X2)) → A__SEL(mark(X1), mark(X2))
MARK(from(X)) → A__FROM(mark(X))
MARK(s(X)) → MARK(X)
A__SEL(s(X), cons(Y, Z)) → A__SEL(mark(X), mark(Z))
MARK(cons(X1, X2)) → MARK(X1)
MARK(first(X1, X2)) → MARK(X2)
A__FROM(X) → MARK(X)
MARK(first(X1, X2)) → A__FIRST(mark(X1), mark(X2))
A__SEL(0, cons(X, Z)) → MARK(X)
A__FIRST(s(X), cons(Y, Z)) → MARK(Y)
A__SEL(s(X), cons(Y, Z)) → MARK(Z)
MARK(sel(X1, X2)) → MARK(X2)
A__SEL(s(X), cons(Y, Z)) → MARK(X)

The TRS R consists of the following rules:

a__from(X) → cons(mark(X), from(s(X)))
a__first(0, Z) → nil
a__first(s(X), cons(Y, Z)) → cons(mark(Y), first(X, Z))
a__sel(0, cons(X, Z)) → mark(X)
a__sel(s(X), cons(Y, Z)) → a__sel(mark(X), mark(Z))
mark(from(X)) → a__from(mark(X))
mark(first(X1, X2)) → a__first(mark(X1), mark(X2))
mark(sel(X1, X2)) → a__sel(mark(X1), mark(X2))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(s(X)) → s(mark(X))
mark(0) → 0
mark(nil) → nil
a__from(X) → from(X)
a__first(X1, X2) → first(X1, X2)
a__sel(X1, X2) → sel(X1, X2)

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(from(x0)), cons(y1, y2)) → A__SEL(a__from(mark(x0)), mark(y2))
A__SEL(s(first(x0, x1)), cons(y1, y2)) → A__SEL(a__first(mark(x0), mark(x1)), 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(s(x0)), cons(y1, y2)) → A__SEL(s(mark(x0)), mark(y2))
A__SEL(s(nil), cons(y1, y2)) → A__SEL(nil, mark(y2))
A__SEL(s(cons(x0, x1)), cons(y1, y2)) → A__SEL(cons(mark(x0), x1), mark(y2))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ Narrowing

        ↳ QDP

          ↳ DependencyGraphProof

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

A__SEL(s(from(x0)), cons(y1, y2)) → A__SEL(a__from(mark(x0)), mark(y2))
MARK(first(X1, X2)) → MARK(X1)
A__SEL(s(first(x0, x1)), cons(y1, y2)) → A__SEL(a__first(mark(x0), mark(x1)), mark(y2))
MARK(sel(X1, X2)) → MARK(X1)
MARK(from(X)) → MARK(X)
MARK(sel(X1, X2)) → A__SEL(mark(X1), mark(X2))
A__SEL(s(s(x0)), cons(y1, y2)) → A__SEL(s(mark(x0)), mark(y2))
MARK(from(X)) → A__FROM(mark(X))
MARK(s(X)) → MARK(X)
MARK(first(X1, X2)) → MARK(X2)
MARK(cons(X1, X2)) → MARK(X1)
MARK(first(X1, X2)) → A__FIRST(mark(X1), mark(X2))
A__FROM(X) → MARK(X)
A__SEL(0, cons(X, Z)) → MARK(X)
A__SEL(s(cons(x0, x1)), cons(y1, y2)) → A__SEL(cons(mark(x0), x1), mark(y2))
A__FIRST(s(X), cons(Y, Z)) → MARK(Y)
A__SEL(s(0), cons(y1, y2)) → A__SEL(0, mark(y2))
A__SEL(s(sel(x0, x1)), cons(y1, y2)) → A__SEL(a__sel(mark(x0), mark(x1)), mark(y2))
A__SEL(s(X), cons(Y, Z)) → MARK(Z)
A__SEL(s(nil), cons(y1, y2)) → A__SEL(nil, mark(y2))
A__SEL(s(X), cons(Y, Z)) → MARK(X)
MARK(sel(X1, X2)) → MARK(X2)

The TRS R consists of the following rules:

a__from(X) → cons(mark(X), from(s(X)))
a__first(0, Z) → nil
a__first(s(X), cons(Y, Z)) → cons(mark(Y), first(X, Z))
a__sel(0, cons(X, Z)) → mark(X)
a__sel(s(X), cons(Y, Z)) → a__sel(mark(X), mark(Z))
mark(from(X)) → a__from(mark(X))
mark(first(X1, X2)) → a__first(mark(X1), mark(X2))
mark(sel(X1, X2)) → a__sel(mark(X1), mark(X2))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(s(X)) → s(mark(X))
mark(0) → 0
mark(nil) → nil
a__from(X) → from(X)
a__first(X1, X2) → first(X1, X2)
a__sel(X1, X2) → sel(X1, X2)

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

      ↳ Narrowing

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ QDP

              ↳ Narrowing

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

A__SEL(s(from(x0)), cons(y1, y2)) → A__SEL(a__from(mark(x0)), mark(y2))
MARK(first(X1, X2)) → MARK(X1)
A__SEL(s(first(x0, x1)), cons(y1, y2)) → A__SEL(a__first(mark(x0), mark(x1)), mark(y2))
MARK(from(X)) → MARK(X)
MARK(sel(X1, X2)) → MARK(X1)
MARK(sel(X1, X2)) → A__SEL(mark(X1), mark(X2))
A__SEL(s(s(x0)), cons(y1, y2)) → A__SEL(s(mark(x0)), mark(y2))
MARK(from(X)) → A__FROM(mark(X))
MARK(s(X)) → MARK(X)
MARK(first(X1, X2)) → MARK(X2)
MARK(cons(X1, X2)) → MARK(X1)
A__FROM(X) → MARK(X)
MARK(first(X1, X2)) → A__FIRST(mark(X1), mark(X2))
A__SEL(0, cons(X, Z)) → MARK(X)
A__FIRST(s(X), cons(Y, Z)) → MARK(Y)
A__SEL(s(0), cons(y1, y2)) → A__SEL(0, mark(y2))
A__SEL(s(sel(x0, x1)), cons(y1, y2)) → A__SEL(a__sel(mark(x0), mark(x1)), mark(y2))
A__SEL(s(X), cons(Y, Z)) → MARK(Z)
A__SEL(s(X), cons(Y, Z)) → MARK(X)
MARK(sel(X1, X2)) → MARK(X2)

The TRS R consists of the following rules:

a__from(X) → cons(mark(X), from(s(X)))
a__first(0, Z) → nil
a__first(s(X), cons(Y, Z)) → cons(mark(Y), first(X, Z))
a__sel(0, cons(X, Z)) → mark(X)
a__sel(s(X), cons(Y, Z)) → a__sel(mark(X), mark(Z))
mark(from(X)) → a__from(mark(X))
mark(first(X1, X2)) → a__first(mark(X1), mark(X2))
mark(sel(X1, X2)) → a__sel(mark(X1), mark(X2))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(s(X)) → s(mark(X))
mark(0) → 0
mark(nil) → nil
a__from(X) → from(X)
a__first(X1, X2) → first(X1, X2)
a__sel(X1, X2) → sel(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule MARK(first(X1, X2)) → A__FIRST(mark(X1), mark(X2)) at position [0] we obtained the following new rules:

MARK(first(sel(x0, x1), y1)) → A__FIRST(a__sel(mark(x0), mark(x1)), mark(y1))
MARK(first(from(x0), y1)) → A__FIRST(a__from(mark(x0)), mark(y1))
MARK(first(s(x0), y1)) → A__FIRST(s(mark(x0)), mark(y1))
MARK(first(first(x0, x1), y1)) → A__FIRST(a__first(mark(x0), mark(x1)), mark(y1))
MARK(first(nil, y1)) → A__FIRST(nil, mark(y1))
MARK(first(cons(x0, x1), y1)) → A__FIRST(cons(mark(x0), x1), mark(y1))
MARK(first(0, y1)) → A__FIRST(0, mark(y1))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ Narrowing

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ QDP

              ↳ Narrowing

                ↳ QDP

                  ↳ DependencyGraphProof

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

MARK(s(X)) → MARK(X)
MARK(cons(X1, X2)) → MARK(X1)
A__SEL(0, cons(X, Z)) → MARK(X)
A__FIRST(s(X), cons(Y, Z)) → MARK(Y)
A__SEL(s(sel(x0, x1)), cons(y1, y2)) → A__SEL(a__sel(mark(x0), mark(x1)), mark(y2))
A__SEL(s(X), cons(Y, Z)) → MARK(Z)
MARK(first(s(x0), y1)) → A__FIRST(s(mark(x0)), mark(y1))
A__SEL(s(X), cons(Y, Z)) → MARK(X)
A__SEL(s(from(x0)), cons(y1, y2)) → A__SEL(a__from(mark(x0)), mark(y2))
MARK(first(X1, X2)) → MARK(X1)
A__SEL(s(first(x0, x1)), cons(y1, y2)) → A__SEL(a__first(mark(x0), mark(x1)), mark(y2))
MARK(from(X)) → MARK(X)
MARK(sel(X1, X2)) → MARK(X1)
MARK(first(sel(x0, x1), y1)) → A__FIRST(a__sel(mark(x0), mark(x1)), mark(y1))
MARK(sel(X1, X2)) → A__SEL(mark(X1), mark(X2))
A__SEL(s(s(x0)), cons(y1, y2)) → A__SEL(s(mark(x0)), mark(y2))
MARK(from(X)) → A__FROM(mark(X))
MARK(first(first(x0, x1), y1)) → A__FIRST(a__first(mark(x0), mark(x1)), mark(y1))
MARK(first(nil, y1)) → A__FIRST(nil, mark(y1))
MARK(first(X1, X2)) → MARK(X2)
MARK(first(cons(x0, x1), y1)) → A__FIRST(cons(mark(x0), x1), mark(y1))
A__FROM(X) → MARK(X)
A__SEL(s(0), cons(y1, y2)) → A__SEL(0, mark(y2))
MARK(first(from(x0), y1)) → A__FIRST(a__from(mark(x0)), mark(y1))
MARK(first(0, y1)) → A__FIRST(0, mark(y1))
MARK(sel(X1, X2)) → MARK(X2)

The TRS R consists of the following rules:

a__from(X) → cons(mark(X), from(s(X)))
a__first(0, Z) → nil
a__first(s(X), cons(Y, Z)) → cons(mark(Y), first(X, Z))
a__sel(0, cons(X, Z)) → mark(X)
a__sel(s(X), cons(Y, Z)) → a__sel(mark(X), mark(Z))
mark(from(X)) → a__from(mark(X))
mark(first(X1, X2)) → a__first(mark(X1), mark(X2))
mark(sel(X1, X2)) → a__sel(mark(X1), mark(X2))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(s(X)) → s(mark(X))
mark(0) → 0
mark(nil) → nil
a__from(X) → from(X)
a__first(X1, X2) → first(X1, X2)
a__sel(X1, X2) → sel(X1, X2)

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ Narrowing

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ QDP

              ↳ Narrowing

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ QDP

                      ↳ Narrowing

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

A__SEL(s(from(x0)), cons(y1, y2)) → A__SEL(a__from(mark(x0)), mark(y2))
MARK(first(X1, X2)) → MARK(X1)
A__SEL(s(first(x0, x1)), cons(y1, y2)) → A__SEL(a__first(mark(x0), mark(x1)), mark(y2))
MARK(from(X)) → MARK(X)
MARK(sel(X1, X2)) → MARK(X1)
MARK(first(sel(x0, x1), y1)) → A__FIRST(a__sel(mark(x0), mark(x1)), mark(y1))
MARK(sel(X1, X2)) → A__SEL(mark(X1), mark(X2))
A__SEL(s(s(x0)), cons(y1, y2)) → A__SEL(s(mark(x0)), mark(y2))
MARK(from(X)) → A__FROM(mark(X))
MARK(s(X)) → MARK(X)
MARK(first(first(x0, x1), y1)) → A__FIRST(a__first(mark(x0), mark(x1)), mark(y1))
MARK(first(X1, X2)) → MARK(X2)
MARK(cons(X1, X2)) → MARK(X1)
A__FROM(X) → MARK(X)
A__SEL(0, cons(X, Z)) → MARK(X)
A__FIRST(s(X), cons(Y, Z)) → MARK(Y)
A__SEL(s(0), cons(y1, y2)) → A__SEL(0, mark(y2))
A__SEL(s(sel(x0, x1)), cons(y1, y2)) → A__SEL(a__sel(mark(x0), mark(x1)), mark(y2))
A__SEL(s(X), cons(Y, Z)) → MARK(Z)
MARK(first(from(x0), y1)) → A__FIRST(a__from(mark(x0)), mark(y1))
MARK(first(s(x0), y1)) → A__FIRST(s(mark(x0)), mark(y1))
A__SEL(s(X), cons(Y, Z)) → MARK(X)
MARK(sel(X1, X2)) → MARK(X2)

The TRS R consists of the following rules:

a__from(X) → cons(mark(X), from(s(X)))
a__first(0, Z) → nil
a__first(s(X), cons(Y, Z)) → cons(mark(Y), first(X, Z))
a__sel(0, cons(X, Z)) → mark(X)
a__sel(s(X), cons(Y, Z)) → a__sel(mark(X), mark(Z))
mark(from(X)) → a__from(mark(X))
mark(first(X1, X2)) → a__first(mark(X1), mark(X2))
mark(sel(X1, X2)) → a__sel(mark(X1), mark(X2))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(s(X)) → s(mark(X))
mark(0) → 0
mark(nil) → nil
a__from(X) → from(X)
a__first(X1, X2) → first(X1, X2)
a__sel(X1, X2) → sel(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule MARK(sel(X1, X2)) → A__SEL(mark(X1), mark(X2)) at position [0] we obtained the following new rules:

MARK(sel(cons(x0, x1), y1)) → A__SEL(cons(mark(x0), x1), mark(y1))
MARK(sel(from(x0), y1)) → A__SEL(a__from(mark(x0)), mark(y1))
MARK(sel(0, y1)) → A__SEL(0, mark(y1))
MARK(sel(first(x0, x1), y1)) → A__SEL(a__first(mark(x0), mark(x1)), mark(y1))
MARK(sel(sel(x0, x1), y1)) → A__SEL(a__sel(mark(x0), mark(x1)), mark(y1))
MARK(sel(s(x0), y1)) → A__SEL(s(mark(x0)), mark(y1))
MARK(sel(nil, y1)) → A__SEL(nil, mark(y1))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ Narrowing

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ QDP

              ↳ Narrowing

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ QDP

                      ↳ Narrowing

                        ↳ QDP

                          ↳ DependencyGraphProof

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

MARK(sel(cons(x0, x1), y1)) → A__SEL(cons(mark(x0), x1), mark(y1))
MARK(sel(from(x0), y1)) → A__SEL(a__from(mark(x0)), mark(y1))
MARK(s(X)) → MARK(X)
MARK(cons(X1, X2)) → MARK(X1)
A__SEL(0, cons(X, Z)) → MARK(X)
A__FIRST(s(X), cons(Y, Z)) → MARK(Y)
A__SEL(s(sel(x0, x1)), cons(y1, y2)) → A__SEL(a__sel(mark(x0), mark(x1)), mark(y2))
A__SEL(s(X), cons(Y, Z)) → MARK(Z)
MARK(first(s(x0), y1)) → A__FIRST(s(mark(x0)), mark(y1))
MARK(sel(0, y1)) → A__SEL(0, mark(y1))
MARK(sel(nil, y1)) → A__SEL(nil, mark(y1))
A__SEL(s(X), cons(Y, Z)) → MARK(X)
A__SEL(s(from(x0)), cons(y1, y2)) → A__SEL(a__from(mark(x0)), mark(y2))
MARK(first(X1, X2)) → MARK(X1)
A__SEL(s(first(x0, x1)), cons(y1, y2)) → A__SEL(a__first(mark(x0), mark(x1)), mark(y2))
MARK(from(X)) → MARK(X)
MARK(sel(X1, X2)) → MARK(X1)
MARK(first(sel(x0, x1), y1)) → A__FIRST(a__sel(mark(x0), mark(x1)), mark(y1))
A__SEL(s(s(x0)), cons(y1, y2)) → A__SEL(s(mark(x0)), mark(y2))
MARK(from(X)) → A__FROM(mark(X))
MARK(first(first(x0, x1), y1)) → A__FIRST(a__first(mark(x0), mark(x1)), mark(y1))
MARK(first(X1, X2)) → MARK(X2)
MARK(sel(s(x0), y1)) → A__SEL(s(mark(x0)), mark(y1))
A__FROM(X) → MARK(X)
A__SEL(s(0), cons(y1, y2)) → A__SEL(0, mark(y2))
MARK(first(from(x0), y1)) → A__FIRST(a__from(mark(x0)), mark(y1))
MARK(sel(sel(x0, x1), y1)) → A__SEL(a__sel(mark(x0), mark(x1)), mark(y1))
MARK(sel(first(x0, x1), y1)) → A__SEL(a__first(mark(x0), mark(x1)), mark(y1))
MARK(sel(X1, X2)) → MARK(X2)

The TRS R consists of the following rules:

a__from(X) → cons(mark(X), from(s(X)))
a__first(0, Z) → nil
a__first(s(X), cons(Y, Z)) → cons(mark(Y), first(X, Z))
a__sel(0, cons(X, Z)) → mark(X)
a__sel(s(X), cons(Y, Z)) → a__sel(mark(X), mark(Z))
mark(from(X)) → a__from(mark(X))
mark(first(X1, X2)) → a__first(mark(X1), mark(X2))
mark(sel(X1, X2)) → a__sel(mark(X1), mark(X2))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(s(X)) → s(mark(X))
mark(0) → 0
mark(nil) → nil
a__from(X) → from(X)
a__first(X1, X2) → first(X1, X2)
a__sel(X1, X2) → sel(X1, X2)

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

      ↳ Narrowing

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ QDP

              ↳ Narrowing

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ QDP

                      ↳ Narrowing

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ QDP

                              ↳ Narrowing

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

MARK(sel(from(x0), y1)) → A__SEL(a__from(mark(x0)), mark(y1))
MARK(s(X)) → MARK(X)
MARK(cons(X1, X2)) → MARK(X1)
A__SEL(0, cons(X, Z)) → MARK(X)
A__FIRST(s(X), cons(Y, Z)) → MARK(Y)
A__SEL(s(X), cons(Y, Z)) → MARK(Z)
A__SEL(s(sel(x0, x1)), cons(y1, y2)) → A__SEL(a__sel(mark(x0), mark(x1)), mark(y2))
MARK(first(s(x0), y1)) → A__FIRST(s(mark(x0)), mark(y1))
MARK(sel(0, y1)) → A__SEL(0, mark(y1))
A__SEL(s(X), cons(Y, Z)) → MARK(X)
MARK(first(X1, X2)) → MARK(X1)
A__SEL(s(from(x0)), cons(y1, y2)) → A__SEL(a__from(mark(x0)), mark(y2))
A__SEL(s(first(x0, x1)), cons(y1, y2)) → A__SEL(a__first(mark(x0), mark(x1)), mark(y2))
MARK(first(sel(x0, x1), y1)) → A__FIRST(a__sel(mark(x0), mark(x1)), mark(y1))
MARK(sel(X1, X2)) → MARK(X1)
MARK(from(X)) → MARK(X)
A__SEL(s(s(x0)), cons(y1, y2)) → A__SEL(s(mark(x0)), mark(y2))
MARK(from(X)) → A__FROM(mark(X))
MARK(first(first(x0, x1), y1)) → A__FIRST(a__first(mark(x0), mark(x1)), mark(y1))
MARK(first(X1, X2)) → MARK(X2)
MARK(sel(s(x0), y1)) → A__SEL(s(mark(x0)), mark(y1))
A__FROM(X) → MARK(X)
A__SEL(s(0), cons(y1, y2)) → A__SEL(0, mark(y2))
MARK(first(from(x0), y1)) → A__FIRST(a__from(mark(x0)), mark(y1))
MARK(sel(first(x0, x1), y1)) → A__SEL(a__first(mark(x0), mark(x1)), mark(y1))
MARK(sel(sel(x0, x1), y1)) → A__SEL(a__sel(mark(x0), mark(x1)), mark(y1))
MARK(sel(X1, X2)) → MARK(X2)

The TRS R consists of the following rules:

a__from(X) → cons(mark(X), from(s(X)))
a__first(0, Z) → nil
a__first(s(X), cons(Y, Z)) → cons(mark(Y), first(X, Z))
a__sel(0, cons(X, Z)) → mark(X)
a__sel(s(X), cons(Y, Z)) → a__sel(mark(X), mark(Z))
mark(from(X)) → a__from(mark(X))
mark(first(X1, X2)) → a__first(mark(X1), mark(X2))
mark(sel(X1, X2)) → a__sel(mark(X1), mark(X2))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(s(X)) → s(mark(X))
mark(0) → 0
mark(nil) → nil
a__from(X) → from(X)
a__first(X1, X2) → first(X1, X2)
a__sel(X1, X2) → sel(X1, X2)

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

MARK(first(sel(y0, y1), s(x0))) → A__FIRST(a__sel(mark(y0), mark(y1)), s(mark(x0)))
MARK(first(sel(y0, y1), from(x0))) → A__FIRST(a__sel(mark(y0), mark(y1)), a__from(mark(x0)))
MARK(first(sel(y0, y1), 0)) → A__FIRST(a__sel(mark(y0), mark(y1)), 0)
MARK(first(sel(y0, y1), nil)) → A__FIRST(a__sel(mark(y0), mark(y1)), nil)
MARK(first(sel(y0, y1), cons(x0, x1))) → A__FIRST(a__sel(mark(y0), mark(y1)), cons(mark(x0), x1))
MARK(first(sel(y0, y1), first(x0, x1))) → A__FIRST(a__sel(mark(y0), mark(y1)), a__first(mark(x0), mark(x1)))
MARK(first(sel(y0, y1), sel(x0, x1))) → A__FIRST(a__sel(mark(y0), mark(y1)), a__sel(mark(x0), mark(x1)))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ Narrowing

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ QDP

              ↳ Narrowing

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ QDP

                      ↳ Narrowing

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ QDP

                              ↳ Narrowing

                                ↳ QDP

                                  ↳ DependencyGraphProof

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

MARK(sel(from(x0), y1)) → A__SEL(a__from(mark(x0)), mark(y1))
MARK(s(X)) → MARK(X)
MARK(cons(X1, X2)) → MARK(X1)
MARK(first(sel(y0, y1), from(x0))) → A__FIRST(a__sel(mark(y0), mark(y1)), a__from(mark(x0)))
A__SEL(0, cons(X, Z)) → MARK(X)
A__FIRST(s(X), cons(Y, Z)) → MARK(Y)
MARK(first(sel(y0, y1), nil)) → A__FIRST(a__sel(mark(y0), mark(y1)), nil)
MARK(first(sel(y0, y1), cons(x0, x1))) → A__FIRST(a__sel(mark(y0), mark(y1)), cons(mark(x0), x1))
A__SEL(s(sel(x0, x1)), cons(y1, y2)) → A__SEL(a__sel(mark(x0), mark(x1)), mark(y2))
A__SEL(s(X), cons(Y, Z)) → MARK(Z)
MARK(first(s(x0), y1)) → A__FIRST(s(mark(x0)), mark(y1))
MARK(sel(0, y1)) → A__SEL(0, mark(y1))
A__SEL(s(X), cons(Y, Z)) → MARK(X)
MARK(first(sel(y0, y1), sel(x0, x1))) → A__FIRST(a__sel(mark(y0), mark(y1)), a__sel(mark(x0), mark(x1)))
A__SEL(s(from(x0)), cons(y1, y2)) → A__SEL(a__from(mark(x0)), mark(y2))
MARK(first(X1, X2)) → MARK(X1)
A__SEL(s(first(x0, x1)), cons(y1, y2)) → A__SEL(a__first(mark(x0), mark(x1)), mark(y2))
MARK(from(X)) → MARK(X)
MARK(sel(X1, X2)) → MARK(X1)
MARK(from(X)) → A__FROM(mark(X))
A__SEL(s(s(x0)), cons(y1, y2)) → A__SEL(s(mark(x0)), mark(y2))
MARK(first(sel(y0, y1), s(x0))) → A__FIRST(a__sel(mark(y0), mark(y1)), s(mark(x0)))
MARK(first(first(x0, x1), y1)) → A__FIRST(a__first(mark(x0), mark(x1)), mark(y1))
MARK(first(X1, X2)) → MARK(X2)
MARK(sel(s(x0), y1)) → A__SEL(s(mark(x0)), mark(y1))
A__FROM(X) → MARK(X)
MARK(first(sel(y0, y1), 0)) → A__FIRST(a__sel(mark(y0), mark(y1)), 0)
MARK(first(from(x0), y1)) → A__FIRST(a__from(mark(x0)), mark(y1))
A__SEL(s(0), cons(y1, y2)) → A__SEL(0, mark(y2))
MARK(first(sel(y0, y1), first(x0, x1))) → A__FIRST(a__sel(mark(y0), mark(y1)), a__first(mark(x0), mark(x1)))
MARK(sel(sel(x0, x1), y1)) → A__SEL(a__sel(mark(x0), mark(x1)), mark(y1))
MARK(sel(first(x0, x1), y1)) → A__SEL(a__first(mark(x0), mark(x1)), mark(y1))
MARK(sel(X1, X2)) → MARK(X2)

The TRS R consists of the following rules:

a__from(X) → cons(mark(X), from(s(X)))
a__first(0, Z) → nil
a__first(s(X), cons(Y, Z)) → cons(mark(Y), first(X, Z))
a__sel(0, cons(X, Z)) → mark(X)
a__sel(s(X), cons(Y, Z)) → a__sel(mark(X), mark(Z))
mark(from(X)) → a__from(mark(X))
mark(first(X1, X2)) → a__first(mark(X1), mark(X2))
mark(sel(X1, X2)) → a__sel(mark(X1), mark(X2))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(s(X)) → s(mark(X))
mark(0) → 0
mark(nil) → nil
a__from(X) → from(X)
a__first(X1, X2) → first(X1, X2)
a__sel(X1, X2) → sel(X1, X2)

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ Narrowing

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ QDP

              ↳ Narrowing

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ QDP

                      ↳ Narrowing

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ QDP

                              ↳ Narrowing

                                ↳ QDP

                                  ↳ DependencyGraphProof

                                    ↳ QDP

                                      ↳ Narrowing

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

MARK(sel(from(x0), y1)) → A__SEL(a__from(mark(x0)), mark(y1))
MARK(s(X)) → MARK(X)
MARK(cons(X1, X2)) → MARK(X1)
MARK(first(sel(y0, y1), from(x0))) → A__FIRST(a__sel(mark(y0), mark(y1)), a__from(mark(x0)))
A__SEL(0, cons(X, Z)) → MARK(X)
A__FIRST(s(X), cons(Y, Z)) → MARK(Y)
MARK(first(sel(y0, y1), cons(x0, x1))) → A__FIRST(a__sel(mark(y0), mark(y1)), cons(mark(x0), x1))
A__SEL(s(X), cons(Y, Z)) → MARK(Z)
A__SEL(s(sel(x0, x1)), cons(y1, y2)) → A__SEL(a__sel(mark(x0), mark(x1)), mark(y2))
MARK(first(s(x0), y1)) → A__FIRST(s(mark(x0)), mark(y1))
MARK(sel(0, y1)) → A__SEL(0, mark(y1))
A__SEL(s(X), cons(Y, Z)) → MARK(X)
MARK(first(sel(y0, y1), sel(x0, x1))) → A__FIRST(a__sel(mark(y0), mark(y1)), a__sel(mark(x0), mark(x1)))
MARK(first(X1, X2)) → MARK(X1)
A__SEL(s(from(x0)), cons(y1, y2)) → A__SEL(a__from(mark(x0)), mark(y2))
A__SEL(s(first(x0, x1)), cons(y1, y2)) → A__SEL(a__first(mark(x0), mark(x1)), mark(y2))
MARK(sel(X1, X2)) → MARK(X1)
MARK(from(X)) → MARK(X)
A__SEL(s(s(x0)), cons(y1, y2)) → A__SEL(s(mark(x0)), mark(y2))
MARK(from(X)) → A__FROM(mark(X))
MARK(first(first(x0, x1), y1)) → A__FIRST(a__first(mark(x0), mark(x1)), mark(y1))
MARK(first(X1, X2)) → MARK(X2)
MARK(sel(s(x0), y1)) → A__SEL(s(mark(x0)), mark(y1))
A__FROM(X) → MARK(X)
MARK(first(from(x0), y1)) → A__FIRST(a__from(mark(x0)), mark(y1))
A__SEL(s(0), cons(y1, y2)) → A__SEL(0, mark(y2))
MARK(first(sel(y0, y1), first(x0, x1))) → A__FIRST(a__sel(mark(y0), mark(y1)), a__first(mark(x0), mark(x1)))
MARK(sel(first(x0, x1), y1)) → A__SEL(a__first(mark(x0), mark(x1)), mark(y1))
MARK(sel(sel(x0, x1), y1)) → A__SEL(a__sel(mark(x0), mark(x1)), mark(y1))
MARK(sel(X1, X2)) → MARK(X2)

The TRS R consists of the following rules:

a__from(X) → cons(mark(X), from(s(X)))
a__first(0, Z) → nil
a__first(s(X), cons(Y, Z)) → cons(mark(Y), first(X, Z))
a__sel(0, cons(X, Z)) → mark(X)
a__sel(s(X), cons(Y, Z)) → a__sel(mark(X), mark(Z))
mark(from(X)) → a__from(mark(X))
mark(first(X1, X2)) → a__first(mark(X1), mark(X2))
mark(sel(X1, X2)) → a__sel(mark(X1), mark(X2))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(s(X)) → s(mark(X))
mark(0) → 0
mark(nil) → nil
a__from(X) → from(X)
a__first(X1, X2) → first(X1, X2)
a__sel(X1, X2) → sel(X1, X2)

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

MARK(first(from(y0), 0)) → A__FIRST(a__from(mark(y0)), 0)
MARK(first(from(y0), nil)) → A__FIRST(a__from(mark(y0)), nil)
MARK(first(from(y0), cons(x0, x1))) → A__FIRST(a__from(mark(y0)), cons(mark(x0), x1))
MARK(first(from(y0), from(x0))) → A__FIRST(a__from(mark(y0)), a__from(mark(x0)))
MARK(first(from(y0), first(x0, x1))) → A__FIRST(a__from(mark(y0)), a__first(mark(x0), mark(x1)))
MARK(first(from(y0), sel(x0, x1))) → A__FIRST(a__from(mark(y0)), a__sel(mark(x0), mark(x1)))
MARK(first(from(y0), s(x0))) → A__FIRST(a__from(mark(y0)), s(mark(x0)))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ Narrowing

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ QDP

              ↳ Narrowing

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ QDP

                      ↳ Narrowing

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ QDP

                              ↳ Narrowing

                                ↳ QDP

                                  ↳ DependencyGraphProof

                                    ↳ QDP

                                      ↳ Narrowing

                                        ↳ QDP

                                          ↳ DependencyGraphProof

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

MARK(sel(from(x0), y1)) → A__SEL(a__from(mark(x0)), mark(y1))
MARK(s(X)) → MARK(X)
MARK(first(from(y0), first(x0, x1))) → A__FIRST(a__from(mark(y0)), a__first(mark(x0), mark(x1)))
MARK(cons(X1, X2)) → MARK(X1)
MARK(first(from(y0), sel(x0, x1))) → A__FIRST(a__from(mark(y0)), a__sel(mark(x0), mark(x1)))
MARK(first(sel(y0, y1), from(x0))) → A__FIRST(a__sel(mark(y0), mark(y1)), a__from(mark(x0)))
A__SEL(0, cons(X, Z)) → MARK(X)
A__FIRST(s(X), cons(Y, Z)) → MARK(Y)
A__SEL(s(sel(x0, x1)), cons(y1, y2)) → A__SEL(a__sel(mark(x0), mark(x1)), mark(y2))
A__SEL(s(X), cons(Y, Z)) → MARK(Z)
MARK(first(sel(y0, y1), cons(x0, x1))) → A__FIRST(a__sel(mark(y0), mark(y1)), cons(mark(x0), x1))
MARK(first(s(x0), y1)) → A__FIRST(s(mark(x0)), mark(y1))
MARK(first(from(y0), cons(x0, x1))) → A__FIRST(a__from(mark(y0)), cons(mark(x0), x1))
MARK(sel(0, y1)) → A__SEL(0, mark(y1))
A__SEL(s(X), cons(Y, Z)) → MARK(X)
MARK(first(sel(y0, y1), sel(x0, x1))) → A__FIRST(a__sel(mark(y0), mark(y1)), a__sel(mark(x0), mark(x1)))
A__SEL(s(from(x0)), cons(y1, y2)) → A__SEL(a__from(mark(x0)), mark(y2))
MARK(first(X1, X2)) → MARK(X1)
A__SEL(s(first(x0, x1)), cons(y1, y2)) → A__SEL(a__first(mark(x0), mark(x1)), mark(y2))
MARK(from(X)) → MARK(X)
MARK(sel(X1, X2)) → MARK(X1)
MARK(from(X)) → A__FROM(mark(X))
A__SEL(s(s(x0)), cons(y1, y2)) → A__SEL(s(mark(x0)), mark(y2))
MARK(first(from(y0), from(x0))) → A__FIRST(a__from(mark(y0)), a__from(mark(x0)))
MARK(first(first(x0, x1), y1)) → A__FIRST(a__first(mark(x0), mark(x1)), mark(y1))
MARK(first(X1, X2)) → MARK(X2)
MARK(sel(s(x0), y1)) → A__SEL(s(mark(x0)), mark(y1))
MARK(first(from(y0), s(x0))) → A__FIRST(a__from(mark(y0)), s(mark(x0)))
A__FROM(X) → MARK(X)
MARK(first(from(y0), 0)) → A__FIRST(a__from(mark(y0)), 0)
A__SEL(s(0), cons(y1, y2)) → A__SEL(0, mark(y2))
MARK(first(from(y0), nil)) → A__FIRST(a__from(mark(y0)), nil)
MARK(first(sel(y0, y1), first(x0, x1))) → A__FIRST(a__sel(mark(y0), mark(y1)), a__first(mark(x0), mark(x1)))
MARK(sel(sel(x0, x1), y1)) → A__SEL(a__sel(mark(x0), mark(x1)), mark(y1))
MARK(sel(first(x0, x1), y1)) → A__SEL(a__first(mark(x0), mark(x1)), mark(y1))
MARK(sel(X1, X2)) → MARK(X2)

The TRS R consists of the following rules:

a__from(X) → cons(mark(X), from(s(X)))
a__first(0, Z) → nil
a__first(s(X), cons(Y, Z)) → cons(mark(Y), first(X, Z))
a__sel(0, cons(X, Z)) → mark(X)
a__sel(s(X), cons(Y, Z)) → a__sel(mark(X), mark(Z))
mark(from(X)) → a__from(mark(X))
mark(first(X1, X2)) → a__first(mark(X1), mark(X2))
mark(sel(X1, X2)) → a__sel(mark(X1), mark(X2))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(s(X)) → s(mark(X))
mark(0) → 0
mark(nil) → nil
a__from(X) → from(X)
a__first(X1, X2) → first(X1, X2)
a__sel(X1, X2) → sel(X1, X2)

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ Narrowing

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ QDP

              ↳ Narrowing

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ QDP

                      ↳ Narrowing

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ QDP

                              ↳ Narrowing

                                ↳ QDP

                                  ↳ DependencyGraphProof

                                    ↳ QDP

                                      ↳ Narrowing

                                        ↳ QDP

                                          ↳ DependencyGraphProof

                                            ↳ QDP

                                              ↳ Narrowing

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

MARK(sel(from(x0), y1)) → A__SEL(a__from(mark(x0)), mark(y1))
MARK(s(X)) → MARK(X)
MARK(first(from(y0), first(x0, x1))) → A__FIRST(a__from(mark(y0)), a__first(mark(x0), mark(x1)))
MARK(cons(X1, X2)) → MARK(X1)
MARK(first(from(y0), sel(x0, x1))) → A__FIRST(a__from(mark(y0)), a__sel(mark(x0), mark(x1)))
MARK(first(sel(y0, y1), from(x0))) → A__FIRST(a__sel(mark(y0), mark(y1)), a__from(mark(x0)))
A__SEL(0, cons(X, Z)) → MARK(X)
A__FIRST(s(X), cons(Y, Z)) → MARK(Y)
MARK(first(sel(y0, y1), cons(x0, x1))) → A__FIRST(a__sel(mark(y0), mark(y1)), cons(mark(x0), x1))
A__SEL(s(X), cons(Y, Z)) → MARK(Z)
A__SEL(s(sel(x0, x1)), cons(y1, y2)) → A__SEL(a__sel(mark(x0), mark(x1)), mark(y2))
MARK(first(s(x0), y1)) → A__FIRST(s(mark(x0)), mark(y1))
MARK(first(from(y0), cons(x0, x1))) → A__FIRST(a__from(mark(y0)), cons(mark(x0), x1))
MARK(sel(0, y1)) → A__SEL(0, mark(y1))
A__SEL(s(X), cons(Y, Z)) → MARK(X)
MARK(first(sel(y0, y1), sel(x0, x1))) → A__FIRST(a__sel(mark(y0), mark(y1)), a__sel(mark(x0), mark(x1)))
MARK(first(X1, X2)) → MARK(X1)
A__SEL(s(from(x0)), cons(y1, y2)) → A__SEL(a__from(mark(x0)), mark(y2))
A__SEL(s(first(x0, x1)), cons(y1, y2)) → A__SEL(a__first(mark(x0), mark(x1)), mark(y2))
MARK(sel(X1, X2)) → MARK(X1)
MARK(from(X)) → MARK(X)
MARK(from(X)) → A__FROM(mark(X))
A__SEL(s(s(x0)), cons(y1, y2)) → A__SEL(s(mark(x0)), mark(y2))
MARK(first(from(y0), from(x0))) → A__FIRST(a__from(mark(y0)), a__from(mark(x0)))
MARK(first(first(x0, x1), y1)) → A__FIRST(a__first(mark(x0), mark(x1)), mark(y1))
MARK(first(X1, X2)) → MARK(X2)
MARK(sel(s(x0), y1)) → A__SEL(s(mark(x0)), mark(y1))
A__FROM(X) → MARK(X)
A__SEL(s(0), cons(y1, y2)) → A__SEL(0, mark(y2))
MARK(first(sel(y0, y1), first(x0, x1))) → A__FIRST(a__sel(mark(y0), mark(y1)), a__first(mark(x0), mark(x1)))
MARK(sel(first(x0, x1), y1)) → A__SEL(a__first(mark(x0), mark(x1)), mark(y1))
MARK(sel(sel(x0, x1), y1)) → A__SEL(a__sel(mark(x0), mark(x1)), mark(y1))
MARK(sel(X1, X2)) → MARK(X2)

The TRS R consists of the following rules:

a__from(X) → cons(mark(X), from(s(X)))
a__first(0, Z) → nil
a__first(s(X), cons(Y, Z)) → cons(mark(Y), first(X, Z))
a__sel(0, cons(X, Z)) → mark(X)
a__sel(s(X), cons(Y, Z)) → a__sel(mark(X), mark(Z))
mark(from(X)) → a__from(mark(X))
mark(first(X1, X2)) → a__first(mark(X1), mark(X2))
mark(sel(X1, X2)) → a__sel(mark(X1), mark(X2))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(s(X)) → s(mark(X))
mark(0) → 0
mark(nil) → nil
a__from(X) → from(X)
a__first(X1, X2) → first(X1, X2)
a__sel(X1, X2) → sel(X1, X2)

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

MARK(first(s(y0), 0)) → A__FIRST(s(mark(y0)), 0)
MARK(first(s(y0), first(x0, x1))) → A__FIRST(s(mark(y0)), a__first(mark(x0), mark(x1)))
MARK(first(s(y0), sel(x0, x1))) → A__FIRST(s(mark(y0)), a__sel(mark(x0), mark(x1)))
MARK(first(s(y0), nil)) → A__FIRST(s(mark(y0)), nil)
MARK(first(s(y0), s(x0))) → A__FIRST(s(mark(y0)), s(mark(x0)))
MARK(first(s(y0), cons(x0, x1))) → A__FIRST(s(mark(y0)), cons(mark(x0), x1))
MARK(first(s(y0), from(x0))) → A__FIRST(s(mark(y0)), a__from(mark(x0)))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ Narrowing

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ QDP

              ↳ Narrowing

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ QDP

                      ↳ Narrowing

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ QDP

                              ↳ Narrowing

                                ↳ QDP

                                  ↳ DependencyGraphProof

                                    ↳ QDP

                                      ↳ Narrowing

                                        ↳ QDP

                                          ↳ DependencyGraphProof

                                            ↳ QDP

                                              ↳ Narrowing

                                                ↳ QDP

                                                  ↳ DependencyGraphProof

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

MARK(sel(from(x0), y1)) → A__SEL(a__from(mark(x0)), mark(y1))
MARK(s(X)) → MARK(X)
MARK(cons(X1, X2)) → MARK(X1)
MARK(first(from(y0), first(x0, x1))) → A__FIRST(a__from(mark(y0)), a__first(mark(x0), mark(x1)))
MARK(first(s(y0), cons(x0, x1))) → A__FIRST(s(mark(y0)), cons(mark(x0), x1))
MARK(first(sel(y0, y1), from(x0))) → A__FIRST(a__sel(mark(y0), mark(y1)), a__from(mark(x0)))
MARK(first(from(y0), sel(x0, x1))) → A__FIRST(a__from(mark(y0)), a__sel(mark(x0), mark(x1)))
A__SEL(0, cons(X, Z)) → MARK(X)
A__FIRST(s(X), cons(Y, Z)) → MARK(Y)
A__SEL(s(sel(x0, x1)), cons(y1, y2)) → A__SEL(a__sel(mark(x0), mark(x1)), mark(y2))
A__SEL(s(X), cons(Y, Z)) → MARK(Z)
MARK(first(sel(y0, y1), cons(x0, x1))) → A__FIRST(a__sel(mark(y0), mark(y1)), cons(mark(x0), x1))
MARK(sel(0, y1)) → A__SEL(0, mark(y1))
MARK(first(from(y0), cons(x0, x1))) → A__FIRST(a__from(mark(y0)), cons(mark(x0), x1))
A__SEL(s(X), cons(Y, Z)) → MARK(X)
MARK(first(sel(y0, y1), sel(x0, x1))) → A__FIRST(a__sel(mark(y0), mark(y1)), a__sel(mark(x0), mark(x1)))
A__SEL(s(from(x0)), cons(y1, y2)) → A__SEL(a__from(mark(x0)), mark(y2))
MARK(first(X1, X2)) → MARK(X1)
A__SEL(s(first(x0, x1)), cons(y1, y2)) → A__SEL(a__first(mark(x0), mark(x1)), mark(y2))
MARK(from(X)) → MARK(X)
MARK(sel(X1, X2)) → MARK(X1)
A__SEL(s(s(x0)), cons(y1, y2)) → A__SEL(s(mark(x0)), mark(y2))
MARK(from(X)) → A__FROM(mark(X))
MARK(first(s(y0), first(x0, x1))) → A__FIRST(s(mark(y0)), a__first(mark(x0), mark(x1)))
MARK(first(s(y0), nil)) → A__FIRST(s(mark(y0)), nil)
MARK(first(s(y0), sel(x0, x1))) → A__FIRST(s(mark(y0)), a__sel(mark(x0), mark(x1)))
MARK(first(first(x0, x1), y1)) → A__FIRST(a__first(mark(x0), mark(x1)), mark(y1))
MARK(first(from(y0), from(x0))) → A__FIRST(a__from(mark(y0)), a__from(mark(x0)))
MARK(first(X1, X2)) → MARK(X2)
MARK(sel(s(x0), y1)) → A__SEL(s(mark(x0)), mark(y1))
MARK(first(s(y0), from(x0))) → A__FIRST(s(mark(y0)), a__from(mark(x0)))
A__FROM(X) → MARK(X)
MARK(first(s(y0), 0)) → A__FIRST(s(mark(y0)), 0)
A__SEL(s(0), cons(y1, y2)) → A__SEL(0, mark(y2))
MARK(first(sel(y0, y1), first(x0, x1))) → A__FIRST(a__sel(mark(y0), mark(y1)), a__first(mark(x0), mark(x1)))
MARK(first(s(y0), s(x0))) → A__FIRST(s(mark(y0)), s(mark(x0)))
MARK(sel(sel(x0, x1), y1)) → A__SEL(a__sel(mark(x0), mark(x1)), mark(y1))
MARK(sel(first(x0, x1), y1)) → A__SEL(a__first(mark(x0), mark(x1)), mark(y1))
MARK(sel(X1, X2)) → MARK(X2)

The TRS R consists of the following rules:

a__from(X) → cons(mark(X), from(s(X)))
a__first(0, Z) → nil
a__first(s(X), cons(Y, Z)) → cons(mark(Y), first(X, Z))
a__sel(0, cons(X, Z)) → mark(X)
a__sel(s(X), cons(Y, Z)) → a__sel(mark(X), mark(Z))
mark(from(X)) → a__from(mark(X))
mark(first(X1, X2)) → a__first(mark(X1), mark(X2))
mark(sel(X1, X2)) → a__sel(mark(X1), mark(X2))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(s(X)) → s(mark(X))
mark(0) → 0
mark(nil) → nil
a__from(X) → from(X)
a__first(X1, X2) → first(X1, X2)
a__sel(X1, X2) → sel(X1, X2)

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ Narrowing

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ QDP

              ↳ Narrowing

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ QDP

                      ↳ Narrowing

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ QDP

                              ↳ Narrowing

                                ↳ QDP

                                  ↳ DependencyGraphProof

                                    ↳ QDP

                                      ↳ Narrowing

                                        ↳ QDP

                                          ↳ DependencyGraphProof

                                            ↳ QDP

                                              ↳ Narrowing

                                                ↳ QDP

                                                  ↳ DependencyGraphProof

                                                    ↳ QDP

                                                      ↳ Narrowing

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

MARK(sel(from(x0), y1)) → A__SEL(a__from(mark(x0)), mark(y1))
MARK(s(X)) → MARK(X)
MARK(first(from(y0), first(x0, x1))) → A__FIRST(a__from(mark(y0)), a__first(mark(x0), mark(x1)))
MARK(cons(X1, X2)) → MARK(X1)
MARK(first(s(y0), cons(x0, x1))) → A__FIRST(s(mark(y0)), cons(mark(x0), x1))
MARK(first(from(y0), sel(x0, x1))) → A__FIRST(a__from(mark(y0)), a__sel(mark(x0), mark(x1)))
MARK(first(sel(y0, y1), from(x0))) → A__FIRST(a__sel(mark(y0), mark(y1)), a__from(mark(x0)))
A__SEL(0, cons(X, Z)) → MARK(X)
A__FIRST(s(X), cons(Y, Z)) → MARK(Y)
MARK(first(sel(y0, y1), cons(x0, x1))) → A__FIRST(a__sel(mark(y0), mark(y1)), cons(mark(x0), x1))
A__SEL(s(X), cons(Y, Z)) → MARK(Z)
A__SEL(s(sel(x0, x1)), cons(y1, y2)) → A__SEL(a__sel(mark(x0), mark(x1)), mark(y2))
MARK(first(from(y0), cons(x0, x1))) → A__FIRST(a__from(mark(y0)), cons(mark(x0), x1))
MARK(sel(0, y1)) → A__SEL(0, mark(y1))
A__SEL(s(X), cons(Y, Z)) → MARK(X)
MARK(first(sel(y0, y1), sel(x0, x1))) → A__FIRST(a__sel(mark(y0), mark(y1)), a__sel(mark(x0), mark(x1)))
MARK(first(X1, X2)) → MARK(X1)
A__SEL(s(from(x0)), cons(y1, y2)) → A__SEL(a__from(mark(x0)), mark(y2))
A__SEL(s(first(x0, x1)), cons(y1, y2)) → A__SEL(a__first(mark(x0), mark(x1)), mark(y2))
MARK(sel(X1, X2)) → MARK(X1)
MARK(from(X)) → MARK(X)
MARK(from(X)) → A__FROM(mark(X))
A__SEL(s(s(x0)), cons(y1, y2)) → A__SEL(s(mark(x0)), mark(y2))
MARK(first(s(y0), first(x0, x1))) → A__FIRST(s(mark(y0)), a__first(mark(x0), mark(x1)))
MARK(first(s(y0), sel(x0, x1))) → A__FIRST(s(mark(y0)), a__sel(mark(x0), mark(x1)))
MARK(first(from(y0), from(x0))) → A__FIRST(a__from(mark(y0)), a__from(mark(x0)))
MARK(first(first(x0, x1), y1)) → A__FIRST(a__first(mark(x0), mark(x1)), mark(y1))
MARK(first(X1, X2)) → MARK(X2)
MARK(sel(s(x0), y1)) → A__SEL(s(mark(x0)), mark(y1))
MARK(first(s(y0), from(x0))) → A__FIRST(s(mark(y0)), a__from(mark(x0)))
A__FROM(X) → MARK(X)
A__SEL(s(0), cons(y1, y2)) → A__SEL(0, mark(y2))
MARK(first(sel(y0, y1), first(x0, x1))) → A__FIRST(a__sel(mark(y0), mark(y1)), a__first(mark(x0), mark(x1)))
MARK(sel(first(x0, x1), y1)) → A__SEL(a__first(mark(x0), mark(x1)), mark(y1))
MARK(sel(sel(x0, x1), y1)) → A__SEL(a__sel(mark(x0), mark(x1)), mark(y1))
MARK(sel(X1, X2)) → MARK(X2)

The TRS R consists of the following rules:

a__from(X) → cons(mark(X), from(s(X)))
a__first(0, Z) → nil
a__first(s(X), cons(Y, Z)) → cons(mark(Y), first(X, Z))
a__sel(0, cons(X, Z)) → mark(X)
a__sel(s(X), cons(Y, Z)) → a__sel(mark(X), mark(Z))
mark(from(X)) → a__from(mark(X))
mark(first(X1, X2)) → a__first(mark(X1), mark(X2))
mark(sel(X1, X2)) → a__sel(mark(X1), mark(X2))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(s(X)) → s(mark(X))
mark(0) → 0
mark(nil) → nil
a__from(X) → from(X)
a__first(X1, X2) → first(X1, X2)
a__sel(X1, X2) → sel(X1, X2)

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

MARK(first(first(y0, y1), cons(x0, x1))) → A__FIRST(a__first(mark(y0), mark(y1)), cons(mark(x0), x1))
MARK(first(first(y0, y1), nil)) → A__FIRST(a__first(mark(y0), mark(y1)), nil)
MARK(first(first(y0, y1), s(x0))) → A__FIRST(a__first(mark(y0), mark(y1)), s(mark(x0)))
MARK(first(first(y0, y1), from(x0))) → A__FIRST(a__first(mark(y0), mark(y1)), a__from(mark(x0)))
MARK(first(first(y0, y1), 0)) → A__FIRST(a__first(mark(y0), mark(y1)), 0)
MARK(first(first(y0, y1), first(x0, x1))) → A__FIRST(a__first(mark(y0), mark(y1)), a__first(mark(x0), mark(x1)))
MARK(first(first(y0, y1), sel(x0, x1))) → A__FIRST(a__first(mark(y0), mark(y1)), a__sel(mark(x0), mark(x1)))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ Narrowing

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ QDP

              ↳ Narrowing

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ QDP

                      ↳ Narrowing

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ QDP

                              ↳ Narrowing

                                ↳ QDP

                                  ↳ DependencyGraphProof

                                    ↳ QDP

                                      ↳ Narrowing

                                        ↳ QDP

                                          ↳ DependencyGraphProof

                                            ↳ QDP

                                              ↳ Narrowing

                                                ↳ QDP

                                                  ↳ DependencyGraphProof

                                                    ↳ QDP

                                                      ↳ Narrowing

                                                        ↳ QDP

                                                          ↳ DependencyGraphProof

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

MARK(sel(from(x0), y1)) → A__SEL(a__from(mark(x0)), mark(y1))
MARK(s(X)) → MARK(X)
MARK(first(first(y0, y1), from(x0))) → A__FIRST(a__first(mark(y0), mark(y1)), a__from(mark(x0)))
MARK(cons(X1, X2)) → MARK(X1)
MARK(first(from(y0), first(x0, x1))) → A__FIRST(a__from(mark(y0)), a__first(mark(x0), mark(x1)))
MARK(first(sel(y0, y1), from(x0))) → A__FIRST(a__sel(mark(y0), mark(y1)), a__from(mark(x0)))
MARK(first(from(y0), sel(x0, x1))) → A__FIRST(a__from(mark(y0)), a__sel(mark(x0), mark(x1)))
MARK(first(s(y0), cons(x0, x1))) → A__FIRST(s(mark(y0)), cons(mark(x0), x1))
A__SEL(0, cons(X, Z)) → MARK(X)
A__FIRST(s(X), cons(Y, Z)) → MARK(Y)
A__SEL(s(sel(x0, x1)), cons(y1, y2)) → A__SEL(a__sel(mark(x0), mark(x1)), mark(y2))
A__SEL(s(X), cons(Y, Z)) → MARK(Z)
MARK(first(sel(y0, y1), cons(x0, x1))) → A__FIRST(a__sel(mark(y0), mark(y1)), cons(mark(x0), x1))
MARK(sel(0, y1)) → A__SEL(0, mark(y1))
MARK(first(from(y0), cons(x0, x1))) → A__FIRST(a__from(mark(y0)), cons(mark(x0), x1))
A__SEL(s(X), cons(Y, Z)) → MARK(X)
MARK(first(sel(y0, y1), sel(x0, x1))) → A__FIRST(a__sel(mark(y0), mark(y1)), a__sel(mark(x0), mark(x1)))
A__SEL(s(from(x0)), cons(y1, y2)) → A__SEL(a__from(mark(x0)), mark(y2))
MARK(first(X1, X2)) → MARK(X1)
MARK(first(first(y0, y1), cons(x0, x1))) → A__FIRST(a__first(mark(y0), mark(y1)), cons(mark(x0), x1))
A__SEL(s(first(x0, x1)), cons(y1, y2)) → A__SEL(a__first(mark(x0), mark(x1)), mark(y2))
MARK(from(X)) → MARK(X)
MARK(sel(X1, X2)) → MARK(X1)
MARK(first(first(y0, y1), nil)) → A__FIRST(a__first(mark(y0), mark(y1)), nil)
A__SEL(s(s(x0)), cons(y1, y2)) → A__SEL(s(mark(x0)), mark(y2))
MARK(from(X)) → A__FROM(mark(X))
MARK(first(first(y0, y1), s(x0))) → A__FIRST(a__first(mark(y0), mark(y1)), s(mark(x0)))
MARK(first(s(y0), first(x0, x1))) → A__FIRST(s(mark(y0)), a__first(mark(x0), mark(x1)))
MARK(first(from(y0), from(x0))) → A__FIRST(a__from(mark(y0)), a__from(mark(x0)))
MARK(first(s(y0), sel(x0, x1))) → A__FIRST(s(mark(y0)), a__sel(mark(x0), mark(x1)))
MARK(first(X1, X2)) → MARK(X2)
MARK(sel(s(x0), y1)) → A__SEL(s(mark(x0)), mark(y1))
A__FROM(X) → MARK(X)
MARK(first(s(y0), from(x0))) → A__FIRST(s(mark(y0)), a__from(mark(x0)))
MARK(first(first(y0, y1), 0)) → A__FIRST(a__first(mark(y0), mark(y1)), 0)
A__SEL(s(0), cons(y1, y2)) → A__SEL(0, mark(y2))
MARK(first(first(y0, y1), first(x0, x1))) → A__FIRST(a__first(mark(y0), mark(y1)), a__first(mark(x0), mark(x1)))
MARK(first(first(y0, y1), sel(x0, x1))) → A__FIRST(a__first(mark(y0), mark(y1)), a__sel(mark(x0), mark(x1)))
MARK(first(sel(y0, y1), first(x0, x1))) → A__FIRST(a__sel(mark(y0), mark(y1)), a__first(mark(x0), mark(x1)))
MARK(sel(sel(x0, x1), y1)) → A__SEL(a__sel(mark(x0), mark(x1)), mark(y1))
MARK(sel(first(x0, x1), y1)) → A__SEL(a__first(mark(x0), mark(x1)), mark(y1))
MARK(sel(X1, X2)) → MARK(X2)

The TRS R consists of the following rules:

a__from(X) → cons(mark(X), from(s(X)))
a__first(0, Z) → nil
a__first(s(X), cons(Y, Z)) → cons(mark(Y), first(X, Z))
a__sel(0, cons(X, Z)) → mark(X)
a__sel(s(X), cons(Y, Z)) → a__sel(mark(X), mark(Z))
mark(from(X)) → a__from(mark(X))
mark(first(X1, X2)) → a__first(mark(X1), mark(X2))
mark(sel(X1, X2)) → a__sel(mark(X1), mark(X2))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(s(X)) → s(mark(X))
mark(0) → 0
mark(nil) → nil
a__from(X) → from(X)
a__first(X1, X2) → first(X1, X2)
a__sel(X1, X2) → sel(X1, X2)

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ Narrowing

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ QDP

              ↳ Narrowing

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ QDP

                      ↳ Narrowing

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ QDP

                              ↳ Narrowing

                                ↳ QDP

                                  ↳ DependencyGraphProof

                                    ↳ QDP

                                      ↳ Narrowing

                                        ↳ QDP

                                          ↳ DependencyGraphProof

                                            ↳ QDP

                                              ↳ Narrowing

                                                ↳ QDP

                                                  ↳ DependencyGraphProof

                                                    ↳ QDP

                                                      ↳ Narrowing

                                                        ↳ QDP

                                                          ↳ DependencyGraphProof

                                                            ↳ QDP

                                                              ↳ Narrowing

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

MARK(sel(from(x0), y1)) → A__SEL(a__from(mark(x0)), mark(y1))
MARK(s(X)) → MARK(X)
MARK(first(first(y0, y1), from(x0))) → A__FIRST(a__first(mark(y0), mark(y1)), a__from(mark(x0)))
MARK(first(from(y0), first(x0, x1))) → A__FIRST(a__from(mark(y0)), a__first(mark(x0), mark(x1)))
MARK(cons(X1, X2)) → MARK(X1)
MARK(first(s(y0), cons(x0, x1))) → A__FIRST(s(mark(y0)), cons(mark(x0), x1))
MARK(first(from(y0), sel(x0, x1))) → A__FIRST(a__from(mark(y0)), a__sel(mark(x0), mark(x1)))
MARK(first(sel(y0, y1), from(x0))) → A__FIRST(a__sel(mark(y0), mark(y1)), a__from(mark(x0)))
A__SEL(0, cons(X, Z)) → MARK(X)
A__FIRST(s(X), cons(Y, Z)) → MARK(Y)
A__SEL(s(X), cons(Y, Z)) → MARK(Z)
A__SEL(s(sel(x0, x1)), cons(y1, y2)) → A__SEL(a__sel(mark(x0), mark(x1)), mark(y2))
MARK(first(sel(y0, y1), cons(x0, x1))) → A__FIRST(a__sel(mark(y0), mark(y1)), cons(mark(x0), x1))
MARK(first(from(y0), cons(x0, x1))) → A__FIRST(a__from(mark(y0)), cons(mark(x0), x1))
MARK(sel(0, y1)) → A__SEL(0, mark(y1))
A__SEL(s(X), cons(Y, Z)) → MARK(X)
MARK(first(sel(y0, y1), sel(x0, x1))) → A__FIRST(a__sel(mark(y0), mark(y1)), a__sel(mark(x0), mark(x1)))
MARK(first(X1, X2)) → MARK(X1)
A__SEL(s(from(x0)), cons(y1, y2)) → A__SEL(a__from(mark(x0)), mark(y2))
MARK(first(first(y0, y1), cons(x0, x1))) → A__FIRST(a__first(mark(y0), mark(y1)), cons(mark(x0), x1))
A__SEL(s(first(x0, x1)), cons(y1, y2)) → A__SEL(a__first(mark(x0), mark(x1)), mark(y2))
MARK(sel(X1, X2)) → MARK(X1)
MARK(from(X)) → MARK(X)
MARK(from(X)) → A__FROM(mark(X))
A__SEL(s(s(x0)), cons(y1, y2)) → A__SEL(s(mark(x0)), mark(y2))
MARK(first(s(y0), first(x0, x1))) → A__FIRST(s(mark(y0)), a__first(mark(x0), mark(x1)))
MARK(first(s(y0), sel(x0, x1))) → A__FIRST(s(mark(y0)), a__sel(mark(x0), mark(x1)))
MARK(first(from(y0), from(x0))) → A__FIRST(a__from(mark(y0)), a__from(mark(x0)))
MARK(first(X1, X2)) → MARK(X2)
MARK(sel(s(x0), y1)) → A__SEL(s(mark(x0)), mark(y1))
MARK(first(s(y0), from(x0))) → A__FIRST(s(mark(y0)), a__from(mark(x0)))
A__FROM(X) → MARK(X)
A__SEL(s(0), cons(y1, y2)) → A__SEL(0, mark(y2))
MARK(first(first(y0, y1), first(x0, x1))) → A__FIRST(a__first(mark(y0), mark(y1)), a__first(mark(x0), mark(x1)))
MARK(first(first(y0, y1), sel(x0, x1))) → A__FIRST(a__first(mark(y0), mark(y1)), a__sel(mark(x0), mark(x1)))
MARK(first(sel(y0, y1), first(x0, x1))) → A__FIRST(a__sel(mark(y0), mark(y1)), a__first(mark(x0), mark(x1)))
MARK(sel(first(x0, x1), y1)) → A__SEL(a__first(mark(x0), mark(x1)), mark(y1))
MARK(sel(sel(x0, x1), y1)) → A__SEL(a__sel(mark(x0), mark(x1)), mark(y1))
MARK(sel(X1, X2)) → MARK(X2)

The TRS R consists of the following rules:

a__from(X) → cons(mark(X), from(s(X)))
a__first(0, Z) → nil
a__first(s(X), cons(Y, Z)) → cons(mark(Y), first(X, Z))
a__sel(0, cons(X, Z)) → mark(X)
a__sel(s(X), cons(Y, Z)) → a__sel(mark(X), mark(Z))
mark(from(X)) → a__from(mark(X))
mark(first(X1, X2)) → a__first(mark(X1), mark(X2))
mark(sel(X1, X2)) → a__sel(mark(X1), mark(X2))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(s(X)) → s(mark(X))
mark(0) → 0
mark(nil) → nil
a__from(X) → from(X)
a__first(X1, X2) → first(X1, X2)
a__sel(X1, X2) → sel(X1, X2)

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

MARK(sel(from(y0), cons(x0, x1))) → A__SEL(a__from(mark(y0)), cons(mark(x0), x1))
MARK(sel(from(y0), from(x0))) → A__SEL(a__from(mark(y0)), a__from(mark(x0)))
MARK(sel(from(y0), sel(x0, x1))) → A__SEL(a__from(mark(y0)), a__sel(mark(x0), mark(x1)))
MARK(sel(from(y0), nil)) → A__SEL(a__from(mark(y0)), nil)
MARK(sel(from(y0), s(x0))) → A__SEL(a__from(mark(y0)), s(mark(x0)))
MARK(sel(from(y0), 0)) → A__SEL(a__from(mark(y0)), 0)
MARK(sel(from(y0), first(x0, x1))) → A__SEL(a__from(mark(y0)), a__first(mark(x0), mark(x1)))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ Narrowing

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ QDP

              ↳ Narrowing

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ QDP

                      ↳ Narrowing

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ QDP

                              ↳ Narrowing

                                ↳ QDP

                                  ↳ DependencyGraphProof

                                    ↳ QDP

                                      ↳ Narrowing

                                        ↳ QDP

                                          ↳ DependencyGraphProof

                                            ↳ QDP

                                              ↳ Narrowing

                                                ↳ QDP

                                                  ↳ DependencyGraphProof

                                                    ↳ QDP

                                                      ↳ Narrowing

                                                        ↳ QDP

                                                          ↳ DependencyGraphProof

                                                            ↳ QDP

                                                              ↳ Narrowing

                                                                ↳ QDP

                                                                  ↳ DependencyGraphProof

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

MARK(sel(from(y0), cons(x0, x1))) → A__SEL(a__from(mark(y0)), cons(mark(x0), x1))
MARK(s(X)) → MARK(X)
MARK(sel(from(y0), nil)) → A__SEL(a__from(mark(y0)), nil)
MARK(cons(X1, X2)) → MARK(X1)
MARK(first(from(y0), first(x0, x1))) → A__FIRST(a__from(mark(y0)), a__first(mark(x0), mark(x1)))
MARK(first(first(y0, y1), from(x0))) → A__FIRST(a__first(mark(y0), mark(y1)), a__from(mark(x0)))
MARK(first(sel(y0, y1), from(x0))) → A__FIRST(a__sel(mark(y0), mark(y1)), a__from(mark(x0)))
MARK(first(from(y0), sel(x0, x1))) → A__FIRST(a__from(mark(y0)), a__sel(mark(x0), mark(x1)))
MARK(first(s(y0), cons(x0, x1))) → A__FIRST(s(mark(y0)), cons(mark(x0), x1))
MARK(sel(from(y0), 0)) → A__SEL(a__from(mark(y0)), 0)
A__SEL(0, cons(X, Z)) → MARK(X)
A__FIRST(s(X), cons(Y, Z)) → MARK(Y)
MARK(first(sel(y0, y1), cons(x0, x1))) → A__FIRST(a__sel(mark(y0), mark(y1)), cons(mark(x0), x1))
A__SEL(s(sel(x0, x1)), cons(y1, y2)) → A__SEL(a__sel(mark(x0), mark(x1)), mark(y2))
A__SEL(s(X), cons(Y, Z)) → MARK(Z)
MARK(sel(0, y1)) → A__SEL(0, mark(y1))
MARK(first(from(y0), cons(x0, x1))) → A__FIRST(a__from(mark(y0)), cons(mark(x0), x1))
MARK(sel(from(y0), s(x0))) → A__SEL(a__from(mark(y0)), s(mark(x0)))
A__SEL(s(X), cons(Y, Z)) → MARK(X)
MARK(first(sel(y0, y1), sel(x0, x1))) → A__FIRST(a__sel(mark(y0), mark(y1)), a__sel(mark(x0), mark(x1)))
A__SEL(s(from(x0)), cons(y1, y2)) → A__SEL(a__from(mark(x0)), mark(y2))
MARK(first(X1, X2)) → MARK(X1)
A__SEL(s(first(x0, x1)), cons(y1, y2)) → A__SEL(a__first(mark(x0), mark(x1)), mark(y2))
MARK(first(first(y0, y1), cons(x0, x1))) → A__FIRST(a__first(mark(y0), mark(y1)), cons(mark(x0), x1))
MARK(from(X)) → MARK(X)
MARK(sel(X1, X2)) → MARK(X1)
A__SEL(s(s(x0)), cons(y1, y2)) → A__SEL(s(mark(x0)), mark(y2))
MARK(from(X)) → A__FROM(mark(X))
MARK(sel(from(y0), from(x0))) → A__SEL(a__from(mark(y0)), a__from(mark(x0)))
MARK(first(s(y0), first(x0, x1))) → A__FIRST(s(mark(y0)), a__first(mark(x0), mark(x1)))
MARK(first(from(y0), from(x0))) → A__FIRST(a__from(mark(y0)), a__from(mark(x0)))
MARK(first(s(y0), sel(x0, x1))) → A__FIRST(s(mark(y0)), a__sel(mark(x0), mark(x1)))
MARK(first(X1, X2)) → MARK(X2)
MARK(sel(s(x0), y1)) → A__SEL(s(mark(x0)), mark(y1))
A__FROM(X) → MARK(X)
MARK(first(s(y0), from(x0))) → A__FIRST(s(mark(y0)), a__from(mark(x0)))
A__SEL(s(0), cons(y1, y2)) → A__SEL(0, mark(y2))
MARK(first(first(y0, y1), first(x0, x1))) → A__FIRST(a__first(mark(y0), mark(y1)), a__first(mark(x0), mark(x1)))
MARK(sel(from(y0), sel(x0, x1))) → A__SEL(a__from(mark(y0)), a__sel(mark(x0), mark(x1)))
MARK(first(sel(y0, y1), first(x0, x1))) → A__FIRST(a__sel(mark(y0), mark(y1)), a__first(mark(x0), mark(x1)))
MARK(first(first(y0, y1), sel(x0, x1))) → A__FIRST(a__first(mark(y0), mark(y1)), a__sel(mark(x0), mark(x1)))
MARK(sel(sel(x0, x1), y1)) → A__SEL(a__sel(mark(x0), mark(x1)), mark(y1))
MARK(sel(first(x0, x1), y1)) → A__SEL(a__first(mark(x0), mark(x1)), mark(y1))
MARK(sel(X1, X2)) → MARK(X2)
MARK(sel(from(y0), first(x0, x1))) → A__SEL(a__from(mark(y0)), a__first(mark(x0), mark(x1)))

The TRS R consists of the following rules:

a__from(X) → cons(mark(X), from(s(X)))
a__first(0, Z) → nil
a__first(s(X), cons(Y, Z)) → cons(mark(Y), first(X, Z))
a__sel(0, cons(X, Z)) → mark(X)
a__sel(s(X), cons(Y, Z)) → a__sel(mark(X), mark(Z))
mark(from(X)) → a__from(mark(X))
mark(first(X1, X2)) → a__first(mark(X1), mark(X2))
mark(sel(X1, X2)) → a__sel(mark(X1), mark(X2))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(s(X)) → s(mark(X))
mark(0) → 0
mark(nil) → nil
a__from(X) → from(X)
a__first(X1, X2) → first(X1, X2)
a__sel(X1, X2) → sel(X1, X2)

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ Narrowing

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ QDP

              ↳ Narrowing

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ QDP

                      ↳ Narrowing

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ QDP

                              ↳ Narrowing

                                ↳ QDP

                                  ↳ DependencyGraphProof

                                    ↳ QDP

                                      ↳ Narrowing

                                        ↳ QDP

                                          ↳ DependencyGraphProof

                                            ↳ QDP

                                              ↳ Narrowing

                                                ↳ QDP

                                                  ↳ DependencyGraphProof

                                                    ↳ QDP

                                                      ↳ Narrowing

                                                        ↳ QDP

                                                          ↳ DependencyGraphProof

                                                            ↳ QDP

                                                              ↳ Narrowing

                                                                ↳ QDP

                                                                  ↳ DependencyGraphProof

                                                                    ↳ QDP

                                                                      ↳ Narrowing

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

MARK(sel(from(y0), cons(x0, x1))) → A__SEL(a__from(mark(y0)), cons(mark(x0), x1))
MARK(s(X)) → MARK(X)
MARK(first(first(y0, y1), from(x0))) → A__FIRST(a__first(mark(y0), mark(y1)), a__from(mark(x0)))
MARK(first(from(y0), first(x0, x1))) → A__FIRST(a__from(mark(y0)), a__first(mark(x0), mark(x1)))
MARK(cons(X1, X2)) → MARK(X1)
MARK(first(s(y0), cons(x0, x1))) → A__FIRST(s(mark(y0)), cons(mark(x0), x1))
MARK(first(from(y0), sel(x0, x1))) → A__FIRST(a__from(mark(y0)), a__sel(mark(x0), mark(x1)))
MARK(first(sel(y0, y1), from(x0))) → A__FIRST(a__sel(mark(y0), mark(y1)), a__from(mark(x0)))
A__SEL(0, cons(X, Z)) → MARK(X)
A__FIRST(s(X), cons(Y, Z)) → MARK(Y)
A__SEL(s(X), cons(Y, Z)) → MARK(Z)
A__SEL(s(sel(x0, x1)), cons(y1, y2)) → A__SEL(a__sel(mark(x0), mark(x1)), mark(y2))
MARK(first(sel(y0, y1), cons(x0, x1))) → A__FIRST(a__sel(mark(y0), mark(y1)), cons(mark(x0), x1))
MARK(first(from(y0), cons(x0, x1))) → A__FIRST(a__from(mark(y0)), cons(mark(x0), x1))
MARK(sel(0, y1)) → A__SEL(0, mark(y1))
A__SEL(s(X), cons(Y, Z)) → MARK(X)
MARK(first(sel(y0, y1), sel(x0, x1))) → A__FIRST(a__sel(mark(y0), mark(y1)), a__sel(mark(x0), mark(x1)))
MARK(first(X1, X2)) → MARK(X1)
A__SEL(s(from(x0)), cons(y1, y2)) → A__SEL(a__from(mark(x0)), mark(y2))
MARK(first(first(y0, y1), cons(x0, x1))) → A__FIRST(a__first(mark(y0), mark(y1)), cons(mark(x0), x1))
A__SEL(s(first(x0, x1)), cons(y1, y2)) → A__SEL(a__first(mark(x0), mark(x1)), mark(y2))
MARK(sel(X1, X2)) → MARK(X1)
MARK(from(X)) → MARK(X)
MARK(from(X)) → A__FROM(mark(X))
A__SEL(s(s(x0)), cons(y1, y2)) → A__SEL(s(mark(x0)), mark(y2))
MARK(sel(from(y0), from(x0))) → A__SEL(a__from(mark(y0)), a__from(mark(x0)))
MARK(first(s(y0), first(x0, x1))) → A__FIRST(s(mark(y0)), a__first(mark(x0), mark(x1)))
MARK(first(s(y0), sel(x0, x1))) → A__FIRST(s(mark(y0)), a__sel(mark(x0), mark(x1)))
MARK(first(from(y0), from(x0))) → A__FIRST(a__from(mark(y0)), a__from(mark(x0)))
MARK(first(X1, X2)) → MARK(X2)
MARK(sel(s(x0), y1)) → A__SEL(s(mark(x0)), mark(y1))
MARK(first(s(y0), from(x0))) → A__FIRST(s(mark(y0)), a__from(mark(x0)))
A__FROM(X) → MARK(X)
A__SEL(s(0), cons(y1, y2)) → A__SEL(0, mark(y2))
MARK(first(first(y0, y1), first(x0, x1))) → A__FIRST(a__first(mark(y0), mark(y1)), a__first(mark(x0), mark(x1)))
MARK(sel(from(y0), sel(x0, x1))) → A__SEL(a__from(mark(y0)), a__sel(mark(x0), mark(x1)))
MARK(first(first(y0, y1), sel(x0, x1))) → A__FIRST(a__first(mark(y0), mark(y1)), a__sel(mark(x0), mark(x1)))
MARK(first(sel(y0, y1), first(x0, x1))) → A__FIRST(a__sel(mark(y0), mark(y1)), a__first(mark(x0), mark(x1)))
MARK(sel(first(x0, x1), y1)) → A__SEL(a__first(mark(x0), mark(x1)), mark(y1))
MARK(sel(sel(x0, x1), y1)) → A__SEL(a__sel(mark(x0), mark(x1)), mark(y1))
MARK(sel(X1, X2)) → MARK(X2)
MARK(sel(from(y0), first(x0, x1))) → A__SEL(a__from(mark(y0)), a__first(mark(x0), mark(x1)))

The TRS R consists of the following rules:

a__from(X) → cons(mark(X), from(s(X)))
a__first(0, Z) → nil
a__first(s(X), cons(Y, Z)) → cons(mark(Y), first(X, Z))
a__sel(0, cons(X, Z)) → mark(X)
a__sel(s(X), cons(Y, Z)) → a__sel(mark(X), mark(Z))
mark(from(X)) → a__from(mark(X))
mark(first(X1, X2)) → a__first(mark(X1), mark(X2))
mark(sel(X1, X2)) → a__sel(mark(X1), mark(X2))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(s(X)) → s(mark(X))
mark(0) → 0
mark(nil) → nil
a__from(X) → from(X)
a__first(X1, X2) → first(X1, X2)
a__sel(X1, X2) → sel(X1, X2)

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

MARK(sel(0, sel(x0, x1))) → A__SEL(0, a__sel(mark(x0), mark(x1)))
MARK(sel(0, cons(x0, x1))) → A__SEL(0, cons(mark(x0), x1))
MARK(sel(0, 0)) → A__SEL(0, 0)
MARK(sel(0, s(x0))) → A__SEL(0, s(mark(x0)))
MARK(sel(0, nil)) → A__SEL(0, nil)
MARK(sel(0, first(x0, x1))) → A__SEL(0, a__first(mark(x0), mark(x1)))
MARK(sel(0, from(x0))) → A__SEL(0, a__from(mark(x0)))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ Narrowing

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ QDP

              ↳ Narrowing

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ QDP

                      ↳ Narrowing

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ QDP

                              ↳ Narrowing

                                ↳ QDP

                                  ↳ DependencyGraphProof

                                    ↳ QDP

                                      ↳ Narrowing

                                        ↳ QDP

                                          ↳ DependencyGraphProof

                                            ↳ QDP

                                              ↳ Narrowing

                                                ↳ QDP

                                                  ↳ DependencyGraphProof

                                                    ↳ QDP

                                                      ↳ Narrowing

                                                        ↳ QDP

                                                          ↳ DependencyGraphProof

                                                            ↳ QDP

                                                              ↳ Narrowing

                                                                ↳ QDP

                                                                  ↳ DependencyGraphProof

                                                                    ↳ QDP

                                                                      ↳ Narrowing

                                                                        ↳ QDP

                                                                          ↳ DependencyGraphProof

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

MARK(first(first(y0, y1), from(x0))) → A__FIRST(a__first(mark(y0), mark(y1)), a__from(mark(x0)))
MARK(first(from(y0), first(x0, x1))) → A__FIRST(a__from(mark(y0)), a__first(mark(x0), mark(x1)))
MARK(cons(X1, X2)) → MARK(X1)
MARK(first(from(y0), sel(x0, x1))) → A__FIRST(a__from(mark(y0)), a__sel(mark(x0), mark(x1)))
MARK(first(sel(y0, y1), from(x0))) → A__FIRST(a__sel(mark(y0), mark(y1)), a__from(mark(x0)))
A__SEL(s(X), cons(Y, Z)) → MARK(Z)
A__SEL(s(sel(x0, x1)), cons(y1, y2)) → A__SEL(a__sel(mark(x0), mark(x1)), mark(y2))
MARK(first(from(y0), cons(x0, x1))) → A__FIRST(a__from(mark(y0)), cons(mark(x0), x1))
MARK(first(sel(y0, y1), sel(x0, x1))) → A__FIRST(a__sel(mark(y0), mark(y1)), a__sel(mark(x0), mark(x1)))
A__SEL(s(from(x0)), cons(y1, y2)) → A__SEL(a__from(mark(x0)), mark(y2))
A__SEL(s(first(x0, x1)), cons(y1, y2)) → A__SEL(a__first(mark(x0), mark(x1)), mark(y2))
MARK(sel(X1, X2)) → MARK(X1)
MARK(first(from(y0), from(x0))) → A__FIRST(a__from(mark(y0)), a__from(mark(x0)))
MARK(first(X1, X2)) → MARK(X2)
MARK(sel(s(x0), y1)) → A__SEL(s(mark(x0)), mark(y1))
MARK(first(s(y0), from(x0))) → A__FIRST(s(mark(y0)), a__from(mark(x0)))
A__FROM(X) → MARK(X)
A__SEL(s(0), cons(y1, y2)) → A__SEL(0, mark(y2))
MARK(first(first(y0, y1), first(x0, x1))) → A__FIRST(a__first(mark(y0), mark(y1)), a__first(mark(x0), mark(x1)))
MARK(sel(from(y0), sel(x0, x1))) → A__SEL(a__from(mark(y0)), a__sel(mark(x0), mark(x1)))
MARK(sel(0, cons(x0, x1))) → A__SEL(0, cons(mark(x0), x1))
MARK(sel(0, 0)) → A__SEL(0, 0)
MARK(first(first(y0, y1), sel(x0, x1))) → A__FIRST(a__first(mark(y0), mark(y1)), a__sel(mark(x0), mark(x1)))
MARK(first(sel(y0, y1), first(x0, x1))) → A__FIRST(a__sel(mark(y0), mark(y1)), a__first(mark(x0), mark(x1)))
MARK(sel(sel(x0, x1), y1)) → A__SEL(a__sel(mark(x0), mark(x1)), mark(y1))
MARK(sel(from(y0), cons(x0, x1))) → A__SEL(a__from(mark(y0)), cons(mark(x0), x1))
MARK(s(X)) → MARK(X)
MARK(first(s(y0), cons(x0, x1))) → A__FIRST(s(mark(y0)), cons(mark(x0), x1))
A__SEL(0, cons(X, Z)) → MARK(X)
MARK(sel(0, from(x0))) → A__SEL(0, a__from(mark(x0)))
A__FIRST(s(X), cons(Y, Z)) → MARK(Y)
MARK(first(sel(y0, y1), cons(x0, x1))) → A__FIRST(a__sel(mark(y0), mark(y1)), cons(mark(x0), x1))
MARK(sel(0, nil)) → A__SEL(0, nil)
A__SEL(s(X), cons(Y, Z)) → MARK(X)
MARK(first(X1, X2)) → MARK(X1)
MARK(first(first(y0, y1), cons(x0, x1))) → A__FIRST(a__first(mark(y0), mark(y1)), cons(mark(x0), x1))
MARK(from(X)) → MARK(X)
MARK(from(X)) → A__FROM(mark(X))
A__SEL(s(s(x0)), cons(y1, y2)) → A__SEL(s(mark(x0)), mark(y2))
MARK(sel(from(y0), from(x0))) → A__SEL(a__from(mark(y0)), a__from(mark(x0)))
MARK(first(s(y0), first(x0, x1))) → A__FIRST(s(mark(y0)), a__first(mark(x0), mark(x1)))
MARK(first(s(y0), sel(x0, x1))) → A__FIRST(s(mark(y0)), a__sel(mark(x0), mark(x1)))
MARK(sel(0, s(x0))) → A__SEL(0, s(mark(x0)))
MARK(sel(0, first(x0, x1))) → A__SEL(0, a__first(mark(x0), mark(x1)))
MARK(sel(0, sel(x0, x1))) → A__SEL(0, a__sel(mark(x0), mark(x1)))
MARK(sel(first(x0, x1), y1)) → A__SEL(a__first(mark(x0), mark(x1)), mark(y1))
MARK(sel(X1, X2)) → MARK(X2)
MARK(sel(from(y0), first(x0, x1))) → A__SEL(a__from(mark(y0)), a__first(mark(x0), mark(x1)))

The TRS R consists of the following rules:

a__from(X) → cons(mark(X), from(s(X)))
a__first(0, Z) → nil
a__first(s(X), cons(Y, Z)) → cons(mark(Y), first(X, Z))
a__sel(0, cons(X, Z)) → mark(X)
a__sel(s(X), cons(Y, Z)) → a__sel(mark(X), mark(Z))
mark(from(X)) → a__from(mark(X))
mark(first(X1, X2)) → a__first(mark(X1), mark(X2))
mark(sel(X1, X2)) → a__sel(mark(X1), mark(X2))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(s(X)) → s(mark(X))
mark(0) → 0
mark(nil) → nil
a__from(X) → from(X)
a__first(X1, X2) → first(X1, X2)
a__sel(X1, X2) → sel(X1, X2)

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ Narrowing

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ QDP

              ↳ Narrowing

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ QDP

                      ↳ Narrowing

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ QDP

                              ↳ Narrowing

                                ↳ QDP

                                  ↳ DependencyGraphProof

                                    ↳ QDP

                                      ↳ Narrowing

                                        ↳ QDP

                                          ↳ DependencyGraphProof

                                            ↳ QDP

                                              ↳ Narrowing

                                                ↳ QDP

                                                  ↳ DependencyGraphProof

                                                    ↳ QDP

                                                      ↳ Narrowing

                                                        ↳ QDP

                                                          ↳ DependencyGraphProof

                                                            ↳ QDP

                                                              ↳ Narrowing

                                                                ↳ QDP

                                                                  ↳ DependencyGraphProof

                                                                    ↳ QDP

                                                                      ↳ Narrowing

                                                                        ↳ QDP

                                                                          ↳ DependencyGraphProof

                                                                            ↳ QDP

                                                                              ↳ Narrowing

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

MARK(sel(from(y0), cons(x0, x1))) → A__SEL(a__from(mark(y0)), cons(mark(x0), x1))
MARK(s(X)) → MARK(X)
MARK(first(first(y0, y1), from(x0))) → A__FIRST(a__first(mark(y0), mark(y1)), a__from(mark(x0)))
MARK(cons(X1, X2)) → MARK(X1)
MARK(first(from(y0), first(x0, x1))) → A__FIRST(a__from(mark(y0)), a__first(mark(x0), mark(x1)))
MARK(first(s(y0), cons(x0, x1))) → A__FIRST(s(mark(y0)), cons(mark(x0), x1))
MARK(first(sel(y0, y1), from(x0))) → A__FIRST(a__sel(mark(y0), mark(y1)), a__from(mark(x0)))
MARK(first(from(y0), sel(x0, x1))) → A__FIRST(a__from(mark(y0)), a__sel(mark(x0), mark(x1)))
MARK(sel(0, from(x0))) → A__SEL(0, a__from(mark(x0)))
A__SEL(0, cons(X, Z)) → MARK(X)
A__FIRST(s(X), cons(Y, Z)) → MARK(Y)
A__SEL(s(sel(x0, x1)), cons(y1, y2)) → A__SEL(a__sel(mark(x0), mark(x1)), mark(y2))
A__SEL(s(X), cons(Y, Z)) → MARK(Z)
MARK(first(sel(y0, y1), cons(x0, x1))) → A__FIRST(a__sel(mark(y0), mark(y1)), cons(mark(x0), x1))
MARK(first(from(y0), cons(x0, x1))) → A__FIRST(a__from(mark(y0)), cons(mark(x0), x1))
A__SEL(s(X), cons(Y, Z)) → MARK(X)
MARK(first(sel(y0, y1), sel(x0, x1))) → A__FIRST(a__sel(mark(y0), mark(y1)), a__sel(mark(x0), mark(x1)))
A__SEL(s(from(x0)), cons(y1, y2)) → A__SEL(a__from(mark(x0)), mark(y2))
MARK(first(X1, X2)) → MARK(X1)
A__SEL(s(first(x0, x1)), cons(y1, y2)) → A__SEL(a__first(mark(x0), mark(x1)), mark(y2))
MARK(first(first(y0, y1), cons(x0, x1))) → A__FIRST(a__first(mark(y0), mark(y1)), cons(mark(x0), x1))
MARK(sel(X1, X2)) → MARK(X1)
MARK(from(X)) → MARK(X)
A__SEL(s(s(x0)), cons(y1, y2)) → A__SEL(s(mark(x0)), mark(y2))
MARK(from(X)) → A__FROM(mark(X))
MARK(sel(from(y0), from(x0))) → A__SEL(a__from(mark(y0)), a__from(mark(x0)))
MARK(first(s(y0), first(x0, x1))) → A__FIRST(s(mark(y0)), a__first(mark(x0), mark(x1)))
MARK(first(s(y0), sel(x0, x1))) → A__FIRST(s(mark(y0)), a__sel(mark(x0), mark(x1)))
MARK(first(from(y0), from(x0))) → A__FIRST(a__from(mark(y0)), a__from(mark(x0)))
MARK(first(X1, X2)) → MARK(X2)
MARK(sel(s(x0), y1)) → A__SEL(s(mark(x0)), mark(y1))
MARK(sel(0, first(x0, x1))) → A__SEL(0, a__first(mark(x0), mark(x1)))
MARK(first(s(y0), from(x0))) → A__FIRST(s(mark(y0)), a__from(mark(x0)))
A__FROM(X) → MARK(X)
A__SEL(s(0), cons(y1, y2)) → A__SEL(0, mark(y2))
MARK(first(first(y0, y1), first(x0, x1))) → A__FIRST(a__first(mark(y0), mark(y1)), a__first(mark(x0), mark(x1)))
MARK(sel(0, sel(x0, x1))) → A__SEL(0, a__sel(mark(x0), mark(x1)))
MARK(sel(0, cons(x0, x1))) → A__SEL(0, cons(mark(x0), x1))
MARK(sel(from(y0), sel(x0, x1))) → A__SEL(a__from(mark(y0)), a__sel(mark(x0), mark(x1)))
MARK(first(first(y0, y1), sel(x0, x1))) → A__FIRST(a__first(mark(y0), mark(y1)), a__sel(mark(x0), mark(x1)))
MARK(first(sel(y0, y1), first(x0, x1))) → A__FIRST(a__sel(mark(y0), mark(y1)), a__first(mark(x0), mark(x1)))
MARK(sel(sel(x0, x1), y1)) → A__SEL(a__sel(mark(x0), mark(x1)), mark(y1))
MARK(sel(first(x0, x1), y1)) → A__SEL(a__first(mark(x0), mark(x1)), mark(y1))
MARK(sel(X1, X2)) → MARK(X2)
MARK(sel(from(y0), first(x0, x1))) → A__SEL(a__from(mark(y0)), a__first(mark(x0), mark(x1)))

The TRS R consists of the following rules:

a__from(X) → cons(mark(X), from(s(X)))
a__first(0, Z) → nil
a__first(s(X), cons(Y, Z)) → cons(mark(Y), first(X, Z))
a__sel(0, cons(X, Z)) → mark(X)
a__sel(s(X), cons(Y, Z)) → a__sel(mark(X), mark(Z))
mark(from(X)) → a__from(mark(X))
mark(first(X1, X2)) → a__first(mark(X1), mark(X2))
mark(sel(X1, X2)) → a__sel(mark(X1), mark(X2))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(s(X)) → s(mark(X))
mark(0) → 0
mark(nil) → nil
a__from(X) → from(X)
a__first(X1, X2) → first(X1, X2)
a__sel(X1, X2) → sel(X1, X2)

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

MARK(sel(first(y0, y1), s(x0))) → A__SEL(a__first(mark(y0), mark(y1)), s(mark(x0)))
MARK(sel(first(y0, y1), sel(x0, x1))) → A__SEL(a__first(mark(y0), mark(y1)), a__sel(mark(x0), mark(x1)))
MARK(sel(first(y0, y1), first(x0, x1))) → A__SEL(a__first(mark(y0), mark(y1)), a__first(mark(x0), mark(x1)))
MARK(sel(first(y0, y1), 0)) → A__SEL(a__first(mark(y0), mark(y1)), 0)
MARK(sel(first(y0, y1), nil)) → A__SEL(a__first(mark(y0), mark(y1)), nil)
MARK(sel(first(y0, y1), cons(x0, x1))) → A__SEL(a__first(mark(y0), mark(y1)), cons(mark(x0), x1))
MARK(sel(first(y0, y1), from(x0))) → A__SEL(a__first(mark(y0), mark(y1)), a__from(mark(x0)))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ Narrowing

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ QDP

              ↳ Narrowing

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ QDP

                      ↳ Narrowing

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ QDP

                              ↳ Narrowing

                                ↳ QDP

                                  ↳ DependencyGraphProof

                                    ↳ QDP

                                      ↳ Narrowing

                                        ↳ QDP

                                          ↳ DependencyGraphProof

                                            ↳ QDP

                                              ↳ Narrowing

                                                ↳ QDP

                                                  ↳ DependencyGraphProof

                                                    ↳ QDP

                                                      ↳ Narrowing

                                                        ↳ QDP

                                                          ↳ DependencyGraphProof

                                                            ↳ QDP

                                                              ↳ Narrowing

                                                                ↳ QDP

                                                                  ↳ DependencyGraphProof

                                                                    ↳ QDP

                                                                      ↳ Narrowing

                                                                        ↳ QDP

                                                                          ↳ DependencyGraphProof

                                                                            ↳ QDP

                                                                              ↳ Narrowing

                                                                                ↳ QDP

                                                                                  ↳ DependencyGraphProof

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

MARK(sel(first(y0, y1), first(x0, x1))) → A__SEL(a__first(mark(y0), mark(y1)), a__first(mark(x0), mark(x1)))
MARK(first(first(y0, y1), from(x0))) → A__FIRST(a__first(mark(y0), mark(y1)), a__from(mark(x0)))
MARK(cons(X1, X2)) → MARK(X1)
MARK(first(from(y0), first(x0, x1))) → A__FIRST(a__from(mark(y0)), a__first(mark(x0), mark(x1)))
MARK(first(sel(y0, y1), from(x0))) → A__FIRST(a__sel(mark(y0), mark(y1)), a__from(mark(x0)))
MARK(first(from(y0), sel(x0, x1))) → A__FIRST(a__from(mark(y0)), a__sel(mark(x0), mark(x1)))
A__SEL(s(sel(x0, x1)), cons(y1, y2)) → A__SEL(a__sel(mark(x0), mark(x1)), mark(y2))
A__SEL(s(X), cons(Y, Z)) → MARK(Z)
MARK(first(from(y0), cons(x0, x1))) → A__FIRST(a__from(mark(y0)), cons(mark(x0), x1))
MARK(first(sel(y0, y1), sel(x0, x1))) → A__FIRST(a__sel(mark(y0), mark(y1)), a__sel(mark(x0), mark(x1)))
A__SEL(s(from(x0)), cons(y1, y2)) → A__SEL(a__from(mark(x0)), mark(y2))
A__SEL(s(first(x0, x1)), cons(y1, y2)) → A__SEL(a__first(mark(x0), mark(x1)), mark(y2))
MARK(sel(X1, X2)) → MARK(X1)
MARK(first(from(y0), from(x0))) → A__FIRST(a__from(mark(y0)), a__from(mark(x0)))
MARK(first(X1, X2)) → MARK(X2)
MARK(sel(s(x0), y1)) → A__SEL(s(mark(x0)), mark(y1))
MARK(first(s(y0), from(x0))) → A__FIRST(s(mark(y0)), a__from(mark(x0)))
A__FROM(X) → MARK(X)
MARK(sel(first(y0, y1), nil)) → A__SEL(a__first(mark(y0), mark(y1)), nil)
A__SEL(s(0), cons(y1, y2)) → A__SEL(0, mark(y2))
MARK(first(first(y0, y1), first(x0, x1))) → A__FIRST(a__first(mark(y0), mark(y1)), a__first(mark(x0), mark(x1)))
MARK(sel(0, cons(x0, x1))) → A__SEL(0, cons(mark(x0), x1))
MARK(sel(from(y0), sel(x0, x1))) → A__SEL(a__from(mark(y0)), a__sel(mark(x0), mark(x1)))
MARK(first(first(y0, y1), sel(x0, x1))) → A__FIRST(a__first(mark(y0), mark(y1)), a__sel(mark(x0), mark(x1)))
MARK(first(sel(y0, y1), first(x0, x1))) → A__FIRST(a__sel(mark(y0), mark(y1)), a__first(mark(x0), mark(x1)))
MARK(sel(sel(x0, x1), y1)) → A__SEL(a__sel(mark(x0), mark(x1)), mark(y1))
MARK(sel(first(y0, y1), cons(x0, x1))) → A__SEL(a__first(mark(y0), mark(y1)), cons(mark(x0), x1))
MARK(sel(from(y0), cons(x0, x1))) → A__SEL(a__from(mark(y0)), cons(mark(x0), x1))
MARK(s(X)) → MARK(X)
MARK(sel(first(y0, y1), sel(x0, x1))) → A__SEL(a__first(mark(y0), mark(y1)), a__sel(mark(x0), mark(x1)))
MARK(first(s(y0), cons(x0, x1))) → A__FIRST(s(mark(y0)), cons(mark(x0), x1))
MARK(sel(0, from(x0))) → A__SEL(0, a__from(mark(x0)))
A__SEL(0, cons(X, Z)) → MARK(X)
A__FIRST(s(X), cons(Y, Z)) → MARK(Y)
MARK(first(sel(y0, y1), cons(x0, x1))) → A__FIRST(a__sel(mark(y0), mark(y1)), cons(mark(x0), x1))
A__SEL(s(X), cons(Y, Z)) → MARK(X)
MARK(first(X1, X2)) → MARK(X1)
MARK(first(first(y0, y1), cons(x0, x1))) → A__FIRST(a__first(mark(y0), mark(y1)), cons(mark(x0), x1))
MARK(from(X)) → MARK(X)
A__SEL(s(s(x0)), cons(y1, y2)) → A__SEL(s(mark(x0)), mark(y2))
MARK(from(X)) → A__FROM(mark(X))
MARK(sel(first(y0, y1), s(x0))) → A__SEL(a__first(mark(y0), mark(y1)), s(mark(x0)))
MARK(sel(from(y0), from(x0))) → A__SEL(a__from(mark(y0)), a__from(mark(x0)))
MARK(first(s(y0), first(x0, x1))) → A__FIRST(s(mark(y0)), a__first(mark(x0), mark(x1)))
MARK(first(s(y0), sel(x0, x1))) → A__FIRST(s(mark(y0)), a__sel(mark(x0), mark(x1)))
MARK(sel(0, first(x0, x1))) → A__SEL(0, a__first(mark(x0), mark(x1)))
MARK(sel(first(y0, y1), 0)) → A__SEL(a__first(mark(y0), mark(y1)), 0)
MARK(sel(0, sel(x0, x1))) → A__SEL(0, a__sel(mark(x0), mark(x1)))
MARK(sel(X1, X2)) → MARK(X2)
MARK(sel(first(y0, y1), from(x0))) → A__SEL(a__first(mark(y0), mark(y1)), a__from(mark(x0)))
MARK(sel(from(y0), first(x0, x1))) → A__SEL(a__from(mark(y0)), a__first(mark(x0), mark(x1)))

The TRS R consists of the following rules:

a__from(X) → cons(mark(X), from(s(X)))
a__first(0, Z) → nil
a__first(s(X), cons(Y, Z)) → cons(mark(Y), first(X, Z))
a__sel(0, cons(X, Z)) → mark(X)
a__sel(s(X), cons(Y, Z)) → a__sel(mark(X), mark(Z))
mark(from(X)) → a__from(mark(X))
mark(first(X1, X2)) → a__first(mark(X1), mark(X2))
mark(sel(X1, X2)) → a__sel(mark(X1), mark(X2))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(s(X)) → s(mark(X))
mark(0) → 0
mark(nil) → nil
a__from(X) → from(X)
a__first(X1, X2) → first(X1, X2)
a__sel(X1, X2) → sel(X1, X2)

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ Narrowing

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ QDP

              ↳ Narrowing

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ QDP

                      ↳ Narrowing

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ QDP

                              ↳ Narrowing

                                ↳ QDP

                                  ↳ DependencyGraphProof

                                    ↳ QDP

                                      ↳ Narrowing

                                        ↳ QDP

                                          ↳ DependencyGraphProof

                                            ↳ QDP

                                              ↳ Narrowing

                                                ↳ QDP

                                                  ↳ DependencyGraphProof

                                                    ↳ QDP

                                                      ↳ Narrowing

                                                        ↳ QDP

                                                          ↳ DependencyGraphProof

                                                            ↳ QDP

                                                              ↳ Narrowing

                                                                ↳ QDP

                                                                  ↳ DependencyGraphProof

                                                                    ↳ QDP

                                                                      ↳ Narrowing

                                                                        ↳ QDP

                                                                          ↳ DependencyGraphProof

                                                                            ↳ QDP

                                                                              ↳ Narrowing

                                                                                ↳ QDP

                                                                                  ↳ DependencyGraphProof

                                                                                    ↳ QDP

                                                                                      ↳ Narrowing

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

MARK(sel(from(y0), cons(x0, x1))) → A__SEL(a__from(mark(y0)), cons(mark(x0), x1))
MARK(s(X)) → MARK(X)
MARK(sel(first(y0, y1), first(x0, x1))) → A__SEL(a__first(mark(y0), mark(y1)), a__first(mark(x0), mark(x1)))
MARK(sel(first(y0, y1), sel(x0, x1))) → A__SEL(a__first(mark(y0), mark(y1)), a__sel(mark(x0), mark(x1)))
MARK(first(first(y0, y1), from(x0))) → A__FIRST(a__first(mark(y0), mark(y1)), a__from(mark(x0)))
MARK(cons(X1, X2)) → MARK(X1)
MARK(first(from(y0), first(x0, x1))) → A__FIRST(a__from(mark(y0)), a__first(mark(x0), mark(x1)))
MARK(first(sel(y0, y1), from(x0))) → A__FIRST(a__sel(mark(y0), mark(y1)), a__from(mark(x0)))
MARK(first(from(y0), sel(x0, x1))) → A__FIRST(a__from(mark(y0)), a__sel(mark(x0), mark(x1)))
MARK(first(s(y0), cons(x0, x1))) → A__FIRST(s(mark(y0)), cons(mark(x0), x1))
MARK(sel(0, from(x0))) → A__SEL(0, a__from(mark(x0)))
A__SEL(0, cons(X, Z)) → MARK(X)
A__FIRST(s(X), cons(Y, Z)) → MARK(Y)
A__SEL(s(sel(x0, x1)), cons(y1, y2)) → A__SEL(a__sel(mark(x0), mark(x1)), mark(y2))
A__SEL(s(X), cons(Y, Z)) → MARK(Z)
MARK(first(sel(y0, y1), cons(x0, x1))) → A__FIRST(a__sel(mark(y0), mark(y1)), cons(mark(x0), x1))
MARK(first(from(y0), cons(x0, x1))) → A__FIRST(a__from(mark(y0)), cons(mark(x0), x1))
A__SEL(s(X), cons(Y, Z)) → MARK(X)
MARK(first(sel(y0, y1), sel(x0, x1))) → A__FIRST(a__sel(mark(y0), mark(y1)), a__sel(mark(x0), mark(x1)))
A__SEL(s(from(x0)), cons(y1, y2)) → A__SEL(a__from(mark(x0)), mark(y2))
MARK(first(X1, X2)) → MARK(X1)
A__SEL(s(first(x0, x1)), cons(y1, y2)) → A__SEL(a__first(mark(x0), mark(x1)), mark(y2))
MARK(first(first(y0, y1), cons(x0, x1))) → A__FIRST(a__first(mark(y0), mark(y1)), cons(mark(x0), x1))
MARK(sel(X1, X2)) → MARK(X1)
MARK(from(X)) → MARK(X)
A__SEL(s(s(x0)), cons(y1, y2)) → A__SEL(s(mark(x0)), mark(y2))
MARK(from(X)) → A__FROM(mark(X))
MARK(sel(from(y0), from(x0))) → A__SEL(a__from(mark(y0)), a__from(mark(x0)))
MARK(first(s(y0), first(x0, x1))) → A__FIRST(s(mark(y0)), a__first(mark(x0), mark(x1)))
MARK(first(from(y0), from(x0))) → A__FIRST(a__from(mark(y0)), a__from(mark(x0)))
MARK(first(s(y0), sel(x0, x1))) → A__FIRST(s(mark(y0)), a__sel(mark(x0), mark(x1)))
MARK(first(X1, X2)) → MARK(X2)
MARK(sel(s(x0), y1)) → A__SEL(s(mark(x0)), mark(y1))
MARK(sel(0, first(x0, x1))) → A__SEL(0, a__first(mark(x0), mark(x1)))
MARK(first(s(y0), from(x0))) → A__FIRST(s(mark(y0)), a__from(mark(x0)))
A__FROM(X) → MARK(X)
A__SEL(s(0), cons(y1, y2)) → A__SEL(0, mark(y2))
MARK(first(first(y0, y1), first(x0, x1))) → A__FIRST(a__first(mark(y0), mark(y1)), a__first(mark(x0), mark(x1)))
MARK(sel(0, sel(x0, x1))) → A__SEL(0, a__sel(mark(x0), mark(x1)))
MARK(sel(0, cons(x0, x1))) → A__SEL(0, cons(mark(x0), x1))
MARK(sel(from(y0), sel(x0, x1))) → A__SEL(a__from(mark(y0)), a__sel(mark(x0), mark(x1)))
MARK(first(first(y0, y1), sel(x0, x1))) → A__FIRST(a__first(mark(y0), mark(y1)), a__sel(mark(x0), mark(x1)))
MARK(first(sel(y0, y1), first(x0, x1))) → A__FIRST(a__sel(mark(y0), mark(y1)), a__first(mark(x0), mark(x1)))
MARK(sel(sel(x0, x1), y1)) → A__SEL(a__sel(mark(x0), mark(x1)), mark(y1))
MARK(sel(first(y0, y1), cons(x0, x1))) → A__SEL(a__first(mark(y0), mark(y1)), cons(mark(x0), x1))
MARK(sel(X1, X2)) → MARK(X2)
MARK(sel(first(y0, y1), from(x0))) → A__SEL(a__first(mark(y0), mark(y1)), a__from(mark(x0)))
MARK(sel(from(y0), first(x0, x1))) → A__SEL(a__from(mark(y0)), a__first(mark(x0), mark(x1)))

The TRS R consists of the following rules:

a__from(X) → cons(mark(X), from(s(X)))
a__first(0, Z) → nil
a__first(s(X), cons(Y, Z)) → cons(mark(Y), first(X, Z))
a__sel(0, cons(X, Z)) → mark(X)
a__sel(s(X), cons(Y, Z)) → a__sel(mark(X), mark(Z))
mark(from(X)) → a__from(mark(X))
mark(first(X1, X2)) → a__first(mark(X1), mark(X2))
mark(sel(X1, X2)) → a__sel(mark(X1), mark(X2))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(s(X)) → s(mark(X))
mark(0) → 0
mark(nil) → nil
a__from(X) → from(X)
a__first(X1, X2) → first(X1, X2)
a__sel(X1, X2) → sel(X1, X2)

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

MARK(sel(sel(y0, y1), sel(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), a__sel(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), cons(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), cons(mark(x0), x1))
MARK(sel(sel(y0, y1), s(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), s(mark(x0)))
MARK(sel(sel(y0, y1), first(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), a__first(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), 0)) → A__SEL(a__sel(mark(y0), mark(y1)), 0)
MARK(sel(sel(y0, y1), from(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__from(mark(x0)))
MARK(sel(sel(y0, y1), nil)) → A__SEL(a__sel(mark(y0), mark(y1)), nil)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ Narrowing

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ QDP

              ↳ Narrowing

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ QDP

                      ↳ Narrowing

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ QDP

                              ↳ Narrowing

                                ↳ QDP

                                  ↳ DependencyGraphProof

                                    ↳ QDP

                                      ↳ Narrowing

                                        ↳ QDP

                                          ↳ DependencyGraphProof

                                            ↳ QDP

                                              ↳ Narrowing

                                                ↳ QDP

                                                  ↳ DependencyGraphProof

                                                    ↳ QDP

                                                      ↳ Narrowing

                                                        ↳ QDP

                                                          ↳ DependencyGraphProof

                                                            ↳ QDP

                                                              ↳ Narrowing

                                                                ↳ QDP

                                                                  ↳ DependencyGraphProof

                                                                    ↳ QDP

                                                                      ↳ Narrowing

                                                                        ↳ QDP

                                                                          ↳ DependencyGraphProof

                                                                            ↳ QDP

                                                                              ↳ Narrowing

                                                                                ↳ QDP

                                                                                  ↳ DependencyGraphProof

                                                                                    ↳ QDP

                                                                                      ↳ Narrowing

                                                                                        ↳ QDP

                                                                                          ↳ DependencyGraphProof

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

MARK(sel(sel(y0, y1), sel(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), a__sel(mark(x0), mark(x1)))
MARK(sel(first(y0, y1), first(x0, x1))) → A__SEL(a__first(mark(y0), mark(y1)), a__first(mark(x0), mark(x1)))
MARK(first(from(y0), first(x0, x1))) → A__FIRST(a__from(mark(y0)), a__first(mark(x0), mark(x1)))
MARK(cons(X1, X2)) → MARK(X1)
MARK(first(first(y0, y1), from(x0))) → A__FIRST(a__first(mark(y0), mark(y1)), a__from(mark(x0)))
MARK(first(from(y0), sel(x0, x1))) → A__FIRST(a__from(mark(y0)), a__sel(mark(x0), mark(x1)))
MARK(first(sel(y0, y1), from(x0))) → A__FIRST(a__sel(mark(y0), mark(y1)), a__from(mark(x0)))
A__SEL(s(X), cons(Y, Z)) → MARK(Z)
A__SEL(s(sel(x0, x1)), cons(y1, y2)) → A__SEL(a__sel(mark(x0), mark(x1)), mark(y2))
MARK(first(from(y0), cons(x0, x1))) → A__FIRST(a__from(mark(y0)), cons(mark(x0), x1))
MARK(first(sel(y0, y1), sel(x0, x1))) → A__FIRST(a__sel(mark(y0), mark(y1)), a__sel(mark(x0), mark(x1)))
A__SEL(s(from(x0)), cons(y1, y2)) → A__SEL(a__from(mark(x0)), mark(y2))
A__SEL(s(first(x0, x1)), cons(y1, y2)) → A__SEL(a__first(mark(x0), mark(x1)), mark(y2))
MARK(sel(X1, X2)) → MARK(X1)
MARK(sel(sel(y0, y1), cons(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), cons(mark(x0), x1))
MARK(first(from(y0), from(x0))) → A__FIRST(a__from(mark(y0)), a__from(mark(x0)))
MARK(first(X1, X2)) → MARK(X2)
MARK(sel(s(x0), y1)) → A__SEL(s(mark(x0)), mark(y1))
A__FROM(X) → MARK(X)
MARK(first(s(y0), from(x0))) → A__FIRST(s(mark(y0)), a__from(mark(x0)))
A__SEL(s(0), cons(y1, y2)) → A__SEL(0, mark(y2))
MARK(first(first(y0, y1), first(x0, x1))) → A__FIRST(a__first(mark(y0), mark(y1)), a__first(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), from(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__from(mark(x0)))
MARK(sel(from(y0), sel(x0, x1))) → A__SEL(a__from(mark(y0)), a__sel(mark(x0), mark(x1)))
MARK(sel(0, cons(x0, x1))) → A__SEL(0, cons(mark(x0), x1))
MARK(first(sel(y0, y1), first(x0, x1))) → A__FIRST(a__sel(mark(y0), mark(y1)), a__first(mark(x0), mark(x1)))
MARK(first(first(y0, y1), sel(x0, x1))) → A__FIRST(a__first(mark(y0), mark(y1)), a__sel(mark(x0), mark(x1)))
MARK(sel(first(y0, y1), cons(x0, x1))) → A__SEL(a__first(mark(y0), mark(y1)), cons(mark(x0), x1))
MARK(sel(from(y0), cons(x0, x1))) → A__SEL(a__from(mark(y0)), cons(mark(x0), x1))
MARK(s(X)) → MARK(X)
MARK(sel(sel(y0, y1), first(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), a__first(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), s(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), s(mark(x0)))
MARK(sel(first(y0, y1), sel(x0, x1))) → A__SEL(a__first(mark(y0), mark(y1)), a__sel(mark(x0), mark(x1)))
MARK(first(s(y0), cons(x0, x1))) → A__FIRST(s(mark(y0)), cons(mark(x0), x1))
A__SEL(0, cons(X, Z)) → MARK(X)
MARK(sel(0, from(x0))) → A__SEL(0, a__from(mark(x0)))
A__FIRST(s(X), cons(Y, Z)) → MARK(Y)
MARK(first(sel(y0, y1), cons(x0, x1))) → A__FIRST(a__sel(mark(y0), mark(y1)), cons(mark(x0), x1))
A__SEL(s(X), cons(Y, Z)) → MARK(X)
MARK(first(X1, X2)) → MARK(X1)
MARK(first(first(y0, y1), cons(x0, x1))) → A__FIRST(a__first(mark(y0), mark(y1)), cons(mark(x0), x1))
MARK(from(X)) → MARK(X)
MARK(from(X)) → A__FROM(mark(X))
A__SEL(s(s(x0)), cons(y1, y2)) → A__SEL(s(mark(x0)), mark(y2))
MARK(first(s(y0), first(x0, x1))) → A__FIRST(s(mark(y0)), a__first(mark(x0), mark(x1)))
MARK(sel(from(y0), from(x0))) → A__SEL(a__from(mark(y0)), a__from(mark(x0)))
MARK(first(s(y0), sel(x0, x1))) → A__FIRST(s(mark(y0)), a__sel(mark(x0), mark(x1)))
MARK(sel(0, first(x0, x1))) → A__SEL(0, a__first(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), 0)) → A__SEL(a__sel(mark(y0), mark(y1)), 0)
MARK(sel(0, sel(x0, x1))) → A__SEL(0, a__sel(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), nil)) → A__SEL(a__sel(mark(y0), mark(y1)), nil)
MARK(sel(X1, X2)) → MARK(X2)
MARK(sel(from(y0), first(x0, x1))) → A__SEL(a__from(mark(y0)), a__first(mark(x0), mark(x1)))
MARK(sel(first(y0, y1), from(x0))) → A__SEL(a__first(mark(y0), mark(y1)), a__from(mark(x0)))

The TRS R consists of the following rules:

a__from(X) → cons(mark(X), from(s(X)))
a__first(0, Z) → nil
a__first(s(X), cons(Y, Z)) → cons(mark(Y), first(X, Z))
a__sel(0, cons(X, Z)) → mark(X)
a__sel(s(X), cons(Y, Z)) → a__sel(mark(X), mark(Z))
mark(from(X)) → a__from(mark(X))
mark(first(X1, X2)) → a__first(mark(X1), mark(X2))
mark(sel(X1, X2)) → a__sel(mark(X1), mark(X2))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(s(X)) → s(mark(X))
mark(0) → 0
mark(nil) → nil
a__from(X) → from(X)
a__first(X1, X2) → first(X1, X2)
a__sel(X1, X2) → sel(X1, X2)

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ Narrowing

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ QDP

              ↳ Narrowing

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ QDP

                      ↳ Narrowing

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ QDP

                              ↳ Narrowing

                                ↳ QDP

                                  ↳ DependencyGraphProof

                                    ↳ QDP

                                      ↳ Narrowing

                                        ↳ QDP

                                          ↳ DependencyGraphProof

                                            ↳ QDP

                                              ↳ Narrowing

                                                ↳ QDP

                                                  ↳ DependencyGraphProof

                                                    ↳ QDP

                                                      ↳ Narrowing

                                                        ↳ QDP

                                                          ↳ DependencyGraphProof

                                                            ↳ QDP

                                                              ↳ Narrowing

                                                                ↳ QDP

                                                                  ↳ DependencyGraphProof

                                                                    ↳ QDP

                                                                      ↳ Narrowing

                                                                        ↳ QDP

                                                                          ↳ DependencyGraphProof

                                                                            ↳ QDP

                                                                              ↳ Narrowing

                                                                                ↳ QDP

                                                                                  ↳ DependencyGraphProof

                                                                                    ↳ QDP

                                                                                      ↳ Narrowing

                                                                                        ↳ QDP

                                                                                          ↳ DependencyGraphProof

                                                                                            ↳ QDP

                                                                                              ↳ Narrowing

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

MARK(sel(sel(y0, y1), sel(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), a__sel(mark(x0), mark(x1)))
MARK(sel(first(y0, y1), first(x0, x1))) → A__SEL(a__first(mark(y0), mark(y1)), a__first(mark(x0), mark(x1)))
MARK(first(from(y0), first(x0, x1))) → A__FIRST(a__from(mark(y0)), a__first(mark(x0), mark(x1)))
MARK(cons(X1, X2)) → MARK(X1)
MARK(first(first(y0, y1), from(x0))) → A__FIRST(a__first(mark(y0), mark(y1)), a__from(mark(x0)))
MARK(first(from(y0), sel(x0, x1))) → A__FIRST(a__from(mark(y0)), a__sel(mark(x0), mark(x1)))
MARK(first(sel(y0, y1), from(x0))) → A__FIRST(a__sel(mark(y0), mark(y1)), a__from(mark(x0)))
A__SEL(s(X), cons(Y, Z)) → MARK(Z)
A__SEL(s(sel(x0, x1)), cons(y1, y2)) → A__SEL(a__sel(mark(x0), mark(x1)), mark(y2))
MARK(first(from(y0), cons(x0, x1))) → A__FIRST(a__from(mark(y0)), cons(mark(x0), x1))
MARK(first(sel(y0, y1), sel(x0, x1))) → A__FIRST(a__sel(mark(y0), mark(y1)), a__sel(mark(x0), mark(x1)))
A__SEL(s(from(x0)), cons(y1, y2)) → A__SEL(a__from(mark(x0)), mark(y2))
A__SEL(s(first(x0, x1)), cons(y1, y2)) → A__SEL(a__first(mark(x0), mark(x1)), mark(y2))
MARK(sel(X1, X2)) → MARK(X1)
MARK(sel(sel(y0, y1), cons(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), cons(mark(x0), x1))
MARK(first(from(y0), from(x0))) → A__FIRST(a__from(mark(y0)), a__from(mark(x0)))
MARK(first(X1, X2)) → MARK(X2)
MARK(sel(s(x0), y1)) → A__SEL(s(mark(x0)), mark(y1))
MARK(first(s(y0), from(x0))) → A__FIRST(s(mark(y0)), a__from(mark(x0)))
A__FROM(X) → MARK(X)
A__SEL(s(0), cons(y1, y2)) → A__SEL(0, mark(y2))
MARK(first(first(y0, y1), first(x0, x1))) → A__FIRST(a__first(mark(y0), mark(y1)), a__first(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), from(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__from(mark(x0)))
MARK(sel(from(y0), sel(x0, x1))) → A__SEL(a__from(mark(y0)), a__sel(mark(x0), mark(x1)))
MARK(sel(0, cons(x0, x1))) → A__SEL(0, cons(mark(x0), x1))
MARK(first(sel(y0, y1), first(x0, x1))) → A__FIRST(a__sel(mark(y0), mark(y1)), a__first(mark(x0), mark(x1)))
MARK(first(first(y0, y1), sel(x0, x1))) → A__FIRST(a__first(mark(y0), mark(y1)), a__sel(mark(x0), mark(x1)))
MARK(sel(first(y0, y1), cons(x0, x1))) → A__SEL(a__first(mark(y0), mark(y1)), cons(mark(x0), x1))
MARK(sel(from(y0), cons(x0, x1))) → A__SEL(a__from(mark(y0)), cons(mark(x0), x1))
MARK(s(X)) → MARK(X)
MARK(sel(sel(y0, y1), first(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), a__first(mark(x0), mark(x1)))
MARK(sel(first(y0, y1), sel(x0, x1))) → A__SEL(a__first(mark(y0), mark(y1)), a__sel(mark(x0), mark(x1)))
MARK(first(s(y0), cons(x0, x1))) → A__FIRST(s(mark(y0)), cons(mark(x0), x1))
A__SEL(0, cons(X, Z)) → MARK(X)
MARK(sel(0, from(x0))) → A__SEL(0, a__from(mark(x0)))
A__FIRST(s(X), cons(Y, Z)) → MARK(Y)
MARK(first(sel(y0, y1), cons(x0, x1))) → A__FIRST(a__sel(mark(y0), mark(y1)), cons(mark(x0), x1))
A__SEL(s(X), cons(Y, Z)) → MARK(X)
MARK(first(X1, X2)) → MARK(X1)
MARK(first(first(y0, y1), cons(x0, x1))) → A__FIRST(a__first(mark(y0), mark(y1)), cons(mark(x0), x1))
MARK(from(X)) → MARK(X)
MARK(from(X)) → A__FROM(mark(X))
A__SEL(s(s(x0)), cons(y1, y2)) → A__SEL(s(mark(x0)), mark(y2))
MARK(first(s(y0), first(x0, x1))) → A__FIRST(s(mark(y0)), a__first(mark(x0), mark(x1)))
MARK(sel(from(y0), from(x0))) → A__SEL(a__from(mark(y0)), a__from(mark(x0)))
MARK(first(s(y0), sel(x0, x1))) → A__FIRST(s(mark(y0)), a__sel(mark(x0), mark(x1)))
MARK(sel(0, first(x0, x1))) → A__SEL(0, a__first(mark(x0), mark(x1)))
MARK(sel(0, sel(x0, x1))) → A__SEL(0, a__sel(mark(x0), mark(x1)))
MARK(sel(X1, X2)) → MARK(X2)
MARK(sel(from(y0), first(x0, x1))) → A__SEL(a__from(mark(y0)), a__first(mark(x0), mark(x1)))
MARK(sel(first(y0, y1), from(x0))) → A__SEL(a__first(mark(y0), mark(y1)), a__from(mark(x0)))

The TRS R consists of the following rules:

a__from(X) → cons(mark(X), from(s(X)))
a__first(0, Z) → nil
a__first(s(X), cons(Y, Z)) → cons(mark(Y), first(X, Z))
a__sel(0, cons(X, Z)) → mark(X)
a__sel(s(X), cons(Y, Z)) → a__sel(mark(X), mark(Z))
mark(from(X)) → a__from(mark(X))
mark(first(X1, X2)) → a__first(mark(X1), mark(X2))
mark(sel(X1, X2)) → a__sel(mark(X1), mark(X2))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(s(X)) → s(mark(X))
mark(0) → 0
mark(nil) → nil
a__from(X) → from(X)
a__first(X1, X2) → first(X1, X2)
a__sel(X1, X2) → sel(X1, X2)

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

MARK(sel(s(y0), first(x0, x1))) → A__SEL(s(mark(y0)), a__first(mark(x0), mark(x1)))
MARK(sel(s(y0), cons(x0, x1))) → A__SEL(s(mark(y0)), cons(mark(x0), x1))
MARK(sel(s(y0), from(x0))) → A__SEL(s(mark(y0)), a__from(mark(x0)))
MARK(sel(s(y0), sel(x0, x1))) → A__SEL(s(mark(y0)), a__sel(mark(x0), mark(x1)))
MARK(sel(s(y0), 0)) → A__SEL(s(mark(y0)), 0)
MARK(sel(s(y0), s(x0))) → A__SEL(s(mark(y0)), s(mark(x0)))
MARK(sel(s(y0), nil)) → A__SEL(s(mark(y0)), nil)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ Narrowing

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ QDP

              ↳ Narrowing

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ QDP

                      ↳ Narrowing

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ QDP

                              ↳ Narrowing

                                ↳ QDP

                                  ↳ DependencyGraphProof

                                    ↳ QDP

                                      ↳ Narrowing

                                        ↳ QDP

                                          ↳ DependencyGraphProof

                                            ↳ QDP

                                              ↳ Narrowing

                                                ↳ QDP

                                                  ↳ DependencyGraphProof

                                                    ↳ QDP

                                                      ↳ Narrowing

                                                        ↳ QDP

                                                          ↳ DependencyGraphProof

                                                            ↳ QDP

                                                              ↳ Narrowing

                                                                ↳ QDP

                                                                  ↳ DependencyGraphProof

                                                                    ↳ QDP

                                                                      ↳ Narrowing

                                                                        ↳ QDP

                                                                          ↳ DependencyGraphProof

                                                                            ↳ QDP

                                                                              ↳ Narrowing

                                                                                ↳ QDP

                                                                                  ↳ DependencyGraphProof

                                                                                    ↳ QDP

                                                                                      ↳ Narrowing

                                                                                        ↳ QDP

                                                                                          ↳ DependencyGraphProof

                                                                                            ↳ QDP

                                                                                              ↳ Narrowing

                                                                                                ↳ QDP

                                                                                                  ↳ DependencyGraphProof

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

MARK(sel(sel(y0, y1), sel(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), a__sel(mark(x0), mark(x1)))
MARK(sel(s(y0), cons(x0, x1))) → A__SEL(s(mark(y0)), cons(mark(x0), x1))
MARK(sel(first(y0, y1), first(x0, x1))) → A__SEL(a__first(mark(y0), mark(y1)), a__first(mark(x0), mark(x1)))
MARK(first(first(y0, y1), from(x0))) → A__FIRST(a__first(mark(y0), mark(y1)), a__from(mark(x0)))
MARK(cons(X1, X2)) → MARK(X1)
MARK(first(from(y0), first(x0, x1))) → A__FIRST(a__from(mark(y0)), a__first(mark(x0), mark(x1)))
MARK(first(sel(y0, y1), from(x0))) → A__FIRST(a__sel(mark(y0), mark(y1)), a__from(mark(x0)))
MARK(first(from(y0), sel(x0, x1))) → A__FIRST(a__from(mark(y0)), a__sel(mark(x0), mark(x1)))
A__SEL(s(sel(x0, x1)), cons(y1, y2)) → A__SEL(a__sel(mark(x0), mark(x1)), mark(y2))
A__SEL(s(X), cons(Y, Z)) → MARK(Z)
MARK(first(from(y0), cons(x0, x1))) → A__FIRST(a__from(mark(y0)), cons(mark(x0), x1))
MARK(sel(s(y0), from(x0))) → A__SEL(s(mark(y0)), a__from(mark(x0)))
MARK(first(sel(y0, y1), sel(x0, x1))) → A__FIRST(a__sel(mark(y0), mark(y1)), a__sel(mark(x0), mark(x1)))
A__SEL(s(from(x0)), cons(y1, y2)) → A__SEL(a__from(mark(x0)), mark(y2))
A__SEL(s(first(x0, x1)), cons(y1, y2)) → A__SEL(a__first(mark(x0), mark(x1)), mark(y2))
MARK(sel(X1, X2)) → MARK(X1)
MARK(sel(sel(y0, y1), cons(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), cons(mark(x0), x1))
MARK(first(from(y0), from(x0))) → A__FIRST(a__from(mark(y0)), a__from(mark(x0)))
MARK(first(X1, X2)) → MARK(X2)
A__FROM(X) → MARK(X)
MARK(first(s(y0), from(x0))) → A__FIRST(s(mark(y0)), a__from(mark(x0)))
A__SEL(s(0), cons(y1, y2)) → A__SEL(0, mark(y2))
MARK(first(first(y0, y1), first(x0, x1))) → A__FIRST(a__first(mark(y0), mark(y1)), a__first(mark(x0), mark(x1)))
MARK(sel(0, cons(x0, x1))) → A__SEL(0, cons(mark(x0), x1))
MARK(sel(from(y0), sel(x0, x1))) → A__SEL(a__from(mark(y0)), a__sel(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), from(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__from(mark(x0)))
MARK(first(first(y0, y1), sel(x0, x1))) → A__FIRST(a__first(mark(y0), mark(y1)), a__sel(mark(x0), mark(x1)))
MARK(first(sel(y0, y1), first(x0, x1))) → A__FIRST(a__sel(mark(y0), mark(y1)), a__first(mark(x0), mark(x1)))
MARK(sel(first(y0, y1), cons(x0, x1))) → A__SEL(a__first(mark(y0), mark(y1)), cons(mark(x0), x1))
MARK(sel(from(y0), cons(x0, x1))) → A__SEL(a__from(mark(y0)), cons(mark(x0), x1))
MARK(s(X)) → MARK(X)
MARK(sel(s(y0), sel(x0, x1))) → A__SEL(s(mark(y0)), a__sel(mark(x0), mark(x1)))
MARK(sel(first(y0, y1), sel(x0, x1))) → A__SEL(a__first(mark(y0), mark(y1)), a__sel(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), first(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), a__first(mark(x0), mark(x1)))
MARK(first(s(y0), cons(x0, x1))) → A__FIRST(s(mark(y0)), cons(mark(x0), x1))
MARK(sel(0, from(x0))) → A__SEL(0, a__from(mark(x0)))
A__SEL(0, cons(X, Z)) → MARK(X)
A__FIRST(s(X), cons(Y, Z)) → MARK(Y)
MARK(first(sel(y0, y1), cons(x0, x1))) → A__FIRST(a__sel(mark(y0), mark(y1)), cons(mark(x0), x1))
MARK(sel(s(y0), s(x0))) → A__SEL(s(mark(y0)), s(mark(x0)))
MARK(sel(s(y0), nil)) → A__SEL(s(mark(y0)), nil)
A__SEL(s(X), cons(Y, Z)) → MARK(X)
MARK(first(X1, X2)) → MARK(X1)
MARK(first(first(y0, y1), cons(x0, x1))) → A__FIRST(a__first(mark(y0), mark(y1)), cons(mark(x0), x1))
MARK(from(X)) → MARK(X)
MARK(sel(s(y0), first(x0, x1))) → A__SEL(s(mark(y0)), a__first(mark(x0), mark(x1)))
A__SEL(s(s(x0)), cons(y1, y2)) → A__SEL(s(mark(x0)), mark(y2))
MARK(from(X)) → A__FROM(mark(X))
MARK(sel(from(y0), from(x0))) → A__SEL(a__from(mark(y0)), a__from(mark(x0)))
MARK(first(s(y0), first(x0, x1))) → A__FIRST(s(mark(y0)), a__first(mark(x0), mark(x1)))
MARK(first(s(y0), sel(x0, x1))) → A__FIRST(s(mark(y0)), a__sel(mark(x0), mark(x1)))
MARK(sel(s(y0), 0)) → A__SEL(s(mark(y0)), 0)
MARK(sel(0, first(x0, x1))) → A__SEL(0, a__first(mark(x0), mark(x1)))
MARK(sel(0, sel(x0, x1))) → A__SEL(0, a__sel(mark(x0), mark(x1)))
MARK(sel(X1, X2)) → MARK(X2)
MARK(sel(first(y0, y1), from(x0))) → A__SEL(a__first(mark(y0), mark(y1)), a__from(mark(x0)))
MARK(sel(from(y0), first(x0, x1))) → A__SEL(a__from(mark(y0)), a__first(mark(x0), mark(x1)))

The TRS R consists of the following rules:

a__from(X) → cons(mark(X), from(s(X)))
a__first(0, Z) → nil
a__first(s(X), cons(Y, Z)) → cons(mark(Y), first(X, Z))
a__sel(0, cons(X, Z)) → mark(X)
a__sel(s(X), cons(Y, Z)) → a__sel(mark(X), mark(Z))
mark(from(X)) → a__from(mark(X))
mark(first(X1, X2)) → a__first(mark(X1), mark(X2))
mark(sel(X1, X2)) → a__sel(mark(X1), mark(X2))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(s(X)) → s(mark(X))
mark(0) → 0
mark(nil) → nil
a__from(X) → from(X)
a__first(X1, X2) → first(X1, X2)
a__sel(X1, X2) → sel(X1, X2)

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ Narrowing

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ QDP

              ↳ Narrowing

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ QDP

                      ↳ Narrowing

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ QDP

                              ↳ Narrowing

                                ↳ QDP

                                  ↳ DependencyGraphProof

                                    ↳ QDP

                                      ↳ Narrowing

                                        ↳ QDP

                                          ↳ DependencyGraphProof

                                            ↳ QDP

                                              ↳ Narrowing

                                                ↳ QDP

                                                  ↳ DependencyGraphProof

                                                    ↳ QDP

                                                      ↳ Narrowing

                                                        ↳ QDP

                                                          ↳ DependencyGraphProof

                                                            ↳ QDP

                                                              ↳ Narrowing

                                                                ↳ QDP

                                                                  ↳ DependencyGraphProof

                                                                    ↳ QDP

                                                                      ↳ Narrowing

                                                                        ↳ QDP

                                                                          ↳ DependencyGraphProof

                                                                            ↳ QDP

                                                                              ↳ Narrowing

                                                                                ↳ QDP

                                                                                  ↳ DependencyGraphProof

                                                                                    ↳ QDP

                                                                                      ↳ Narrowing

                                                                                        ↳ QDP

                                                                                          ↳ DependencyGraphProof

                                                                                            ↳ QDP

                                                                                              ↳ Narrowing

                                                                                                ↳ QDP

                                                                                                  ↳ DependencyGraphProof

                                                                                                    ↳ QDP

                                                                                                      ↳ Narrowing

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

MARK(sel(sel(y0, y1), sel(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), a__sel(mark(x0), mark(x1)))
MARK(sel(s(y0), cons(x0, x1))) → A__SEL(s(mark(y0)), cons(mark(x0), x1))
MARK(sel(first(y0, y1), first(x0, x1))) → A__SEL(a__first(mark(y0), mark(y1)), a__first(mark(x0), mark(x1)))
MARK(first(first(y0, y1), from(x0))) → A__FIRST(a__first(mark(y0), mark(y1)), a__from(mark(x0)))
MARK(first(from(y0), first(x0, x1))) → A__FIRST(a__from(mark(y0)), a__first(mark(x0), mark(x1)))
MARK(cons(X1, X2)) → MARK(X1)
MARK(first(from(y0), sel(x0, x1))) → A__FIRST(a__from(mark(y0)), a__sel(mark(x0), mark(x1)))
MARK(first(sel(y0, y1), from(x0))) → A__FIRST(a__sel(mark(y0), mark(y1)), a__from(mark(x0)))
A__SEL(s(X), cons(Y, Z)) → MARK(Z)
A__SEL(s(sel(x0, x1)), cons(y1, y2)) → A__SEL(a__sel(mark(x0), mark(x1)), mark(y2))
MARK(first(from(y0), cons(x0, x1))) → A__FIRST(a__from(mark(y0)), cons(mark(x0), x1))
MARK(sel(s(y0), from(x0))) → A__SEL(s(mark(y0)), a__from(mark(x0)))
MARK(first(sel(y0, y1), sel(x0, x1))) → A__FIRST(a__sel(mark(y0), mark(y1)), a__sel(mark(x0), mark(x1)))
A__SEL(s(from(x0)), cons(y1, y2)) → A__SEL(a__from(mark(x0)), mark(y2))
A__SEL(s(first(x0, x1)), cons(y1, y2)) → A__SEL(a__first(mark(x0), mark(x1)), mark(y2))
MARK(sel(X1, X2)) → MARK(X1)
MARK(sel(sel(y0, y1), cons(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), cons(mark(x0), x1))
MARK(first(from(y0), from(x0))) → A__FIRST(a__from(mark(y0)), a__from(mark(x0)))
MARK(first(X1, X2)) → MARK(X2)
MARK(first(s(y0), from(x0))) → A__FIRST(s(mark(y0)), a__from(mark(x0)))
A__FROM(X) → MARK(X)
A__SEL(s(0), cons(y1, y2)) → A__SEL(0, mark(y2))
MARK(first(first(y0, y1), first(x0, x1))) → A__FIRST(a__first(mark(y0), mark(y1)), a__first(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), from(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__from(mark(x0)))
MARK(sel(from(y0), sel(x0, x1))) → A__SEL(a__from(mark(y0)), a__sel(mark(x0), mark(x1)))
MARK(sel(0, cons(x0, x1))) → A__SEL(0, cons(mark(x0), x1))
MARK(first(sel(y0, y1), first(x0, x1))) → A__FIRST(a__sel(mark(y0), mark(y1)), a__first(mark(x0), mark(x1)))
MARK(first(first(y0, y1), sel(x0, x1))) → A__FIRST(a__first(mark(y0), mark(y1)), a__sel(mark(x0), mark(x1)))
MARK(sel(first(y0, y1), cons(x0, x1))) → A__SEL(a__first(mark(y0), mark(y1)), cons(mark(x0), x1))
MARK(sel(from(y0), cons(x0, x1))) → A__SEL(a__from(mark(y0)), cons(mark(x0), x1))
MARK(s(X)) → MARK(X)
MARK(sel(s(y0), sel(x0, x1))) → A__SEL(s(mark(y0)), a__sel(mark(x0), mark(x1)))
MARK(sel(first(y0, y1), sel(x0, x1))) → A__SEL(a__first(mark(y0), mark(y1)), a__sel(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), first(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), a__first(mark(x0), mark(x1)))
MARK(first(s(y0), cons(x0, x1))) → A__FIRST(s(mark(y0)), cons(mark(x0), x1))
A__SEL(0, cons(X, Z)) → MARK(X)
MARK(sel(0, from(x0))) → A__SEL(0, a__from(mark(x0)))
A__FIRST(s(X), cons(Y, Z)) → MARK(Y)
MARK(first(sel(y0, y1), cons(x0, x1))) → A__FIRST(a__sel(mark(y0), mark(y1)), cons(mark(x0), x1))
A__SEL(s(X), cons(Y, Z)) → MARK(X)
MARK(first(X1, X2)) → MARK(X1)
MARK(first(first(y0, y1), cons(x0, x1))) → A__FIRST(a__first(mark(y0), mark(y1)), cons(mark(x0), x1))
MARK(from(X)) → MARK(X)
MARK(sel(s(y0), first(x0, x1))) → A__SEL(s(mark(y0)), a__first(mark(x0), mark(x1)))
MARK(from(X)) → A__FROM(mark(X))
A__SEL(s(s(x0)), cons(y1, y2)) → A__SEL(s(mark(x0)), mark(y2))
MARK(first(s(y0), first(x0, x1))) → A__FIRST(s(mark(y0)), a__first(mark(x0), mark(x1)))
MARK(sel(from(y0), from(x0))) → A__SEL(a__from(mark(y0)), a__from(mark(x0)))
MARK(first(s(y0), sel(x0, x1))) → A__FIRST(s(mark(y0)), a__sel(mark(x0), mark(x1)))
MARK(sel(0, first(x0, x1))) → A__SEL(0, a__first(mark(x0), mark(x1)))
MARK(sel(0, sel(x0, x1))) → A__SEL(0, a__sel(mark(x0), mark(x1)))
MARK(sel(X1, X2)) → MARK(X2)
MARK(sel(from(y0), first(x0, x1))) → A__SEL(a__from(mark(y0)), a__first(mark(x0), mark(x1)))
MARK(sel(first(y0, y1), from(x0))) → A__SEL(a__first(mark(y0), mark(y1)), a__from(mark(x0)))

The TRS R consists of the following rules:

a__from(X) → cons(mark(X), from(s(X)))
a__first(0, Z) → nil
a__first(s(X), cons(Y, Z)) → cons(mark(Y), first(X, Z))
a__sel(0, cons(X, Z)) → mark(X)
a__sel(s(X), cons(Y, Z)) → a__sel(mark(X), mark(Z))
mark(from(X)) → a__from(mark(X))
mark(first(X1, X2)) → a__first(mark(X1), mark(X2))
mark(sel(X1, X2)) → a__sel(mark(X1), mark(X2))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(s(X)) → s(mark(X))
mark(0) → 0
mark(nil) → nil
a__from(X) → from(X)
a__first(X1, X2) → first(X1, X2)
a__sel(X1, X2) → sel(X1, X2)

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(mark(x0)), mark(y2)) at position [1] we obtained the following new rules:

A__SEL(s(from(y0)), cons(y1, cons(x0, x1))) → A__SEL(a__from(mark(y0)), cons(mark(x0), x1))
A__SEL(s(from(y0)), cons(y1, nil)) → A__SEL(a__from(mark(y0)), nil)
A__SEL(s(from(y0)), cons(y1, sel(x0, x1))) → A__SEL(a__from(mark(y0)), a__sel(mark(x0), mark(x1)))
A__SEL(s(from(y0)), cons(y1, from(x0))) → A__SEL(a__from(mark(y0)), a__from(mark(x0)))
A__SEL(s(from(y0)), cons(y1, 0)) → A__SEL(a__from(mark(y0)), 0)
A__SEL(s(from(y0)), cons(y1, first(x0, x1))) → A__SEL(a__from(mark(y0)), a__first(mark(x0), mark(x1)))
A__SEL(s(from(y0)), cons(y1, s(x0))) → A__SEL(a__from(mark(y0)), s(mark(x0)))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ Narrowing

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ QDP

              ↳ Narrowing

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ QDP

                      ↳ Narrowing

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ QDP

                              ↳ Narrowing

                                ↳ QDP

                                  ↳ DependencyGraphProof

                                    ↳ QDP

                                      ↳ Narrowing

                                        ↳ QDP

                                          ↳ DependencyGraphProof

                                            ↳ QDP

                                              ↳ Narrowing

                                                ↳ QDP

                                                  ↳ DependencyGraphProof

                                                    ↳ QDP

                                                      ↳ Narrowing

                                                        ↳ QDP

                                                          ↳ DependencyGraphProof

                                                            ↳ QDP

                                                              ↳ Narrowing

                                                                ↳ QDP

                                                                  ↳ DependencyGraphProof

                                                                    ↳ QDP

                                                                      ↳ Narrowing

                                                                        ↳ QDP

                                                                          ↳ DependencyGraphProof

                                                                            ↳ QDP

                                                                              ↳ Narrowing

                                                                                ↳ QDP

                                                                                  ↳ DependencyGraphProof

                                                                                    ↳ QDP

                                                                                      ↳ Narrowing

                                                                                        ↳ QDP

                                                                                          ↳ DependencyGraphProof

                                                                                            ↳ QDP

                                                                                              ↳ Narrowing

                                                                                                ↳ QDP

                                                                                                  ↳ DependencyGraphProof

                                                                                                    ↳ QDP

                                                                                                      ↳ Narrowing

                                                                                                        ↳ QDP

                                                                                                          ↳ DependencyGraphProof

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

MARK(sel(sel(y0, y1), sel(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), a__sel(mark(x0), mark(x1)))
A__SEL(s(from(y0)), cons(y1, nil)) → A__SEL(a__from(mark(y0)), nil)
MARK(sel(s(y0), cons(x0, x1))) → A__SEL(s(mark(y0)), cons(mark(x0), x1))
MARK(sel(first(y0, y1), first(x0, x1))) → A__SEL(a__first(mark(y0), mark(y1)), a__first(mark(x0), mark(x1)))
MARK(cons(X1, X2)) → MARK(X1)
MARK(first(from(y0), first(x0, x1))) → A__FIRST(a__from(mark(y0)), a__first(mark(x0), mark(x1)))
MARK(first(first(y0, y1), from(x0))) → A__FIRST(a__first(mark(y0), mark(y1)), a__from(mark(x0)))
MARK(first(sel(y0, y1), from(x0))) → A__FIRST(a__sel(mark(y0), mark(y1)), a__from(mark(x0)))
MARK(first(from(y0), sel(x0, x1))) → A__FIRST(a__from(mark(y0)), a__sel(mark(x0), mark(x1)))
A__SEL(s(sel(x0, x1)), cons(y1, y2)) → A__SEL(a__sel(mark(x0), mark(x1)), mark(y2))
A__SEL(s(X), cons(Y, Z)) → MARK(Z)
MARK(first(from(y0), cons(x0, x1))) → A__FIRST(a__from(mark(y0)), cons(mark(x0), x1))
MARK(sel(s(y0), from(x0))) → A__SEL(s(mark(y0)), a__from(mark(x0)))
MARK(first(sel(y0, y1), sel(x0, x1))) → A__FIRST(a__sel(mark(y0), mark(y1)), a__sel(mark(x0), mark(x1)))
A__SEL(s(first(x0, x1)), cons(y1, y2)) → A__SEL(a__first(mark(x0), mark(x1)), mark(y2))
MARK(sel(X1, X2)) → MARK(X1)
MARK(sel(sel(y0, y1), cons(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), cons(mark(x0), x1))
MARK(first(from(y0), from(x0))) → A__FIRST(a__from(mark(y0)), a__from(mark(x0)))
A__SEL(s(from(y0)), cons(y1, first(x0, x1))) → A__SEL(a__from(mark(y0)), a__first(mark(x0), mark(x1)))
MARK(first(X1, X2)) → MARK(X2)
A__FROM(X) → MARK(X)
MARK(first(s(y0), from(x0))) → A__FIRST(s(mark(y0)), a__from(mark(x0)))
A__SEL(s(0), cons(y1, y2)) → A__SEL(0, mark(y2))
MARK(first(first(y0, y1), first(x0, x1))) → A__FIRST(a__first(mark(y0), mark(y1)), a__first(mark(x0), mark(x1)))
A__SEL(s(from(y0)), cons(y1, sel(x0, x1))) → A__SEL(a__from(mark(y0)), a__sel(mark(x0), mark(x1)))
A__SEL(s(from(y0)), cons(y1, from(x0))) → A__SEL(a__from(mark(y0)), a__from(mark(x0)))
MARK(sel(0, cons(x0, x1))) → A__SEL(0, cons(mark(x0), x1))
MARK(sel(from(y0), sel(x0, x1))) → A__SEL(a__from(mark(y0)), a__sel(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), from(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__from(mark(x0)))
A__SEL(s(from(y0)), cons(y1, 0)) → A__SEL(a__from(mark(y0)), 0)
MARK(first(first(y0, y1), sel(x0, x1))) → A__FIRST(a__first(mark(y0), mark(y1)), a__sel(mark(x0), mark(x1)))
MARK(first(sel(y0, y1), first(x0, x1))) → A__FIRST(a__sel(mark(y0), mark(y1)), a__first(mark(x0), mark(x1)))
MARK(sel(first(y0, y1), cons(x0, x1))) → A__SEL(a__first(mark(y0), mark(y1)), cons(mark(x0), x1))
MARK(sel(from(y0), cons(x0, x1))) → A__SEL(a__from(mark(y0)), cons(mark(x0), x1))
MARK(s(X)) → MARK(X)
MARK(sel(sel(y0, y1), first(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), a__first(mark(x0), mark(x1)))
MARK(sel(first(y0, y1), sel(x0, x1))) → A__SEL(a__first(mark(y0), mark(y1)), a__sel(mark(x0), mark(x1)))
MARK(sel(s(y0), sel(x0, x1))) → A__SEL(s(mark(y0)), a__sel(mark(x0), mark(x1)))
MARK(first(s(y0), cons(x0, x1))) → A__FIRST(s(mark(y0)), cons(mark(x0), x1))
MARK(sel(0, from(x0))) → A__SEL(0, a__from(mark(x0)))
A__SEL(0, cons(X, Z)) → MARK(X)
A__SEL(s(from(y0)), cons(y1, cons(x0, x1))) → A__SEL(a__from(mark(y0)), cons(mark(x0), x1))
A__FIRST(s(X), cons(Y, Z)) → MARK(Y)
MARK(first(sel(y0, y1), cons(x0, x1))) → A__FIRST(a__sel(mark(y0), mark(y1)), cons(mark(x0), x1))
A__SEL(s(from(y0)), cons(y1, s(x0))) → A__SEL(a__from(mark(y0)), s(mark(x0)))
A__SEL(s(X), cons(Y, Z)) → MARK(X)
MARK(first(X1, X2)) → MARK(X1)
MARK(first(first(y0, y1), cons(x0, x1))) → A__FIRST(a__first(mark(y0), mark(y1)), cons(mark(x0), x1))
MARK(from(X)) → MARK(X)
A__SEL(s(s(x0)), cons(y1, y2)) → A__SEL(s(mark(x0)), mark(y2))
MARK(from(X)) → A__FROM(mark(X))
MARK(sel(s(y0), first(x0, x1))) → A__SEL(s(mark(y0)), a__first(mark(x0), mark(x1)))
MARK(sel(from(y0), from(x0))) → A__SEL(a__from(mark(y0)), a__from(mark(x0)))
MARK(first(s(y0), first(x0, x1))) → A__FIRST(s(mark(y0)), a__first(mark(x0), mark(x1)))
MARK(first(s(y0), sel(x0, x1))) → A__FIRST(s(mark(y0)), a__sel(mark(x0), mark(x1)))
MARK(sel(0, first(x0, x1))) → A__SEL(0, a__first(mark(x0), mark(x1)))
MARK(sel(0, sel(x0, x1))) → A__SEL(0, a__sel(mark(x0), mark(x1)))
MARK(sel(X1, X2)) → MARK(X2)
MARK(sel(first(y0, y1), from(x0))) → A__SEL(a__first(mark(y0), mark(y1)), a__from(mark(x0)))
MARK(sel(from(y0), first(x0, x1))) → A__SEL(a__from(mark(y0)), a__first(mark(x0), mark(x1)))

The TRS R consists of the following rules:

a__from(X) → cons(mark(X), from(s(X)))
a__first(0, Z) → nil
a__first(s(X), cons(Y, Z)) → cons(mark(Y), first(X, Z))
a__sel(0, cons(X, Z)) → mark(X)
a__sel(s(X), cons(Y, Z)) → a__sel(mark(X), mark(Z))
mark(from(X)) → a__from(mark(X))
mark(first(X1, X2)) → a__first(mark(X1), mark(X2))
mark(sel(X1, X2)) → a__sel(mark(X1), mark(X2))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(s(X)) → s(mark(X))
mark(0) → 0
mark(nil) → nil
a__from(X) → from(X)
a__first(X1, X2) → first(X1, X2)
a__sel(X1, X2) → sel(X1, X2)

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ Narrowing

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ QDP

              ↳ Narrowing

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ QDP

                      ↳ Narrowing

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ QDP

                              ↳ Narrowing

                                ↳ QDP

                                  ↳ DependencyGraphProof

                                    ↳ QDP

                                      ↳ Narrowing

                                        ↳ QDP

                                          ↳ DependencyGraphProof

                                            ↳ QDP

                                              ↳ Narrowing

                                                ↳ QDP

                                                  ↳ DependencyGraphProof

                                                    ↳ QDP

                                                      ↳ Narrowing

                                                        ↳ QDP

                                                          ↳ DependencyGraphProof

                                                            ↳ QDP

                                                              ↳ Narrowing

                                                                ↳ QDP

                                                                  ↳ DependencyGraphProof

                                                                    ↳ QDP

                                                                      ↳ Narrowing

                                                                        ↳ QDP

                                                                          ↳ DependencyGraphProof

                                                                            ↳ QDP

                                                                              ↳ Narrowing

                                                                                ↳ QDP

                                                                                  ↳ DependencyGraphProof

                                                                                    ↳ QDP

                                                                                      ↳ Narrowing

                                                                                        ↳ QDP

                                                                                          ↳ DependencyGraphProof

                                                                                            ↳ QDP

                                                                                              ↳ Narrowing

                                                                                                ↳ QDP

                                                                                                  ↳ DependencyGraphProof

                                                                                                    ↳ QDP

                                                                                                      ↳ Narrowing

                                                                                                        ↳ QDP

                                                                                                          ↳ DependencyGraphProof

                                                                                                            ↳ QDP

                                                                                                              ↳ Narrowing

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

MARK(sel(sel(y0, y1), sel(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), a__sel(mark(x0), mark(x1)))
MARK(sel(s(y0), cons(x0, x1))) → A__SEL(s(mark(y0)), cons(mark(x0), x1))
MARK(sel(first(y0, y1), first(x0, x1))) → A__SEL(a__first(mark(y0), mark(y1)), a__first(mark(x0), mark(x1)))
MARK(first(first(y0, y1), from(x0))) → A__FIRST(a__first(mark(y0), mark(y1)), a__from(mark(x0)))
MARK(first(from(y0), first(x0, x1))) → A__FIRST(a__from(mark(y0)), a__first(mark(x0), mark(x1)))
MARK(cons(X1, X2)) → MARK(X1)
MARK(first(from(y0), sel(x0, x1))) → A__FIRST(a__from(mark(y0)), a__sel(mark(x0), mark(x1)))
MARK(first(sel(y0, y1), from(x0))) → A__FIRST(a__sel(mark(y0), mark(y1)), a__from(mark(x0)))
A__SEL(s(X), cons(Y, Z)) → MARK(Z)
A__SEL(s(sel(x0, x1)), cons(y1, y2)) → A__SEL(a__sel(mark(x0), mark(x1)), mark(y2))
MARK(first(from(y0), cons(x0, x1))) → A__FIRST(a__from(mark(y0)), cons(mark(x0), x1))
MARK(sel(s(y0), from(x0))) → A__SEL(s(mark(y0)), a__from(mark(x0)))
MARK(first(sel(y0, y1), sel(x0, x1))) → A__FIRST(a__sel(mark(y0), mark(y1)), a__sel(mark(x0), mark(x1)))
A__SEL(s(first(x0, x1)), cons(y1, y2)) → A__SEL(a__first(mark(x0), mark(x1)), mark(y2))
MARK(sel(X1, X2)) → MARK(X1)
MARK(sel(sel(y0, y1), cons(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), cons(mark(x0), x1))
MARK(first(from(y0), from(x0))) → A__FIRST(a__from(mark(y0)), a__from(mark(x0)))
MARK(first(X1, X2)) → MARK(X2)
A__SEL(s(from(y0)), cons(y1, first(x0, x1))) → A__SEL(a__from(mark(y0)), a__first(mark(x0), mark(x1)))
MARK(first(s(y0), from(x0))) → A__FIRST(s(mark(y0)), a__from(mark(x0)))
A__FROM(X) → MARK(X)
A__SEL(s(0), cons(y1, y2)) → A__SEL(0, mark(y2))
MARK(first(first(y0, y1), first(x0, x1))) → A__FIRST(a__first(mark(y0), mark(y1)), a__first(mark(x0), mark(x1)))
A__SEL(s(from(y0)), cons(y1, sel(x0, x1))) → A__SEL(a__from(mark(y0)), a__sel(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), from(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__from(mark(x0)))
MARK(sel(from(y0), sel(x0, x1))) → A__SEL(a__from(mark(y0)), a__sel(mark(x0), mark(x1)))
A__SEL(s(from(y0)), cons(y1, from(x0))) → A__SEL(a__from(mark(y0)), a__from(mark(x0)))
MARK(sel(0, cons(x0, x1))) → A__SEL(0, cons(mark(x0), x1))
MARK(first(sel(y0, y1), first(x0, x1))) → A__FIRST(a__sel(mark(y0), mark(y1)), a__first(mark(x0), mark(x1)))
MARK(first(first(y0, y1), sel(x0, x1))) → A__FIRST(a__first(mark(y0), mark(y1)), a__sel(mark(x0), mark(x1)))
MARK(sel(first(y0, y1), cons(x0, x1))) → A__SEL(a__first(mark(y0), mark(y1)), cons(mark(x0), x1))
MARK(sel(from(y0), cons(x0, x1))) → A__SEL(a__from(mark(y0)), cons(mark(x0), x1))
MARK(s(X)) → MARK(X)
MARK(sel(s(y0), sel(x0, x1))) → A__SEL(s(mark(y0)), a__sel(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), first(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), a__first(mark(x0), mark(x1)))
MARK(sel(first(y0, y1), sel(x0, x1))) → A__SEL(a__first(mark(y0), mark(y1)), a__sel(mark(x0), mark(x1)))
MARK(first(s(y0), cons(x0, x1))) → A__FIRST(s(mark(y0)), cons(mark(x0), x1))
A__SEL(0, cons(X, Z)) → MARK(X)
MARK(sel(0, from(x0))) → A__SEL(0, a__from(mark(x0)))
A__FIRST(s(X), cons(Y, Z)) → MARK(Y)
A__SEL(s(from(y0)), cons(y1, cons(x0, x1))) → A__SEL(a__from(mark(y0)), cons(mark(x0), x1))
MARK(first(sel(y0, y1), cons(x0, x1))) → A__FIRST(a__sel(mark(y0), mark(y1)), cons(mark(x0), x1))
A__SEL(s(X), cons(Y, Z)) → MARK(X)
MARK(first(X1, X2)) → MARK(X1)
MARK(first(first(y0, y1), cons(x0, x1))) → A__FIRST(a__first(mark(y0), mark(y1)), cons(mark(x0), x1))
MARK(from(X)) → MARK(X)
MARK(sel(s(y0), first(x0, x1))) → A__SEL(s(mark(y0)), a__first(mark(x0), mark(x1)))
MARK(from(X)) → A__FROM(mark(X))
A__SEL(s(s(x0)), cons(y1, y2)) → A__SEL(s(mark(x0)), mark(y2))
MARK(first(s(y0), first(x0, x1))) → A__FIRST(s(mark(y0)), a__first(mark(x0), mark(x1)))
MARK(sel(from(y0), from(x0))) → A__SEL(a__from(mark(y0)), a__from(mark(x0)))
MARK(first(s(y0), sel(x0, x1))) → A__FIRST(s(mark(y0)), a__sel(mark(x0), mark(x1)))
MARK(sel(0, first(x0, x1))) → A__SEL(0, a__first(mark(x0), mark(x1)))
MARK(sel(0, sel(x0, x1))) → A__SEL(0, a__sel(mark(x0), mark(x1)))
MARK(sel(X1, X2)) → MARK(X2)
MARK(sel(from(y0), first(x0, x1))) → A__SEL(a__from(mark(y0)), a__first(mark(x0), mark(x1)))
MARK(sel(first(y0, y1), from(x0))) → A__SEL(a__first(mark(y0), mark(y1)), a__from(mark(x0)))

The TRS R consists of the following rules:

a__from(X) → cons(mark(X), from(s(X)))
a__first(0, Z) → nil
a__first(s(X), cons(Y, Z)) → cons(mark(Y), first(X, Z))
a__sel(0, cons(X, Z)) → mark(X)
a__sel(s(X), cons(Y, Z)) → a__sel(mark(X), mark(Z))
mark(from(X)) → a__from(mark(X))
mark(first(X1, X2)) → a__first(mark(X1), mark(X2))
mark(sel(X1, X2)) → a__sel(mark(X1), mark(X2))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(s(X)) → s(mark(X))
mark(0) → 0
mark(nil) → nil
a__from(X) → from(X)
a__first(X1, X2) → first(X1, X2)
a__sel(X1, X2) → sel(X1, X2)

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

A__SEL(s(first(y0, y1)), cons(y2, nil)) → A__SEL(a__first(mark(y0), mark(y1)), nil)
A__SEL(s(first(y0, y1)), cons(y2, 0)) → A__SEL(a__first(mark(y0), mark(y1)), 0)
A__SEL(s(first(y0, y1)), cons(y2, cons(x0, x1))) → A__SEL(a__first(mark(y0), mark(y1)), cons(mark(x0), x1))
A__SEL(s(first(y0, y1)), cons(y2, sel(x0, x1))) → A__SEL(a__first(mark(y0), mark(y1)), a__sel(mark(x0), mark(x1)))
A__SEL(s(first(y0, y1)), cons(y2, from(x0))) → A__SEL(a__first(mark(y0), mark(y1)), a__from(mark(x0)))
A__SEL(s(first(y0, y1)), cons(y2, s(x0))) → A__SEL(a__first(mark(y0), mark(y1)), s(mark(x0)))
A__SEL(s(first(y0, y1)), cons(y2, first(x0, x1))) → A__SEL(a__first(mark(y0), mark(y1)), a__first(mark(x0), mark(x1)))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ Narrowing

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ QDP

              ↳ Narrowing

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ QDP

                      ↳ Narrowing

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ QDP

                              ↳ Narrowing

                                ↳ QDP

                                  ↳ DependencyGraphProof

                                    ↳ QDP

                                      ↳ Narrowing

                                        ↳ QDP

                                          ↳ DependencyGraphProof

                                            ↳ QDP

                                              ↳ Narrowing

                                                ↳ QDP

                                                  ↳ DependencyGraphProof

                                                    ↳ QDP

                                                      ↳ Narrowing

                                                        ↳ QDP

                                                          ↳ DependencyGraphProof

                                                            ↳ QDP

                                                              ↳ Narrowing

                                                                ↳ QDP

                                                                  ↳ DependencyGraphProof

                                                                    ↳ QDP

                                                                      ↳ Narrowing

                                                                        ↳ QDP

                                                                          ↳ DependencyGraphProof

                                                                            ↳ QDP

                                                                              ↳ Narrowing

                                                                                ↳ QDP

                                                                                  ↳ DependencyGraphProof

                                                                                    ↳ QDP

                                                                                      ↳ Narrowing

                                                                                        ↳ QDP

                                                                                          ↳ DependencyGraphProof

                                                                                            ↳ QDP

                                                                                              ↳ Narrowing

                                                                                                ↳ QDP

                                                                                                  ↳ DependencyGraphProof

                                                                                                    ↳ QDP

                                                                                                      ↳ Narrowing

                                                                                                        ↳ QDP

                                                                                                          ↳ DependencyGraphProof

                                                                                                            ↳ QDP

                                                                                                              ↳ Narrowing

                                                                                                                ↳ QDP

                                                                                                                  ↳ DependencyGraphProof

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

MARK(sel(sel(y0, y1), sel(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), a__sel(mark(x0), mark(x1)))
MARK(sel(s(y0), cons(x0, x1))) → A__SEL(s(mark(y0)), cons(mark(x0), x1))
MARK(sel(first(y0, y1), first(x0, x1))) → A__SEL(a__first(mark(y0), mark(y1)), a__first(mark(x0), mark(x1)))
MARK(cons(X1, X2)) → MARK(X1)
MARK(first(from(y0), first(x0, x1))) → A__FIRST(a__from(mark(y0)), a__first(mark(x0), mark(x1)))
MARK(first(first(y0, y1), from(x0))) → A__FIRST(a__first(mark(y0), mark(y1)), a__from(mark(x0)))
MARK(first(sel(y0, y1), from(x0))) → A__FIRST(a__sel(mark(y0), mark(y1)), a__from(mark(x0)))
MARK(first(from(y0), sel(x0, x1))) → A__FIRST(a__from(mark(y0)), a__sel(mark(x0), mark(x1)))
A__SEL(s(first(y0, y1)), cons(y2, 0)) → A__SEL(a__first(mark(y0), mark(y1)), 0)
A__SEL(s(sel(x0, x1)), cons(y1, y2)) → A__SEL(a__sel(mark(x0), mark(x1)), mark(y2))
A__SEL(s(X), cons(Y, Z)) → MARK(Z)
MARK(first(from(y0), cons(x0, x1))) → A__FIRST(a__from(mark(y0)), cons(mark(x0), x1))
MARK(sel(s(y0), from(x0))) → A__SEL(s(mark(y0)), a__from(mark(x0)))
MARK(first(sel(y0, y1), sel(x0, x1))) → A__FIRST(a__sel(mark(y0), mark(y1)), a__sel(mark(x0), mark(x1)))
MARK(sel(X1, X2)) → MARK(X1)
MARK(sel(sel(y0, y1), cons(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), cons(mark(x0), x1))
MARK(first(from(y0), from(x0))) → A__FIRST(a__from(mark(y0)), a__from(mark(x0)))
A__SEL(s(from(y0)), cons(y1, first(x0, x1))) → A__SEL(a__from(mark(y0)), a__first(mark(x0), mark(x1)))
MARK(first(X1, X2)) → MARK(X2)
A__FROM(X) → MARK(X)
MARK(first(s(y0), from(x0))) → A__FIRST(s(mark(y0)), a__from(mark(x0)))
A__SEL(s(first(y0, y1)), cons(y2, cons(x0, x1))) → A__SEL(a__first(mark(y0), mark(y1)), cons(mark(x0), x1))
A__SEL(s(0), cons(y1, y2)) → A__SEL(0, mark(y2))
MARK(first(first(y0, y1), first(x0, x1))) → A__FIRST(a__first(mark(y0), mark(y1)), a__first(mark(x0), mark(x1)))
A__SEL(s(from(y0)), cons(y1, sel(x0, x1))) → A__SEL(a__from(mark(y0)), a__sel(mark(x0), mark(x1)))
MARK(sel(0, cons(x0, x1))) → A__SEL(0, cons(mark(x0), x1))
A__SEL(s(from(y0)), cons(y1, from(x0))) → A__SEL(a__from(mark(y0)), a__from(mark(x0)))
MARK(sel(from(y0), sel(x0, x1))) → A__SEL(a__from(mark(y0)), a__sel(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), from(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__from(mark(x0)))
A__SEL(s(first(y0, y1)), cons(y2, from(x0))) → A__SEL(a__first(mark(y0), mark(y1)), a__from(mark(x0)))
MARK(first(first(y0, y1), sel(x0, x1))) → A__FIRST(a__first(mark(y0), mark(y1)), a__sel(mark(x0), mark(x1)))
MARK(first(sel(y0, y1), first(x0, x1))) → A__FIRST(a__sel(mark(y0), mark(y1)), a__first(mark(x0), mark(x1)))
A__SEL(s(first(y0, y1)), cons(y2, s(x0))) → A__SEL(a__first(mark(y0), mark(y1)), s(mark(x0)))
MARK(sel(first(y0, y1), cons(x0, x1))) → A__SEL(a__first(mark(y0), mark(y1)), cons(mark(x0), x1))
MARK(sel(from(y0), cons(x0, x1))) → A__SEL(a__from(mark(y0)), cons(mark(x0), x1))
MARK(s(X)) → MARK(X)
MARK(sel(first(y0, y1), sel(x0, x1))) → A__SEL(a__first(mark(y0), mark(y1)), a__sel(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), first(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), a__first(mark(x0), mark(x1)))
MARK(sel(s(y0), sel(x0, x1))) → A__SEL(s(mark(y0)), a__sel(mark(x0), mark(x1)))
MARK(first(s(y0), cons(x0, x1))) → A__FIRST(s(mark(y0)), cons(mark(x0), x1))
MARK(sel(0, from(x0))) → A__SEL(0, a__from(mark(x0)))
A__SEL(0, cons(X, Z)) → MARK(X)
A__SEL(s(from(y0)), cons(y1, cons(x0, x1))) → A__SEL(a__from(mark(y0)), cons(mark(x0), x1))
A__FIRST(s(X), cons(Y, Z)) → MARK(Y)
MARK(first(sel(y0, y1), cons(x0, x1))) → A__FIRST(a__sel(mark(y0), mark(y1)), cons(mark(x0), x1))
A__SEL(s(X), cons(Y, Z)) → MARK(X)
MARK(first(X1, X2)) → MARK(X1)
A__SEL(s(first(y0, y1)), cons(y2, nil)) → A__SEL(a__first(mark(y0), mark(y1)), nil)
MARK(first(first(y0, y1), cons(x0, x1))) → A__FIRST(a__first(mark(y0), mark(y1)), cons(mark(x0), x1))
MARK(from(X)) → MARK(X)
A__SEL(s(s(x0)), cons(y1, y2)) → A__SEL(s(mark(x0)), mark(y2))
MARK(from(X)) → A__FROM(mark(X))
MARK(sel(s(y0), first(x0, x1))) → A__SEL(s(mark(y0)), a__first(mark(x0), mark(x1)))
MARK(sel(from(y0), from(x0))) → A__SEL(a__from(mark(y0)), a__from(mark(x0)))
MARK(first(s(y0), first(x0, x1))) → A__FIRST(s(mark(y0)), a__first(mark(x0), mark(x1)))
MARK(first(s(y0), sel(x0, x1))) → A__FIRST(s(mark(y0)), a__sel(mark(x0), mark(x1)))
MARK(sel(0, first(x0, x1))) → A__SEL(0, a__first(mark(x0), mark(x1)))
MARK(sel(0, sel(x0, x1))) → A__SEL(0, a__sel(mark(x0), mark(x1)))
A__SEL(s(first(y0, y1)), cons(y2, sel(x0, x1))) → A__SEL(a__first(mark(y0), mark(y1)), a__sel(mark(x0), mark(x1)))
MARK(sel(X1, X2)) → MARK(X2)
A__SEL(s(first(y0, y1)), cons(y2, first(x0, x1))) → A__SEL(a__first(mark(y0), mark(y1)), a__first(mark(x0), mark(x1)))
MARK(sel(first(y0, y1), from(x0))) → A__SEL(a__first(mark(y0), mark(y1)), a__from(mark(x0)))
MARK(sel(from(y0), first(x0, x1))) → A__SEL(a__from(mark(y0)), a__first(mark(x0), mark(x1)))

The TRS R consists of the following rules:

a__from(X) → cons(mark(X), from(s(X)))
a__first(0, Z) → nil
a__first(s(X), cons(Y, Z)) → cons(mark(Y), first(X, Z))
a__sel(0, cons(X, Z)) → mark(X)
a__sel(s(X), cons(Y, Z)) → a__sel(mark(X), mark(Z))
mark(from(X)) → a__from(mark(X))
mark(first(X1, X2)) → a__first(mark(X1), mark(X2))
mark(sel(X1, X2)) → a__sel(mark(X1), mark(X2))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(s(X)) → s(mark(X))
mark(0) → 0
mark(nil) → nil
a__from(X) → from(X)
a__first(X1, X2) → first(X1, X2)
a__sel(X1, X2) → sel(X1, X2)

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ Narrowing

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ QDP

              ↳ Narrowing

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ QDP

                      ↳ Narrowing

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ QDP

                              ↳ Narrowing

                                ↳ QDP

                                  ↳ DependencyGraphProof

                                    ↳ QDP

                                      ↳ Narrowing

                                        ↳ QDP

                                          ↳ DependencyGraphProof

                                            ↳ QDP

                                              ↳ Narrowing

                                                ↳ QDP

                                                  ↳ DependencyGraphProof

                                                    ↳ QDP

                                                      ↳ Narrowing

                                                        ↳ QDP

                                                          ↳ DependencyGraphProof

                                                            ↳ QDP

                                                              ↳ Narrowing

                                                                ↳ QDP

                                                                  ↳ DependencyGraphProof

                                                                    ↳ QDP

                                                                      ↳ Narrowing

                                                                        ↳ QDP

                                                                          ↳ DependencyGraphProof

                                                                            ↳ QDP

                                                                              ↳ Narrowing

                                                                                ↳ QDP

                                                                                  ↳ DependencyGraphProof

                                                                                    ↳ QDP

                                                                                      ↳ Narrowing

                                                                                        ↳ QDP

                                                                                          ↳ DependencyGraphProof

                                                                                            ↳ QDP

                                                                                              ↳ Narrowing

                                                                                                ↳ QDP

                                                                                                  ↳ DependencyGraphProof

                                                                                                    ↳ QDP

                                                                                                      ↳ Narrowing

                                                                                                        ↳ QDP

                                                                                                          ↳ DependencyGraphProof

                                                                                                            ↳ QDP

                                                                                                              ↳ Narrowing

                                                                                                                ↳ QDP

                                                                                                                  ↳ DependencyGraphProof

                                                                                                                    ↳ QDP

                                                                                                                      ↳ Narrowing

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

MARK(sel(sel(y0, y1), sel(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), a__sel(mark(x0), mark(x1)))
MARK(sel(s(y0), cons(x0, x1))) → A__SEL(s(mark(y0)), cons(mark(x0), x1))
MARK(sel(first(y0, y1), first(x0, x1))) → A__SEL(a__first(mark(y0), mark(y1)), a__first(mark(x0), mark(x1)))
MARK(first(first(y0, y1), from(x0))) → A__FIRST(a__first(mark(y0), mark(y1)), a__from(mark(x0)))
MARK(first(from(y0), first(x0, x1))) → A__FIRST(a__from(mark(y0)), a__first(mark(x0), mark(x1)))
MARK(cons(X1, X2)) → MARK(X1)
MARK(first(from(y0), sel(x0, x1))) → A__FIRST(a__from(mark(y0)), a__sel(mark(x0), mark(x1)))
MARK(first(sel(y0, y1), from(x0))) → A__FIRST(a__sel(mark(y0), mark(y1)), a__from(mark(x0)))
A__SEL(s(X), cons(Y, Z)) → MARK(Z)
A__SEL(s(sel(x0, x1)), cons(y1, y2)) → A__SEL(a__sel(mark(x0), mark(x1)), mark(y2))
MARK(first(from(y0), cons(x0, x1))) → A__FIRST(a__from(mark(y0)), cons(mark(x0), x1))
MARK(sel(s(y0), from(x0))) → A__SEL(s(mark(y0)), a__from(mark(x0)))
MARK(first(sel(y0, y1), sel(x0, x1))) → A__FIRST(a__sel(mark(y0), mark(y1)), a__sel(mark(x0), mark(x1)))
MARK(sel(X1, X2)) → MARK(X1)
MARK(sel(sel(y0, y1), cons(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), cons(mark(x0), x1))
MARK(first(from(y0), from(x0))) → A__FIRST(a__from(mark(y0)), a__from(mark(x0)))
MARK(first(X1, X2)) → MARK(X2)
A__SEL(s(from(y0)), cons(y1, first(x0, x1))) → A__SEL(a__from(mark(y0)), a__first(mark(x0), mark(x1)))
MARK(first(s(y0), from(x0))) → A__FIRST(s(mark(y0)), a__from(mark(x0)))
A__FROM(X) → MARK(X)
A__SEL(s(first(y0, y1)), cons(y2, cons(x0, x1))) → A__SEL(a__first(mark(y0), mark(y1)), cons(mark(x0), x1))
A__SEL(s(0), cons(y1, y2)) → A__SEL(0, mark(y2))
MARK(first(first(y0, y1), first(x0, x1))) → A__FIRST(a__first(mark(y0), mark(y1)), a__first(mark(x0), mark(x1)))
A__SEL(s(from(y0)), cons(y1, sel(x0, x1))) → A__SEL(a__from(mark(y0)), a__sel(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), from(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__from(mark(x0)))
MARK(sel(from(y0), sel(x0, x1))) → A__SEL(a__from(mark(y0)), a__sel(mark(x0), mark(x1)))
A__SEL(s(from(y0)), cons(y1, from(x0))) → A__SEL(a__from(mark(y0)), a__from(mark(x0)))
MARK(sel(0, cons(x0, x1))) → A__SEL(0, cons(mark(x0), x1))
A__SEL(s(first(y0, y1)), cons(y2, from(x0))) → A__SEL(a__first(mark(y0), mark(y1)), a__from(mark(x0)))
MARK(first(sel(y0, y1), first(x0, x1))) → A__FIRST(a__sel(mark(y0), mark(y1)), a__first(mark(x0), mark(x1)))
MARK(first(first(y0, y1), sel(x0, x1))) → A__FIRST(a__first(mark(y0), mark(y1)), a__sel(mark(x0), mark(x1)))
MARK(sel(first(y0, y1), cons(x0, x1))) → A__SEL(a__first(mark(y0), mark(y1)), cons(mark(x0), x1))
MARK(sel(from(y0), cons(x0, x1))) → A__SEL(a__from(mark(y0)), cons(mark(x0), x1))
MARK(s(X)) → MARK(X)
MARK(sel(s(y0), sel(x0, x1))) → A__SEL(s(mark(y0)), a__sel(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), first(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), a__first(mark(x0), mark(x1)))
MARK(sel(first(y0, y1), sel(x0, x1))) → A__SEL(a__first(mark(y0), mark(y1)), a__sel(mark(x0), mark(x1)))
MARK(first(s(y0), cons(x0, x1))) → A__FIRST(s(mark(y0)), cons(mark(x0), x1))
A__SEL(0, cons(X, Z)) → MARK(X)
MARK(sel(0, from(x0))) → A__SEL(0, a__from(mark(x0)))
A__FIRST(s(X), cons(Y, Z)) → MARK(Y)
A__SEL(s(from(y0)), cons(y1, cons(x0, x1))) → A__SEL(a__from(mark(y0)), cons(mark(x0), x1))
MARK(first(sel(y0, y1), cons(x0, x1))) → A__FIRST(a__sel(mark(y0), mark(y1)), cons(mark(x0), x1))
A__SEL(s(X), cons(Y, Z)) → MARK(X)
MARK(first(X1, X2)) → MARK(X1)
MARK(first(first(y0, y1), cons(x0, x1))) → A__FIRST(a__first(mark(y0), mark(y1)), cons(mark(x0), x1))
MARK(from(X)) → MARK(X)
MARK(sel(s(y0), first(x0, x1))) → A__SEL(s(mark(y0)), a__first(mark(x0), mark(x1)))
MARK(from(X)) → A__FROM(mark(X))
A__SEL(s(s(x0)), cons(y1, y2)) → A__SEL(s(mark(x0)), mark(y2))
MARK(first(s(y0), first(x0, x1))) → A__FIRST(s(mark(y0)), a__first(mark(x0), mark(x1)))
MARK(sel(from(y0), from(x0))) → A__SEL(a__from(mark(y0)), a__from(mark(x0)))
MARK(first(s(y0), sel(x0, x1))) → A__FIRST(s(mark(y0)), a__sel(mark(x0), mark(x1)))
MARK(sel(0, first(x0, x1))) → A__SEL(0, a__first(mark(x0), mark(x1)))
MARK(sel(0, sel(x0, x1))) → A__SEL(0, a__sel(mark(x0), mark(x1)))
A__SEL(s(first(y0, y1)), cons(y2, sel(x0, x1))) → A__SEL(a__first(mark(y0), mark(y1)), a__sel(mark(x0), mark(x1)))
MARK(sel(X1, X2)) → MARK(X2)
MARK(sel(first(y0, y1), from(x0))) → A__SEL(a__first(mark(y0), mark(y1)), a__from(mark(x0)))
MARK(sel(from(y0), first(x0, x1))) → A__SEL(a__from(mark(y0)), a__first(mark(x0), mark(x1)))
A__SEL(s(first(y0, y1)), cons(y2, first(x0, x1))) → A__SEL(a__first(mark(y0), mark(y1)), a__first(mark(x0), mark(x1)))

The TRS R consists of the following rules:

a__from(X) → cons(mark(X), from(s(X)))
a__first(0, Z) → nil
a__first(s(X), cons(Y, Z)) → cons(mark(Y), first(X, Z))
a__sel(0, cons(X, Z)) → mark(X)
a__sel(s(X), cons(Y, Z)) → a__sel(mark(X), mark(Z))
mark(from(X)) → a__from(mark(X))
mark(first(X1, X2)) → a__first(mark(X1), mark(X2))
mark(sel(X1, X2)) → a__sel(mark(X1), mark(X2))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(s(X)) → s(mark(X))
mark(0) → 0
mark(nil) → nil
a__from(X) → from(X)
a__first(X1, X2) → first(X1, X2)
a__sel(X1, X2) → sel(X1, X2)

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

A__SEL(s(sel(y0, y1)), cons(y2, sel(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), a__sel(mark(x0), mark(x1)))
A__SEL(s(sel(y0, y1)), cons(y2, from(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__from(mark(x0)))
A__SEL(s(sel(y0, y1)), cons(y2, cons(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), cons(mark(x0), x1))
A__SEL(s(sel(y0, y1)), cons(y2, s(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), s(mark(x0)))
A__SEL(s(sel(y0, y1)), cons(y2, 0)) → A__SEL(a__sel(mark(y0), mark(y1)), 0)
A__SEL(s(sel(y0, y1)), cons(y2, nil)) → A__SEL(a__sel(mark(y0), mark(y1)), nil)
A__SEL(s(sel(y0, y1)), cons(y2, first(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), a__first(mark(x0), mark(x1)))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ Narrowing

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ QDP

              ↳ Narrowing

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ QDP

                      ↳ Narrowing

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ QDP

                              ↳ Narrowing

                                ↳ QDP

                                  ↳ DependencyGraphProof

                                    ↳ QDP

                                      ↳ Narrowing

                                        ↳ QDP

                                          ↳ DependencyGraphProof

                                            ↳ QDP

                                              ↳ Narrowing

                                                ↳ QDP

                                                  ↳ DependencyGraphProof

                                                    ↳ QDP

                                                      ↳ Narrowing

                                                        ↳ QDP

                                                          ↳ DependencyGraphProof

                                                            ↳ QDP

                                                              ↳ Narrowing

                                                                ↳ QDP

                                                                  ↳ DependencyGraphProof

                                                                    ↳ QDP

                                                                      ↳ Narrowing

                                                                        ↳ QDP

                                                                          ↳ DependencyGraphProof

                                                                            ↳ QDP

                                                                              ↳ Narrowing

                                                                                ↳ QDP

                                                                                  ↳ DependencyGraphProof

                                                                                    ↳ QDP

                                                                                      ↳ Narrowing

                                                                                        ↳ QDP

                                                                                          ↳ DependencyGraphProof

                                                                                            ↳ QDP

                                                                                              ↳ Narrowing

                                                                                                ↳ QDP

                                                                                                  ↳ DependencyGraphProof

                                                                                                    ↳ QDP

                                                                                                      ↳ Narrowing

                                                                                                        ↳ QDP

                                                                                                          ↳ DependencyGraphProof

                                                                                                            ↳ QDP

                                                                                                              ↳ Narrowing

                                                                                                                ↳ QDP

                                                                                                                  ↳ DependencyGraphProof

                                                                                                                    ↳ QDP

                                                                                                                      ↳ Narrowing

                                                                                                                        ↳ QDP

                                                                                                                          ↳ DependencyGraphProof

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

MARK(sel(sel(y0, y1), sel(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), a__sel(mark(x0), mark(x1)))
MARK(sel(s(y0), cons(x0, x1))) → A__SEL(s(mark(y0)), cons(mark(x0), x1))
A__SEL(s(sel(y0, y1)), cons(y2, from(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__from(mark(x0)))
MARK(sel(first(y0, y1), first(x0, x1))) → A__SEL(a__first(mark(y0), mark(y1)), a__first(mark(x0), mark(x1)))
MARK(cons(X1, X2)) → MARK(X1)
MARK(first(from(y0), first(x0, x1))) → A__FIRST(a__from(mark(y0)), a__first(mark(x0), mark(x1)))
MARK(first(first(y0, y1), from(x0))) → A__FIRST(a__first(mark(y0), mark(y1)), a__from(mark(x0)))
MARK(first(sel(y0, y1), from(x0))) → A__FIRST(a__sel(mark(y0), mark(y1)), a__from(mark(x0)))
MARK(first(from(y0), sel(x0, x1))) → A__FIRST(a__from(mark(y0)), a__sel(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(X), cons(Y, Z)) → MARK(Z)
MARK(first(from(y0), cons(x0, x1))) → A__FIRST(a__from(mark(y0)), cons(mark(x0), x1))
MARK(sel(s(y0), from(x0))) → A__SEL(s(mark(y0)), a__from(mark(x0)))
MARK(first(sel(y0, y1), sel(x0, x1))) → A__FIRST(a__sel(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)))
MARK(sel(X1, X2)) → MARK(X1)
MARK(sel(sel(y0, y1), cons(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), cons(mark(x0), x1))
MARK(first(from(y0), from(x0))) → A__FIRST(a__from(mark(y0)), a__from(mark(x0)))
A__SEL(s(from(y0)), cons(y1, first(x0, x1))) → A__SEL(a__from(mark(y0)), a__first(mark(x0), mark(x1)))
MARK(first(X1, X2)) → MARK(X2)
A__SEL(s(sel(y0, y1)), cons(y2, cons(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), cons(mark(x0), x1))
A__FROM(X) → MARK(X)
MARK(first(s(y0), from(x0))) → A__FIRST(s(mark(y0)), a__from(mark(x0)))
A__SEL(s(first(y0, y1)), cons(y2, cons(x0, x1))) → A__SEL(a__first(mark(y0), mark(y1)), cons(mark(x0), x1))
A__SEL(s(sel(y0, y1)), cons(y2, nil)) → A__SEL(a__sel(mark(y0), mark(y1)), nil)
A__SEL(s(0), cons(y1, y2)) → A__SEL(0, mark(y2))
MARK(first(first(y0, y1), first(x0, x1))) → A__FIRST(a__first(mark(y0), mark(y1)), a__first(mark(x0), mark(x1)))
A__SEL(s(from(y0)), cons(y1, sel(x0, x1))) → A__SEL(a__from(mark(y0)), a__sel(mark(x0), mark(x1)))
MARK(sel(0, cons(x0, x1))) → A__SEL(0, cons(mark(x0), x1))
A__SEL(s(from(y0)), cons(y1, from(x0))) → A__SEL(a__from(mark(y0)), a__from(mark(x0)))
MARK(sel(from(y0), sel(x0, x1))) → A__SEL(a__from(mark(y0)), a__sel(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), from(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__from(mark(x0)))
A__SEL(s(first(y0, y1)), cons(y2, from(x0))) → A__SEL(a__first(mark(y0), mark(y1)), a__from(mark(x0)))
MARK(first(first(y0, y1), sel(x0, x1))) → A__FIRST(a__first(mark(y0), mark(y1)), a__sel(mark(x0), mark(x1)))
MARK(first(sel(y0, y1), first(x0, x1))) → A__FIRST(a__sel(mark(y0), mark(y1)), a__first(mark(x0), mark(x1)))
MARK(sel(first(y0, y1), cons(x0, x1))) → A__SEL(a__first(mark(y0), mark(y1)), cons(mark(x0), x1))
MARK(sel(from(y0), cons(x0, x1))) → A__SEL(a__from(mark(y0)), cons(mark(x0), x1))
MARK(s(X)) → MARK(X)
MARK(sel(first(y0, y1), sel(x0, x1))) → A__SEL(a__first(mark(y0), mark(y1)), a__sel(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), first(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), a__first(mark(x0), mark(x1)))
MARK(sel(s(y0), sel(x0, x1))) → A__SEL(s(mark(y0)), a__sel(mark(x0), mark(x1)))
MARK(first(s(y0), cons(x0, x1))) → A__FIRST(s(mark(y0)), cons(mark(x0), x1))
MARK(sel(0, from(x0))) → A__SEL(0, a__from(mark(x0)))
A__SEL(0, cons(X, Z)) → MARK(X)
A__SEL(s(from(y0)), cons(y1, cons(x0, x1))) → A__SEL(a__from(mark(y0)), cons(mark(x0), x1))
A__FIRST(s(X), cons(Y, Z)) → MARK(Y)
MARK(first(sel(y0, y1), cons(x0, x1))) → A__FIRST(a__sel(mark(y0), mark(y1)), cons(mark(x0), x1))
A__SEL(s(X), cons(Y, Z)) → MARK(X)
MARK(first(X1, X2)) → MARK(X1)
MARK(first(first(y0, y1), cons(x0, x1))) → A__FIRST(a__first(mark(y0), mark(y1)), cons(mark(x0), x1))
MARK(from(X)) → MARK(X)
A__SEL(s(s(x0)), cons(y1, y2)) → A__SEL(s(mark(x0)), mark(y2))
MARK(from(X)) → A__FROM(mark(X))
MARK(sel(s(y0), first(x0, x1))) → A__SEL(s(mark(y0)), a__first(mark(x0), mark(x1)))
MARK(sel(from(y0), from(x0))) → A__SEL(a__from(mark(y0)), a__from(mark(x0)))
MARK(first(s(y0), first(x0, x1))) → A__FIRST(s(mark(y0)), a__first(mark(x0), mark(x1)))
MARK(first(s(y0), sel(x0, x1))) → A__FIRST(s(mark(y0)), a__sel(mark(x0), mark(x1)))
MARK(sel(0, first(x0, x1))) → A__SEL(0, a__first(mark(x0), mark(x1)))
A__SEL(s(sel(y0, y1)), cons(y2, s(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), s(mark(x0)))
MARK(sel(0, sel(x0, x1))) → A__SEL(0, a__sel(mark(x0), mark(x1)))
A__SEL(s(sel(y0, y1)), cons(y2, first(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), a__first(mark(x0), mark(x1)))
A__SEL(s(first(y0, y1)), cons(y2, sel(x0, x1))) → A__SEL(a__first(mark(y0), mark(y1)), a__sel(mark(x0), mark(x1)))
MARK(sel(X1, X2)) → MARK(X2)
A__SEL(s(first(y0, y1)), cons(y2, first(x0, x1))) → A__SEL(a__first(mark(y0), mark(y1)), a__first(mark(x0), mark(x1)))
MARK(sel(from(y0), first(x0, x1))) → A__SEL(a__from(mark(y0)), a__first(mark(x0), mark(x1)))
MARK(sel(first(y0, y1), from(x0))) → A__SEL(a__first(mark(y0), mark(y1)), a__from(mark(x0)))

The TRS R consists of the following rules:

a__from(X) → cons(mark(X), from(s(X)))
a__first(0, Z) → nil
a__first(s(X), cons(Y, Z)) → cons(mark(Y), first(X, Z))
a__sel(0, cons(X, Z)) → mark(X)
a__sel(s(X), cons(Y, Z)) → a__sel(mark(X), mark(Z))
mark(from(X)) → a__from(mark(X))
mark(first(X1, X2)) → a__first(mark(X1), mark(X2))
mark(sel(X1, X2)) → a__sel(mark(X1), mark(X2))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(s(X)) → s(mark(X))
mark(0) → 0
mark(nil) → nil
a__from(X) → from(X)
a__first(X1, X2) → first(X1, X2)
a__sel(X1, X2) → sel(X1, X2)

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ Narrowing

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ QDP

              ↳ Narrowing

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ QDP

                      ↳ Narrowing

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ QDP

                              ↳ Narrowing

                                ↳ QDP

                                  ↳ DependencyGraphProof

                                    ↳ QDP

                                      ↳ Narrowing

                                        ↳ QDP

                                          ↳ DependencyGraphProof

                                            ↳ QDP

                                              ↳ Narrowing

                                                ↳ QDP

                                                  ↳ DependencyGraphProof

                                                    ↳ QDP

                                                      ↳ Narrowing

                                                        ↳ QDP

                                                          ↳ DependencyGraphProof

                                                            ↳ QDP

                                                              ↳ Narrowing

                                                                ↳ QDP

                                                                  ↳ DependencyGraphProof

                                                                    ↳ QDP

                                                                      ↳ Narrowing

                                                                        ↳ QDP

                                                                          ↳ DependencyGraphProof

                                                                            ↳ QDP

                                                                              ↳ Narrowing

                                                                                ↳ QDP

                                                                                  ↳ DependencyGraphProof

                                                                                    ↳ QDP

                                                                                      ↳ Narrowing

                                                                                        ↳ QDP

                                                                                          ↳ DependencyGraphProof

                                                                                            ↳ QDP

                                                                                              ↳ Narrowing

                                                                                                ↳ QDP

                                                                                                  ↳ DependencyGraphProof

                                                                                                    ↳ QDP

                                                                                                      ↳ Narrowing

                                                                                                        ↳ QDP

                                                                                                          ↳ DependencyGraphProof

                                                                                                            ↳ QDP

                                                                                                              ↳ Narrowing

                                                                                                                ↳ QDP

                                                                                                                  ↳ DependencyGraphProof

                                                                                                                    ↳ QDP

                                                                                                                      ↳ Narrowing

                                                                                                                        ↳ QDP

                                                                                                                          ↳ DependencyGraphProof

                                                                                                                            ↳ QDP

                                                                                                                              ↳ Narrowing

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

MARK(sel(sel(y0, y1), sel(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), a__sel(mark(x0), mark(x1)))
MARK(sel(s(y0), cons(x0, x1))) → A__SEL(s(mark(y0)), cons(mark(x0), x1))
A__SEL(s(sel(y0, y1)), cons(y2, from(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__from(mark(x0)))
MARK(sel(first(y0, y1), first(x0, x1))) → A__SEL(a__first(mark(y0), mark(y1)), a__first(mark(x0), mark(x1)))
MARK(first(first(y0, y1), from(x0))) → A__FIRST(a__first(mark(y0), mark(y1)), a__from(mark(x0)))
MARK(first(from(y0), first(x0, x1))) → A__FIRST(a__from(mark(y0)), a__first(mark(x0), mark(x1)))
MARK(cons(X1, X2)) → MARK(X1)
MARK(first(from(y0), sel(x0, x1))) → A__FIRST(a__from(mark(y0)), a__sel(mark(x0), mark(x1)))
MARK(first(sel(y0, y1), from(x0))) → A__FIRST(a__sel(mark(y0), mark(y1)), a__from(mark(x0)))
A__SEL(s(X), cons(Y, Z)) → MARK(Z)
MARK(first(from(y0), cons(x0, x1))) → A__FIRST(a__from(mark(y0)), cons(mark(x0), x1))
MARK(sel(s(y0), from(x0))) → A__SEL(s(mark(y0)), a__from(mark(x0)))
MARK(first(sel(y0, y1), sel(x0, x1))) → A__FIRST(a__sel(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)))
MARK(sel(X1, X2)) → MARK(X1)
MARK(sel(sel(y0, y1), cons(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), cons(mark(x0), x1))
MARK(first(from(y0), from(x0))) → A__FIRST(a__from(mark(y0)), a__from(mark(x0)))
MARK(first(X1, X2)) → MARK(X2)
A__SEL(s(from(y0)), cons(y1, first(x0, x1))) → A__SEL(a__from(mark(y0)), a__first(mark(x0), mark(x1)))
A__SEL(s(sel(y0, y1)), cons(y2, cons(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), cons(mark(x0), x1))
MARK(first(s(y0), from(x0))) → A__FIRST(s(mark(y0)), a__from(mark(x0)))
A__FROM(X) → MARK(X)
A__SEL(s(first(y0, y1)), cons(y2, cons(x0, x1))) → A__SEL(a__first(mark(y0), mark(y1)), cons(mark(x0), x1))
A__SEL(s(0), cons(y1, y2)) → A__SEL(0, mark(y2))
MARK(first(first(y0, y1), first(x0, x1))) → A__FIRST(a__first(mark(y0), mark(y1)), a__first(mark(x0), mark(x1)))
A__SEL(s(from(y0)), cons(y1, sel(x0, x1))) → A__SEL(a__from(mark(y0)), a__sel(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), from(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__from(mark(x0)))
MARK(sel(from(y0), sel(x0, x1))) → A__SEL(a__from(mark(y0)), a__sel(mark(x0), mark(x1)))
A__SEL(s(from(y0)), cons(y1, from(x0))) → A__SEL(a__from(mark(y0)), a__from(mark(x0)))
MARK(sel(0, cons(x0, x1))) → A__SEL(0, cons(mark(x0), x1))
A__SEL(s(first(y0, y1)), cons(y2, from(x0))) → A__SEL(a__first(mark(y0), mark(y1)), a__from(mark(x0)))
MARK(first(sel(y0, y1), first(x0, x1))) → A__FIRST(a__sel(mark(y0), mark(y1)), a__first(mark(x0), mark(x1)))
MARK(first(first(y0, y1), sel(x0, x1))) → A__FIRST(a__first(mark(y0), mark(y1)), a__sel(mark(x0), mark(x1)))
MARK(sel(first(y0, y1), cons(x0, x1))) → A__SEL(a__first(mark(y0), mark(y1)), cons(mark(x0), x1))
MARK(sel(from(y0), cons(x0, x1))) → A__SEL(a__from(mark(y0)), cons(mark(x0), x1))
MARK(s(X)) → MARK(X)
MARK(sel(s(y0), sel(x0, x1))) → A__SEL(s(mark(y0)), a__sel(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), first(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), a__first(mark(x0), mark(x1)))
MARK(sel(first(y0, y1), sel(x0, x1))) → A__SEL(a__first(mark(y0), mark(y1)), a__sel(mark(x0), mark(x1)))
MARK(first(s(y0), cons(x0, x1))) → A__FIRST(s(mark(y0)), cons(mark(x0), x1))
MARK(sel(0, from(x0))) → A__SEL(0, a__from(mark(x0)))
A__SEL(0, cons(X, Z)) → MARK(X)
A__FIRST(s(X), cons(Y, Z)) → MARK(Y)
A__SEL(s(from(y0)), cons(y1, cons(x0, x1))) → A__SEL(a__from(mark(y0)), cons(mark(x0), x1))
MARK(first(sel(y0, y1), cons(x0, x1))) → A__FIRST(a__sel(mark(y0), mark(y1)), cons(mark(x0), x1))
A__SEL(s(X), cons(Y, Z)) → MARK(X)
MARK(first(X1, X2)) → MARK(X1)
MARK(first(first(y0, y1), cons(x0, x1))) → A__FIRST(a__first(mark(y0), mark(y1)), cons(mark(x0), x1))
MARK(from(X)) → MARK(X)
MARK(sel(s(y0), first(x0, x1))) → A__SEL(s(mark(y0)), a__first(mark(x0), mark(x1)))
MARK(from(X)) → A__FROM(mark(X))
A__SEL(s(s(x0)), cons(y1, y2)) → A__SEL(s(mark(x0)), mark(y2))
MARK(first(s(y0), first(x0, x1))) → A__FIRST(s(mark(y0)), a__first(mark(x0), mark(x1)))
MARK(sel(from(y0), from(x0))) → A__SEL(a__from(mark(y0)), a__from(mark(x0)))
MARK(first(s(y0), sel(x0, x1))) → A__FIRST(s(mark(y0)), a__sel(mark(x0), mark(x1)))
MARK(sel(0, first(x0, x1))) → A__SEL(0, a__first(mark(x0), mark(x1)))
MARK(sel(0, sel(x0, x1))) → A__SEL(0, a__sel(mark(x0), mark(x1)))
A__SEL(s(first(y0, y1)), cons(y2, sel(x0, x1))) → A__SEL(a__first(mark(y0), mark(y1)), a__sel(mark(x0), mark(x1)))
A__SEL(s(sel(y0, y1)), cons(y2, first(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), a__first(mark(x0), mark(x1)))
MARK(sel(X1, X2)) → MARK(X2)
MARK(sel(first(y0, y1), from(x0))) → A__SEL(a__first(mark(y0), mark(y1)), a__from(mark(x0)))
MARK(sel(from(y0), first(x0, x1))) → A__SEL(a__from(mark(y0)), a__first(mark(x0), mark(x1)))
A__SEL(s(first(y0, y1)), cons(y2, first(x0, x1))) → A__SEL(a__first(mark(y0), mark(y1)), a__first(mark(x0), mark(x1)))

The TRS R consists of the following rules:

a__from(X) → cons(mark(X), from(s(X)))
a__first(0, Z) → nil
a__first(s(X), cons(Y, Z)) → cons(mark(Y), first(X, Z))
a__sel(0, cons(X, Z)) → mark(X)
a__sel(s(X), cons(Y, Z)) → a__sel(mark(X), mark(Z))
mark(from(X)) → a__from(mark(X))
mark(first(X1, X2)) → a__first(mark(X1), mark(X2))
mark(sel(X1, X2)) → a__sel(mark(X1), mark(X2))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(s(X)) → s(mark(X))
mark(0) → 0
mark(nil) → nil
a__from(X) → from(X)
a__first(X1, X2) → first(X1, X2)
a__sel(X1, X2) → sel(X1, X2)

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

A__SEL(s(0), cons(y0, first(x0, x1))) → A__SEL(0, a__first(mark(x0), mark(x1)))
A__SEL(s(0), cons(y0, nil)) → A__SEL(0, nil)
A__SEL(s(0), cons(y0, s(x0))) → A__SEL(0, s(mark(x0)))
A__SEL(s(0), cons(y0, from(x0))) → A__SEL(0, a__from(mark(x0)))
A__SEL(s(0), cons(y0, cons(x0, x1))) → A__SEL(0, cons(mark(x0), x1))
A__SEL(s(0), cons(y0, sel(x0, x1))) → A__SEL(0, a__sel(mark(x0), mark(x1)))
A__SEL(s(0), cons(y0, 0)) → A__SEL(0, 0)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ Narrowing

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ QDP

              ↳ Narrowing

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ QDP

                      ↳ Narrowing

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ QDP

                              ↳ Narrowing

                                ↳ QDP

                                  ↳ DependencyGraphProof

                                    ↳ QDP

                                      ↳ Narrowing

                                        ↳ QDP

                                          ↳ DependencyGraphProof

                                            ↳ QDP

                                              ↳ Narrowing

                                                ↳ QDP

                                                  ↳ DependencyGraphProof

                                                    ↳ QDP

                                                      ↳ Narrowing

                                                        ↳ QDP

                                                          ↳ DependencyGraphProof

                                                            ↳ QDP

                                                              ↳ Narrowing

                                                                ↳ QDP

                                                                  ↳ DependencyGraphProof

                                                                    ↳ QDP

                                                                      ↳ Narrowing

                                                                        ↳ QDP

                                                                          ↳ DependencyGraphProof

                                                                            ↳ QDP

                                                                              ↳ Narrowing

                                                                                ↳ QDP

                                                                                  ↳ DependencyGraphProof

                                                                                    ↳ QDP

                                                                                      ↳ Narrowing

                                                                                        ↳ QDP

                                                                                          ↳ DependencyGraphProof

                                                                                            ↳ QDP

                                                                                              ↳ Narrowing

                                                                                                ↳ QDP

                                                                                                  ↳ DependencyGraphProof

                                                                                                    ↳ QDP

                                                                                                      ↳ Narrowing

                                                                                                        ↳ QDP

                                                                                                          ↳ DependencyGraphProof

                                                                                                            ↳ QDP

                                                                                                              ↳ Narrowing

                                                                                                                ↳ QDP

                                                                                                                  ↳ DependencyGraphProof

                                                                                                                    ↳ QDP

                                                                                                                      ↳ Narrowing

                                                                                                                        ↳ QDP

                                                                                                                          ↳ DependencyGraphProof

                                                                                                                            ↳ QDP

                                                                                                                              ↳ Narrowing

                                                                                                                                ↳ QDP

                                                                                                                                  ↳ DependencyGraphProof

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

MARK(sel(sel(y0, y1), sel(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), a__sel(mark(x0), mark(x1)))
MARK(sel(s(y0), cons(x0, x1))) → A__SEL(s(mark(y0)), cons(mark(x0), x1))
A__SEL(s(sel(y0, y1)), cons(y2, from(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__from(mark(x0)))
A__SEL(s(0), cons(y0, s(x0))) → A__SEL(0, s(mark(x0)))
MARK(sel(first(y0, y1), first(x0, x1))) → A__SEL(a__first(mark(y0), mark(y1)), a__first(mark(x0), mark(x1)))
MARK(cons(X1, X2)) → MARK(X1)
MARK(first(from(y0), first(x0, x1))) → A__FIRST(a__from(mark(y0)), a__first(mark(x0), mark(x1)))
MARK(first(first(y0, y1), from(x0))) → A__FIRST(a__first(mark(y0), mark(y1)), a__from(mark(x0)))
MARK(first(sel(y0, y1), from(x0))) → A__FIRST(a__sel(mark(y0), mark(y1)), a__from(mark(x0)))
MARK(first(from(y0), sel(x0, x1))) → A__FIRST(a__from(mark(y0)), a__sel(mark(x0), mark(x1)))
A__SEL(s(0), cons(y0, first(x0, x1))) → A__SEL(0, a__first(mark(x0), mark(x1)))
A__SEL(s(X), cons(Y, Z)) → MARK(Z)
MARK(first(from(y0), cons(x0, x1))) → A__FIRST(a__from(mark(y0)), cons(mark(x0), x1))
MARK(sel(s(y0), from(x0))) → A__SEL(s(mark(y0)), a__from(mark(x0)))
MARK(first(sel(y0, y1), sel(x0, x1))) → A__FIRST(a__sel(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)))
MARK(sel(X1, X2)) → MARK(X1)
MARK(sel(sel(y0, y1), cons(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), cons(mark(x0), x1))
MARK(first(from(y0), from(x0))) → A__FIRST(a__from(mark(y0)), a__from(mark(x0)))
A__SEL(s(from(y0)), cons(y1, first(x0, x1))) → A__SEL(a__from(mark(y0)), a__first(mark(x0), mark(x1)))
MARK(first(X1, X2)) → MARK(X2)
A__SEL(s(sel(y0, y1)), cons(y2, cons(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), cons(mark(x0), x1))
A__FROM(X) → MARK(X)
MARK(first(s(y0), from(x0))) → A__FIRST(s(mark(y0)), a__from(mark(x0)))
A__SEL(s(first(y0, y1)), cons(y2, cons(x0, x1))) → A__SEL(a__first(mark(y0), mark(y1)), cons(mark(x0), x1))
MARK(first(first(y0, y1), first(x0, x1))) → A__FIRST(a__first(mark(y0), mark(y1)), a__first(mark(x0), mark(x1)))
A__SEL(s(from(y0)), cons(y1, sel(x0, x1))) → A__SEL(a__from(mark(y0)), a__sel(mark(x0), mark(x1)))
A__SEL(s(0), cons(y0, nil)) → A__SEL(0, nil)
MARK(sel(0, cons(x0, x1))) → A__SEL(0, cons(mark(x0), x1))
A__SEL(s(from(y0)), cons(y1, from(x0))) → A__SEL(a__from(mark(y0)), a__from(mark(x0)))
MARK(sel(from(y0), sel(x0, x1))) → A__SEL(a__from(mark(y0)), a__sel(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), from(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__from(mark(x0)))
A__SEL(s(first(y0, y1)), cons(y2, from(x0))) → A__SEL(a__first(mark(y0), mark(y1)), a__from(mark(x0)))
MARK(first(first(y0, y1), sel(x0, x1))) → A__FIRST(a__first(mark(y0), mark(y1)), a__sel(mark(x0), mark(x1)))
MARK(first(sel(y0, y1), first(x0, x1))) → A__FIRST(a__sel(mark(y0), mark(y1)), a__first(mark(x0), mark(x1)))
MARK(sel(first(y0, y1), cons(x0, x1))) → A__SEL(a__first(mark(y0), mark(y1)), cons(mark(x0), x1))
A__SEL(s(0), cons(y0, sel(x0, x1))) → A__SEL(0, a__sel(mark(x0), mark(x1)))
MARK(sel(from(y0), cons(x0, x1))) → A__SEL(a__from(mark(y0)), cons(mark(x0), x1))
MARK(s(X)) → MARK(X)
MARK(sel(first(y0, y1), sel(x0, x1))) → A__SEL(a__first(mark(y0), mark(y1)), a__sel(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), first(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), a__first(mark(x0), mark(x1)))
MARK(sel(s(y0), sel(x0, x1))) → A__SEL(s(mark(y0)), a__sel(mark(x0), mark(x1)))
MARK(first(s(y0), cons(x0, x1))) → A__FIRST(s(mark(y0)), cons(mark(x0), x1))
A__SEL(0, cons(X, Z)) → MARK(X)
MARK(sel(0, from(x0))) → A__SEL(0, a__from(mark(x0)))
A__SEL(s(from(y0)), cons(y1, cons(x0, x1))) → A__SEL(a__from(mark(y0)), cons(mark(x0), x1))
A__FIRST(s(X), cons(Y, Z)) → MARK(Y)
MARK(first(sel(y0, y1), cons(x0, x1))) → A__FIRST(a__sel(mark(y0), mark(y1)), cons(mark(x0), x1))
A__SEL(s(0), cons(y0, cons(x0, x1))) → A__SEL(0, cons(mark(x0), x1))
A__SEL(s(X), cons(Y, Z)) → MARK(X)
MARK(first(X1, X2)) → MARK(X1)
MARK(first(first(y0, y1), cons(x0, x1))) → A__FIRST(a__first(mark(y0), mark(y1)), cons(mark(x0), x1))
MARK(from(X)) → MARK(X)
A__SEL(s(s(x0)), cons(y1, y2)) → A__SEL(s(mark(x0)), mark(y2))
MARK(from(X)) → A__FROM(mark(X))
MARK(sel(s(y0), first(x0, x1))) → A__SEL(s(mark(y0)), a__first(mark(x0), mark(x1)))
MARK(sel(from(y0), from(x0))) → A__SEL(a__from(mark(y0)), a__from(mark(x0)))
MARK(first(s(y0), first(x0, x1))) → A__FIRST(s(mark(y0)), a__first(mark(x0), mark(x1)))
MARK(first(s(y0), sel(x0, x1))) → A__FIRST(s(mark(y0)), a__sel(mark(x0), mark(x1)))
A__SEL(s(0), cons(y0, 0)) → A__SEL(0, 0)
MARK(sel(0, first(x0, x1))) → A__SEL(0, a__first(mark(x0), mark(x1)))
MARK(sel(0, sel(x0, x1))) → A__SEL(0, a__sel(mark(x0), mark(x1)))
A__SEL(s(sel(y0, y1)), cons(y2, first(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), a__first(mark(x0), mark(x1)))
A__SEL(s(first(y0, y1)), cons(y2, sel(x0, x1))) → A__SEL(a__first(mark(y0), mark(y1)), a__sel(mark(x0), mark(x1)))
A__SEL(s(0), cons(y0, from(x0))) → A__SEL(0, a__from(mark(x0)))
MARK(sel(X1, X2)) → MARK(X2)
A__SEL(s(first(y0, y1)), cons(y2, first(x0, x1))) → A__SEL(a__first(mark(y0), mark(y1)), a__first(mark(x0), mark(x1)))
MARK(sel(from(y0), first(x0, x1))) → A__SEL(a__from(mark(y0)), a__first(mark(x0), mark(x1)))
MARK(sel(first(y0, y1), from(x0))) → A__SEL(a__first(mark(y0), mark(y1)), a__from(mark(x0)))

The TRS R consists of the following rules:

a__from(X) → cons(mark(X), from(s(X)))
a__first(0, Z) → nil
a__first(s(X), cons(Y, Z)) → cons(mark(Y), first(X, Z))
a__sel(0, cons(X, Z)) → mark(X)
a__sel(s(X), cons(Y, Z)) → a__sel(mark(X), mark(Z))
mark(from(X)) → a__from(mark(X))
mark(first(X1, X2)) → a__first(mark(X1), mark(X2))
mark(sel(X1, X2)) → a__sel(mark(X1), mark(X2))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(s(X)) → s(mark(X))
mark(0) → 0
mark(nil) → nil
a__from(X) → from(X)
a__first(X1, X2) → first(X1, X2)
a__sel(X1, X2) → sel(X1, X2)

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ Narrowing

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ QDP

              ↳ Narrowing

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ QDP

                      ↳ Narrowing

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ QDP

                              ↳ Narrowing

                                ↳ QDP

                                  ↳ DependencyGraphProof

                                    ↳ QDP

                                      ↳ Narrowing

                                        ↳ QDP

                                          ↳ DependencyGraphProof

                                            ↳ QDP

                                              ↳ Narrowing

                                                ↳ QDP

                                                  ↳ DependencyGraphProof

                                                    ↳ QDP

                                                      ↳ Narrowing

                                                        ↳ QDP

                                                          ↳ DependencyGraphProof

                                                            ↳ QDP

                                                              ↳ Narrowing

                                                                ↳ QDP

                                                                  ↳ DependencyGraphProof

                                                                    ↳ QDP

                                                                      ↳ Narrowing

                                                                        ↳ QDP

                                                                          ↳ DependencyGraphProof

                                                                            ↳ QDP

                                                                              ↳ Narrowing

                                                                                ↳ QDP

                                                                                  ↳ DependencyGraphProof

                                                                                    ↳ QDP

                                                                                      ↳ Narrowing

                                                                                        ↳ QDP

                                                                                          ↳ DependencyGraphProof

                                                                                            ↳ QDP

                                                                                              ↳ Narrowing

                                                                                                ↳ QDP

                                                                                                  ↳ DependencyGraphProof

                                                                                                    ↳ QDP

                                                                                                      ↳ Narrowing

                                                                                                        ↳ QDP

                                                                                                          ↳ DependencyGraphProof

                                                                                                            ↳ QDP

                                                                                                              ↳ Narrowing

                                                                                                                ↳ QDP

                                                                                                                  ↳ DependencyGraphProof

                                                                                                                    ↳ QDP

                                                                                                                      ↳ Narrowing

                                                                                                                        ↳ QDP

                                                                                                                          ↳ DependencyGraphProof

                                                                                                                            ↳ QDP

                                                                                                                              ↳ Narrowing

                                                                                                                                ↳ QDP

                                                                                                                                  ↳ DependencyGraphProof

                                                                                                                                    ↳ QDP

                                                                                                                                      ↳ Narrowing

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

MARK(sel(sel(y0, y1), sel(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), a__sel(mark(x0), mark(x1)))
MARK(sel(s(y0), cons(x0, x1))) → A__SEL(s(mark(y0)), cons(mark(x0), x1))
A__SEL(s(sel(y0, y1)), cons(y2, from(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__from(mark(x0)))
MARK(sel(first(y0, y1), first(x0, x1))) → A__SEL(a__first(mark(y0), mark(y1)), a__first(mark(x0), mark(x1)))
MARK(first(first(y0, y1), from(x0))) → A__FIRST(a__first(mark(y0), mark(y1)), a__from(mark(x0)))
MARK(first(from(y0), first(x0, x1))) → A__FIRST(a__from(mark(y0)), a__first(mark(x0), mark(x1)))
MARK(cons(X1, X2)) → MARK(X1)
MARK(first(from(y0), sel(x0, x1))) → A__FIRST(a__from(mark(y0)), a__sel(mark(x0), mark(x1)))
MARK(first(sel(y0, y1), from(x0))) → A__FIRST(a__sel(mark(y0), mark(y1)), a__from(mark(x0)))
A__SEL(s(0), cons(y0, first(x0, x1))) → A__SEL(0, a__first(mark(x0), mark(x1)))
A__SEL(s(X), cons(Y, Z)) → MARK(Z)
MARK(first(from(y0), cons(x0, x1))) → A__FIRST(a__from(mark(y0)), cons(mark(x0), x1))
MARK(sel(s(y0), from(x0))) → A__SEL(s(mark(y0)), a__from(mark(x0)))
MARK(first(sel(y0, y1), sel(x0, x1))) → A__FIRST(a__sel(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)))
MARK(sel(X1, X2)) → MARK(X1)
MARK(sel(sel(y0, y1), cons(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), cons(mark(x0), x1))
MARK(first(from(y0), from(x0))) → A__FIRST(a__from(mark(y0)), a__from(mark(x0)))
MARK(first(X1, X2)) → MARK(X2)
A__SEL(s(from(y0)), cons(y1, first(x0, x1))) → A__SEL(a__from(mark(y0)), a__first(mark(x0), mark(x1)))
A__SEL(s(sel(y0, y1)), cons(y2, cons(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), cons(mark(x0), x1))
MARK(first(s(y0), from(x0))) → A__FIRST(s(mark(y0)), a__from(mark(x0)))
A__FROM(X) → MARK(X)
A__SEL(s(first(y0, y1)), cons(y2, cons(x0, x1))) → A__SEL(a__first(mark(y0), mark(y1)), cons(mark(x0), x1))
MARK(first(first(y0, y1), first(x0, x1))) → A__FIRST(a__first(mark(y0), mark(y1)), a__first(mark(x0), mark(x1)))
A__SEL(s(from(y0)), cons(y1, sel(x0, x1))) → A__SEL(a__from(mark(y0)), a__sel(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), from(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__from(mark(x0)))
MARK(sel(0, cons(x0, x1))) → A__SEL(0, cons(mark(x0), x1))
MARK(sel(from(y0), sel(x0, x1))) → A__SEL(a__from(mark(y0)), a__sel(mark(x0), mark(x1)))
A__SEL(s(from(y0)), cons(y1, from(x0))) → A__SEL(a__from(mark(y0)), a__from(mark(x0)))
A__SEL(s(first(y0, y1)), cons(y2, from(x0))) → A__SEL(a__first(mark(y0), mark(y1)), a__from(mark(x0)))
MARK(first(sel(y0, y1), first(x0, x1))) → A__FIRST(a__sel(mark(y0), mark(y1)), a__first(mark(x0), mark(x1)))
MARK(first(first(y0, y1), sel(x0, x1))) → A__FIRST(a__first(mark(y0), mark(y1)), a__sel(mark(x0), mark(x1)))
MARK(sel(first(y0, y1), cons(x0, x1))) → A__SEL(a__first(mark(y0), mark(y1)), cons(mark(x0), x1))
A__SEL(s(0), cons(y0, sel(x0, x1))) → A__SEL(0, a__sel(mark(x0), mark(x1)))
MARK(sel(from(y0), cons(x0, x1))) → A__SEL(a__from(mark(y0)), cons(mark(x0), x1))
MARK(s(X)) → MARK(X)
MARK(sel(s(y0), sel(x0, x1))) → A__SEL(s(mark(y0)), a__sel(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), first(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), a__first(mark(x0), mark(x1)))
MARK(sel(first(y0, y1), sel(x0, x1))) → A__SEL(a__first(mark(y0), mark(y1)), a__sel(mark(x0), mark(x1)))
MARK(first(s(y0), cons(x0, x1))) → A__FIRST(s(mark(y0)), cons(mark(x0), x1))
MARK(sel(0, from(x0))) → A__SEL(0, a__from(mark(x0)))
A__SEL(0, cons(X, Z)) → MARK(X)
A__FIRST(s(X), cons(Y, Z)) → MARK(Y)
A__SEL(s(from(y0)), cons(y1, cons(x0, x1))) → A__SEL(a__from(mark(y0)), cons(mark(x0), x1))
MARK(first(sel(y0, y1), cons(x0, x1))) → A__FIRST(a__sel(mark(y0), mark(y1)), cons(mark(x0), x1))
A__SEL(s(0), cons(y0, cons(x0, x1))) → A__SEL(0, cons(mark(x0), x1))
A__SEL(s(X), cons(Y, Z)) → MARK(X)
MARK(first(X1, X2)) → MARK(X1)
MARK(first(first(y0, y1), cons(x0, x1))) → A__FIRST(a__first(mark(y0), mark(y1)), cons(mark(x0), x1))
MARK(from(X)) → MARK(X)
MARK(sel(s(y0), first(x0, x1))) → A__SEL(s(mark(y0)), a__first(mark(x0), mark(x1)))
MARK(from(X)) → A__FROM(mark(X))
A__SEL(s(s(x0)), cons(y1, y2)) → A__SEL(s(mark(x0)), mark(y2))
MARK(first(s(y0), first(x0, x1))) → A__FIRST(s(mark(y0)), a__first(mark(x0), mark(x1)))
MARK(sel(from(y0), from(x0))) → A__SEL(a__from(mark(y0)), a__from(mark(x0)))
MARK(first(s(y0), sel(x0, x1))) → A__FIRST(s(mark(y0)), a__sel(mark(x0), mark(x1)))
MARK(sel(0, first(x0, x1))) → A__SEL(0, a__first(mark(x0), mark(x1)))
MARK(sel(0, sel(x0, x1))) → A__SEL(0, a__sel(mark(x0), mark(x1)))
A__SEL(s(first(y0, y1)), cons(y2, sel(x0, x1))) → A__SEL(a__first(mark(y0), mark(y1)), a__sel(mark(x0), mark(x1)))
A__SEL(s(sel(y0, y1)), cons(y2, first(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), a__first(mark(x0), mark(x1)))
A__SEL(s(0), cons(y0, from(x0))) → A__SEL(0, a__from(mark(x0)))
MARK(sel(X1, X2)) → MARK(X2)
MARK(sel(first(y0, y1), from(x0))) → A__SEL(a__first(mark(y0), mark(y1)), a__from(mark(x0)))
MARK(sel(from(y0), first(x0, x1))) → A__SEL(a__from(mark(y0)), a__first(mark(x0), mark(x1)))
A__SEL(s(first(y0, y1)), cons(y2, first(x0, x1))) → A__SEL(a__first(mark(y0), mark(y1)), a__first(mark(x0), mark(x1)))

The TRS R consists of the following rules:

a__from(X) → cons(mark(X), from(s(X)))
a__first(0, Z) → nil
a__first(s(X), cons(Y, Z)) → cons(mark(Y), first(X, Z))
a__sel(0, cons(X, Z)) → mark(X)
a__sel(s(X), cons(Y, Z)) → a__sel(mark(X), mark(Z))
mark(from(X)) → a__from(mark(X))
mark(first(X1, X2)) → a__first(mark(X1), mark(X2))
mark(sel(X1, X2)) → a__sel(mark(X1), mark(X2))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(s(X)) → s(mark(X))
mark(0) → 0
mark(nil) → nil
a__from(X) → from(X)
a__first(X1, X2) → first(X1, X2)
a__sel(X1, X2) → sel(X1, X2)

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(mark(x0)), mark(y2)) at position [1] we obtained the following new rules:

A__SEL(s(s(y0)), cons(y1, s(x0))) → A__SEL(s(mark(y0)), s(mark(x0)))
A__SEL(s(s(y0)), cons(y1, sel(x0, x1))) → A__SEL(s(mark(y0)), a__sel(mark(x0), mark(x1)))
A__SEL(s(s(y0)), cons(y1, 0)) → A__SEL(s(mark(y0)), 0)
A__SEL(s(s(y0)), cons(y1, nil)) → A__SEL(s(mark(y0)), nil)
A__SEL(s(s(y0)), cons(y1, first(x0, x1))) → A__SEL(s(mark(y0)), a__first(mark(x0), mark(x1)))
A__SEL(s(s(y0)), cons(y1, cons(x0, x1))) → A__SEL(s(mark(y0)), cons(mark(x0), x1))
A__SEL(s(s(y0)), cons(y1, from(x0))) → A__SEL(s(mark(y0)), a__from(mark(x0)))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ Narrowing

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ QDP

              ↳ Narrowing

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ QDP

                      ↳ Narrowing

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ QDP

                              ↳ Narrowing

                                ↳ QDP

                                  ↳ DependencyGraphProof

                                    ↳ QDP

                                      ↳ Narrowing

                                        ↳ QDP

                                          ↳ DependencyGraphProof

                                            ↳ QDP

                                              ↳ Narrowing

                                                ↳ QDP

                                                  ↳ DependencyGraphProof

                                                    ↳ QDP

                                                      ↳ Narrowing

                                                        ↳ QDP

                                                          ↳ DependencyGraphProof

                                                            ↳ QDP

                                                              ↳ Narrowing

                                                                ↳ QDP

                                                                  ↳ DependencyGraphProof

                                                                    ↳ QDP

                                                                      ↳ Narrowing

                                                                        ↳ QDP

                                                                          ↳ DependencyGraphProof

                                                                            ↳ QDP

                                                                              ↳ Narrowing

                                                                                ↳ QDP

                                                                                  ↳ DependencyGraphProof

                                                                                    ↳ QDP

                                                                                      ↳ Narrowing

                                                                                        ↳ QDP

                                                                                          ↳ DependencyGraphProof

                                                                                            ↳ QDP

                                                                                              ↳ Narrowing

                                                                                                ↳ QDP

                                                                                                  ↳ DependencyGraphProof

                                                                                                    ↳ QDP

                                                                                                      ↳ Narrowing

                                                                                                        ↳ QDP

                                                                                                          ↳ DependencyGraphProof

                                                                                                            ↳ QDP

                                                                                                              ↳ Narrowing

                                                                                                                ↳ QDP

                                                                                                                  ↳ DependencyGraphProof

                                                                                                                    ↳ QDP

                                                                                                                      ↳ Narrowing

                                                                                                                        ↳ QDP

                                                                                                                          ↳ DependencyGraphProof

                                                                                                                            ↳ QDP

                                                                                                                              ↳ Narrowing

                                                                                                                                ↳ QDP

                                                                                                                                  ↳ DependencyGraphProof

                                                                                                                                    ↳ QDP

                                                                                                                                      ↳ Narrowing

                                                                                                                                        ↳ QDP

                                                                                                                                          ↳ DependencyGraphProof

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

A__SEL(s(s(y0)), cons(y1, s(x0))) → A__SEL(s(mark(y0)), s(mark(x0)))
MARK(sel(sel(y0, y1), sel(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), a__sel(mark(x0), mark(x1)))
MARK(sel(s(y0), cons(x0, x1))) → A__SEL(s(mark(y0)), cons(mark(x0), x1))
A__SEL(s(sel(y0, y1)), cons(y2, from(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__from(mark(x0)))
MARK(sel(first(y0, y1), first(x0, x1))) → A__SEL(a__first(mark(y0), mark(y1)), a__first(mark(x0), mark(x1)))
MARK(cons(X1, X2)) → MARK(X1)
MARK(first(from(y0), first(x0, x1))) → A__FIRST(a__from(mark(y0)), a__first(mark(x0), mark(x1)))
MARK(first(first(y0, y1), from(x0))) → A__FIRST(a__first(mark(y0), mark(y1)), a__from(mark(x0)))
MARK(first(sel(y0, y1), from(x0))) → A__FIRST(a__sel(mark(y0), mark(y1)), a__from(mark(x0)))
MARK(first(from(y0), sel(x0, x1))) → A__FIRST(a__from(mark(y0)), a__sel(mark(x0), mark(x1)))
A__SEL(s(0), cons(y0, first(x0, x1))) → A__SEL(0, a__first(mark(x0), mark(x1)))
A__SEL(s(X), cons(Y, Z)) → MARK(Z)
MARK(first(from(y0), cons(x0, x1))) → A__FIRST(a__from(mark(y0)), cons(mark(x0), x1))
A__SEL(s(s(y0)), cons(y1, nil)) → A__SEL(s(mark(y0)), nil)
MARK(sel(s(y0), from(x0))) → A__SEL(s(mark(y0)), a__from(mark(x0)))
MARK(first(sel(y0, y1), sel(x0, x1))) → A__FIRST(a__sel(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(s(y0)), cons(y1, sel(x0, x1))) → A__SEL(s(mark(y0)), a__sel(mark(x0), mark(x1)))
MARK(sel(X1, X2)) → MARK(X1)
MARK(sel(sel(y0, y1), cons(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), cons(mark(x0), x1))
MARK(first(from(y0), from(x0))) → A__FIRST(a__from(mark(y0)), a__from(mark(x0)))
A__SEL(s(s(y0)), cons(y1, first(x0, x1))) → A__SEL(s(mark(y0)), a__first(mark(x0), mark(x1)))
A__SEL(s(from(y0)), cons(y1, first(x0, x1))) → A__SEL(a__from(mark(y0)), a__first(mark(x0), mark(x1)))
MARK(first(X1, X2)) → MARK(X2)
A__SEL(s(sel(y0, y1)), cons(y2, cons(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), cons(mark(x0), x1))
A__SEL(s(s(y0)), cons(y1, cons(x0, x1))) → A__SEL(s(mark(y0)), cons(mark(x0), x1))
A__FROM(X) → MARK(X)
MARK(first(s(y0), from(x0))) → A__FIRST(s(mark(y0)), a__from(mark(x0)))
A__SEL(s(first(y0, y1)), cons(y2, cons(x0, x1))) → A__SEL(a__first(mark(y0), mark(y1)), cons(mark(x0), x1))
A__SEL(s(s(y0)), cons(y1, 0)) → A__SEL(s(mark(y0)), 0)
MARK(first(first(y0, y1), first(x0, x1))) → A__FIRST(a__first(mark(y0), mark(y1)), a__first(mark(x0), mark(x1)))
A__SEL(s(from(y0)), cons(y1, sel(x0, x1))) → A__SEL(a__from(mark(y0)), a__sel(mark(x0), mark(x1)))
A__SEL(s(from(y0)), cons(y1, from(x0))) → A__SEL(a__from(mark(y0)), a__from(mark(x0)))
MARK(sel(from(y0), sel(x0, x1))) → A__SEL(a__from(mark(y0)), a__sel(mark(x0), mark(x1)))
MARK(sel(0, cons(x0, x1))) → A__SEL(0, cons(mark(x0), x1))
MARK(sel(sel(y0, y1), from(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__from(mark(x0)))
A__SEL(s(first(y0, y1)), cons(y2, from(x0))) → A__SEL(a__first(mark(y0), mark(y1)), a__from(mark(x0)))
MARK(first(first(y0, y1), sel(x0, x1))) → A__FIRST(a__first(mark(y0), mark(y1)), a__sel(mark(x0), mark(x1)))
MARK(first(sel(y0, y1), first(x0, x1))) → A__FIRST(a__sel(mark(y0), mark(y1)), a__first(mark(x0), mark(x1)))
MARK(sel(first(y0, y1), cons(x0, x1))) → A__SEL(a__first(mark(y0), mark(y1)), cons(mark(x0), x1))
A__SEL(s(0), cons(y0, sel(x0, x1))) → A__SEL(0, a__sel(mark(x0), mark(x1)))
MARK(sel(from(y0), cons(x0, x1))) → A__SEL(a__from(mark(y0)), cons(mark(x0), x1))
MARK(s(X)) → MARK(X)
MARK(sel(first(y0, y1), sel(x0, x1))) → A__SEL(a__first(mark(y0), mark(y1)), a__sel(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), first(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), a__first(mark(x0), mark(x1)))
MARK(sel(s(y0), sel(x0, x1))) → A__SEL(s(mark(y0)), a__sel(mark(x0), mark(x1)))
MARK(first(s(y0), cons(x0, x1))) → A__FIRST(s(mark(y0)), cons(mark(x0), x1))
A__SEL(0, cons(X, Z)) → MARK(X)
MARK(sel(0, from(x0))) → A__SEL(0, a__from(mark(x0)))
A__SEL(s(from(y0)), cons(y1, cons(x0, x1))) → A__SEL(a__from(mark(y0)), cons(mark(x0), x1))
A__FIRST(s(X), cons(Y, Z)) → MARK(Y)
MARK(first(sel(y0, y1), cons(x0, x1))) → A__FIRST(a__sel(mark(y0), mark(y1)), cons(mark(x0), x1))
A__SEL(s(0), cons(y0, cons(x0, x1))) → A__SEL(0, cons(mark(x0), x1))
A__SEL(s(s(y0)), cons(y1, from(x0))) → A__SEL(s(mark(y0)), a__from(mark(x0)))
A__SEL(s(X), cons(Y, Z)) → MARK(X)
MARK(first(X1, X2)) → MARK(X1)
MARK(first(first(y0, y1), cons(x0, x1))) → A__FIRST(a__first(mark(y0), mark(y1)), cons(mark(x0), x1))
MARK(from(X)) → MARK(X)
MARK(from(X)) → A__FROM(mark(X))
MARK(sel(s(y0), first(x0, x1))) → A__SEL(s(mark(y0)), a__first(mark(x0), mark(x1)))
MARK(sel(from(y0), from(x0))) → A__SEL(a__from(mark(y0)), a__from(mark(x0)))
MARK(first(s(y0), first(x0, x1))) → A__FIRST(s(mark(y0)), a__first(mark(x0), mark(x1)))
MARK(first(s(y0), sel(x0, x1))) → A__FIRST(s(mark(y0)), a__sel(mark(x0), mark(x1)))
MARK(sel(0, first(x0, x1))) → A__SEL(0, a__first(mark(x0), mark(x1)))
MARK(sel(0, sel(x0, x1))) → A__SEL(0, a__sel(mark(x0), mark(x1)))
A__SEL(s(sel(y0, y1)), cons(y2, first(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), a__first(mark(x0), mark(x1)))
A__SEL(s(first(y0, y1)), cons(y2, sel(x0, x1))) → A__SEL(a__first(mark(y0), mark(y1)), a__sel(mark(x0), mark(x1)))
A__SEL(s(0), cons(y0, from(x0))) → A__SEL(0, a__from(mark(x0)))
MARK(sel(X1, X2)) → MARK(X2)
A__SEL(s(first(y0, y1)), cons(y2, first(x0, x1))) → A__SEL(a__first(mark(y0), mark(y1)), a__first(mark(x0), mark(x1)))
MARK(sel(from(y0), first(x0, x1))) → A__SEL(a__from(mark(y0)), a__first(mark(x0), mark(x1)))
MARK(sel(first(y0, y1), from(x0))) → A__SEL(a__first(mark(y0), mark(y1)), a__from(mark(x0)))

The TRS R consists of the following rules:

a__from(X) → cons(mark(X), from(s(X)))
a__first(0, Z) → nil
a__first(s(X), cons(Y, Z)) → cons(mark(Y), first(X, Z))
a__sel(0, cons(X, Z)) → mark(X)
a__sel(s(X), cons(Y, Z)) → a__sel(mark(X), mark(Z))
mark(from(X)) → a__from(mark(X))
mark(first(X1, X2)) → a__first(mark(X1), mark(X2))
mark(sel(X1, X2)) → a__sel(mark(X1), mark(X2))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(s(X)) → s(mark(X))
mark(0) → 0
mark(nil) → nil
a__from(X) → from(X)
a__first(X1, X2) → first(X1, X2)
a__sel(X1, X2) → sel(X1, X2)

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ Narrowing

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ QDP

              ↳ Narrowing

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ QDP

                      ↳ Narrowing

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ QDP

                              ↳ Narrowing

                                ↳ QDP

                                  ↳ DependencyGraphProof

                                    ↳ QDP

                                      ↳ Narrowing

                                        ↳ QDP

                                          ↳ DependencyGraphProof

                                            ↳ QDP

                                              ↳ Narrowing

                                                ↳ QDP

                                                  ↳ DependencyGraphProof

                                                    ↳ QDP

                                                      ↳ Narrowing

                                                        ↳ QDP

                                                          ↳ DependencyGraphProof

                                                            ↳ QDP

                                                              ↳ Narrowing

                                                                ↳ QDP

                                                                  ↳ DependencyGraphProof

                                                                    ↳ QDP

                                                                      ↳ Narrowing

                                                                        ↳ QDP

                                                                          ↳ DependencyGraphProof

                                                                            ↳ QDP

                                                                              ↳ Narrowing

                                                                                ↳ QDP

                                                                                  ↳ DependencyGraphProof

                                                                                    ↳ QDP

                                                                                      ↳ Narrowing

                                                                                        ↳ QDP

                                                                                          ↳ DependencyGraphProof

                                                                                            ↳ QDP

                                                                                              ↳ 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

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

MARK(sel(sel(y0, y1), sel(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), a__sel(mark(x0), mark(x1)))
MARK(sel(s(y0), cons(x0, x1))) → A__SEL(s(mark(y0)), cons(mark(x0), x1))
A__SEL(s(sel(y0, y1)), cons(y2, from(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__from(mark(x0)))
MARK(sel(first(y0, y1), first(x0, x1))) → A__SEL(a__first(mark(y0), mark(y1)), a__first(mark(x0), mark(x1)))
MARK(first(first(y0, y1), from(x0))) → A__FIRST(a__first(mark(y0), mark(y1)), a__from(mark(x0)))
MARK(first(from(y0), first(x0, x1))) → A__FIRST(a__from(mark(y0)), a__first(mark(x0), mark(x1)))
MARK(cons(X1, X2)) → MARK(X1)
MARK(first(from(y0), sel(x0, x1))) → A__FIRST(a__from(mark(y0)), a__sel(mark(x0), mark(x1)))
MARK(first(sel(y0, y1), from(x0))) → A__FIRST(a__sel(mark(y0), mark(y1)), a__from(mark(x0)))
A__SEL(s(0), cons(y0, first(x0, x1))) → A__SEL(0, a__first(mark(x0), mark(x1)))
A__SEL(s(X), cons(Y, Z)) → MARK(Z)
MARK(first(from(y0), cons(x0, x1))) → A__FIRST(a__from(mark(y0)), cons(mark(x0), x1))
MARK(sel(s(y0), from(x0))) → A__SEL(s(mark(y0)), a__from(mark(x0)))
MARK(first(sel(y0, y1), sel(x0, x1))) → A__FIRST(a__sel(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(s(y0)), cons(y1, sel(x0, x1))) → A__SEL(s(mark(y0)), a__sel(mark(x0), mark(x1)))
MARK(sel(X1, X2)) → MARK(X1)
MARK(sel(sel(y0, y1), cons(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), cons(mark(x0), x1))
MARK(first(from(y0), from(x0))) → A__FIRST(a__from(mark(y0)), a__from(mark(x0)))
A__SEL(s(s(y0)), cons(y1, first(x0, x1))) → A__SEL(s(mark(y0)), a__first(mark(x0), mark(x1)))
MARK(first(X1, X2)) → MARK(X2)
A__SEL(s(from(y0)), cons(y1, first(x0, x1))) → A__SEL(a__from(mark(y0)), a__first(mark(x0), mark(x1)))
A__SEL(s(sel(y0, y1)), cons(y2, cons(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), cons(mark(x0), x1))
A__SEL(s(s(y0)), cons(y1, cons(x0, x1))) → A__SEL(s(mark(y0)), cons(mark(x0), x1))
MARK(first(s(y0), from(x0))) → A__FIRST(s(mark(y0)), a__from(mark(x0)))
A__FROM(X) → MARK(X)
A__SEL(s(first(y0, y1)), cons(y2, cons(x0, x1))) → A__SEL(a__first(mark(y0), mark(y1)), cons(mark(x0), x1))
MARK(first(first(y0, y1), first(x0, x1))) → A__FIRST(a__first(mark(y0), mark(y1)), a__first(mark(x0), mark(x1)))
A__SEL(s(from(y0)), cons(y1, sel(x0, x1))) → A__SEL(a__from(mark(y0)), a__sel(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), from(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__from(mark(x0)))
MARK(sel(0, cons(x0, x1))) → A__SEL(0, cons(mark(x0), x1))
MARK(sel(from(y0), sel(x0, x1))) → A__SEL(a__from(mark(y0)), a__sel(mark(x0), mark(x1)))
A__SEL(s(from(y0)), cons(y1, from(x0))) → A__SEL(a__from(mark(y0)), a__from(mark(x0)))
A__SEL(s(first(y0, y1)), cons(y2, from(x0))) → A__SEL(a__first(mark(y0), mark(y1)), a__from(mark(x0)))
MARK(first(sel(y0, y1), first(x0, x1))) → A__FIRST(a__sel(mark(y0), mark(y1)), a__first(mark(x0), mark(x1)))
MARK(first(first(y0, y1), sel(x0, x1))) → A__FIRST(a__first(mark(y0), mark(y1)), a__sel(mark(x0), mark(x1)))
MARK(sel(first(y0, y1), cons(x0, x1))) → A__SEL(a__first(mark(y0), mark(y1)), cons(mark(x0), x1))
A__SEL(s(0), cons(y0, sel(x0, x1))) → A__SEL(0, a__sel(mark(x0), mark(x1)))
MARK(sel(from(y0), cons(x0, x1))) → A__SEL(a__from(mark(y0)), cons(mark(x0), x1))
MARK(s(X)) → MARK(X)
MARK(sel(s(y0), sel(x0, x1))) → A__SEL(s(mark(y0)), a__sel(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), first(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), a__first(mark(x0), mark(x1)))
MARK(sel(first(y0, y1), sel(x0, x1))) → A__SEL(a__first(mark(y0), mark(y1)), a__sel(mark(x0), mark(x1)))
MARK(first(s(y0), cons(x0, x1))) → A__FIRST(s(mark(y0)), cons(mark(x0), x1))
MARK(sel(0, from(x0))) → A__SEL(0, a__from(mark(x0)))
A__SEL(0, cons(X, Z)) → MARK(X)
A__FIRST(s(X), cons(Y, Z)) → MARK(Y)
A__SEL(s(from(y0)), cons(y1, cons(x0, x1))) → A__SEL(a__from(mark(y0)), cons(mark(x0), x1))
MARK(first(sel(y0, y1), cons(x0, x1))) → A__FIRST(a__sel(mark(y0), mark(y1)), cons(mark(x0), x1))
A__SEL(s(0), cons(y0, cons(x0, x1))) → A__SEL(0, cons(mark(x0), x1))
A__SEL(s(s(y0)), cons(y1, from(x0))) → A__SEL(s(mark(y0)), a__from(mark(x0)))
A__SEL(s(X), cons(Y, Z)) → MARK(X)
MARK(first(X1, X2)) → MARK(X1)
MARK(first(first(y0, y1), cons(x0, x1))) → A__FIRST(a__first(mark(y0), mark(y1)), cons(mark(x0), x1))
MARK(from(X)) → MARK(X)
MARK(sel(s(y0), first(x0, x1))) → A__SEL(s(mark(y0)), a__first(mark(x0), mark(x1)))
MARK(from(X)) → A__FROM(mark(X))
MARK(first(s(y0), first(x0, x1))) → A__FIRST(s(mark(y0)), a__first(mark(x0), mark(x1)))
MARK(sel(from(y0), from(x0))) → A__SEL(a__from(mark(y0)), a__from(mark(x0)))
MARK(first(s(y0), sel(x0, x1))) → A__FIRST(s(mark(y0)), a__sel(mark(x0), mark(x1)))
MARK(sel(0, first(x0, x1))) → A__SEL(0, a__first(mark(x0), mark(x1)))
MARK(sel(0, sel(x0, x1))) → A__SEL(0, a__sel(mark(x0), mark(x1)))
A__SEL(s(first(y0, y1)), cons(y2, sel(x0, x1))) → A__SEL(a__first(mark(y0), mark(y1)), a__sel(mark(x0), mark(x1)))
A__SEL(s(sel(y0, y1)), cons(y2, first(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), a__first(mark(x0), mark(x1)))
A__SEL(s(0), cons(y0, from(x0))) → A__SEL(0, a__from(mark(x0)))
MARK(sel(X1, X2)) → MARK(X2)
MARK(sel(first(y0, y1), from(x0))) → A__SEL(a__first(mark(y0), mark(y1)), a__from(mark(x0)))
MARK(sel(from(y0), first(x0, x1))) → A__SEL(a__from(mark(y0)), a__first(mark(x0), mark(x1)))
A__SEL(s(first(y0, y1)), cons(y2, first(x0, x1))) → A__SEL(a__first(mark(y0), mark(y1)), a__first(mark(x0), mark(x1)))

The TRS R consists of the following rules:

a__from(X) → cons(mark(X), from(s(X)))
a__first(0, Z) → nil
a__first(s(X), cons(Y, Z)) → cons(mark(Y), first(X, Z))
a__sel(0, cons(X, Z)) → mark(X)
a__sel(s(X), cons(Y, Z)) → a__sel(mark(X), mark(Z))
mark(from(X)) → a__from(mark(X))
mark(first(X1, X2)) → a__first(mark(X1), mark(X2))
mark(sel(X1, X2)) → a__sel(mark(X1), mark(X2))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(s(X)) → s(mark(X))
mark(0) → 0
mark(nil) → nil
a__from(X) → from(X)
a__first(X1, X2) → first(X1, X2)
a__sel(X1, X2) → sel(X1, X2)

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(first(first(y0, y1), from(x0))) → A__FIRST(a__first(mark(y0), mark(y1)), a__from(mark(x0)))
MARK(first(from(y0), first(x0, x1))) → A__FIRST(a__from(mark(y0)), a__first(mark(x0), mark(x1)))
MARK(first(from(y0), sel(x0, x1))) → A__FIRST(a__from(mark(y0)), a__sel(mark(x0), mark(x1)))
MARK(first(from(y0), cons(x0, x1))) → A__FIRST(a__from(mark(y0)), cons(mark(x0), x1))
MARK(first(from(y0), from(x0))) → A__FIRST(a__from(mark(y0)), a__from(mark(x0)))
MARK(first(first(y0, y1), first(x0, x1))) → A__FIRST(a__first(mark(y0), mark(y1)), a__first(mark(x0), mark(x1)))
MARK(first(first(y0, y1), sel(x0, x1))) → A__FIRST(a__first(mark(y0), mark(y1)), a__sel(mark(x0), mark(x1)))
MARK(first(first(y0, y1), cons(x0, x1))) → A__FIRST(a__first(mark(y0), mark(y1)), cons(mark(x0), x1))

The remaining pairs can at least be oriented weakly.

MARK(sel(sel(y0, y1), sel(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), a__sel(mark(x0), mark(x1)))
MARK(sel(s(y0), cons(x0, x1))) → A__SEL(s(mark(y0)), cons(mark(x0), x1))
A__SEL(s(sel(y0, y1)), cons(y2, from(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__from(mark(x0)))
MARK(sel(first(y0, y1), first(x0, x1))) → A__SEL(a__first(mark(y0), mark(y1)), a__first(mark(x0), mark(x1)))
MARK(cons(X1, X2)) → MARK(X1)
MARK(first(sel(y0, y1), from(x0))) → A__FIRST(a__sel(mark(y0), mark(y1)), a__from(mark(x0)))
A__SEL(s(0), cons(y0, first(x0, x1))) → A__SEL(0, a__first(mark(x0), mark(x1)))
A__SEL(s(X), cons(Y, Z)) → MARK(Z)
MARK(sel(s(y0), from(x0))) → A__SEL(s(mark(y0)), a__from(mark(x0)))
MARK(first(sel(y0, y1), sel(x0, x1))) → A__FIRST(a__sel(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(s(y0)), cons(y1, sel(x0, x1))) → A__SEL(s(mark(y0)), a__sel(mark(x0), mark(x1)))
MARK(sel(X1, X2)) → MARK(X1)
MARK(sel(sel(y0, y1), cons(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), cons(mark(x0), x1))
A__SEL(s(s(y0)), cons(y1, first(x0, x1))) → A__SEL(s(mark(y0)), a__first(mark(x0), mark(x1)))
MARK(first(X1, X2)) → MARK(X2)
A__SEL(s(from(y0)), cons(y1, first(x0, x1))) → A__SEL(a__from(mark(y0)), a__first(mark(x0), mark(x1)))
A__SEL(s(sel(y0, y1)), cons(y2, cons(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), cons(mark(x0), x1))
A__SEL(s(s(y0)), cons(y1, cons(x0, x1))) → A__SEL(s(mark(y0)), cons(mark(x0), x1))
MARK(first(s(y0), from(x0))) → A__FIRST(s(mark(y0)), a__from(mark(x0)))
A__FROM(X) → MARK(X)
A__SEL(s(first(y0, y1)), cons(y2, cons(x0, x1))) → A__SEL(a__first(mark(y0), mark(y1)), cons(mark(x0), x1))
A__SEL(s(from(y0)), cons(y1, sel(x0, x1))) → A__SEL(a__from(mark(y0)), a__sel(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), from(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__from(mark(x0)))
MARK(sel(0, cons(x0, x1))) → A__SEL(0, cons(mark(x0), x1))
MARK(sel(from(y0), sel(x0, x1))) → A__SEL(a__from(mark(y0)), a__sel(mark(x0), mark(x1)))
A__SEL(s(from(y0)), cons(y1, from(x0))) → A__SEL(a__from(mark(y0)), a__from(mark(x0)))
A__SEL(s(first(y0, y1)), cons(y2, from(x0))) → A__SEL(a__first(mark(y0), mark(y1)), a__from(mark(x0)))
MARK(first(sel(y0, y1), first(x0, x1))) → A__FIRST(a__sel(mark(y0), mark(y1)), a__first(mark(x0), mark(x1)))
MARK(sel(first(y0, y1), cons(x0, x1))) → A__SEL(a__first(mark(y0), mark(y1)), cons(mark(x0), x1))
A__SEL(s(0), cons(y0, sel(x0, x1))) → A__SEL(0, a__sel(mark(x0), mark(x1)))
MARK(sel(from(y0), cons(x0, x1))) → A__SEL(a__from(mark(y0)), cons(mark(x0), x1))
MARK(s(X)) → MARK(X)
MARK(sel(s(y0), sel(x0, x1))) → A__SEL(s(mark(y0)), a__sel(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), first(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), a__first(mark(x0), mark(x1)))
MARK(sel(first(y0, y1), sel(x0, x1))) → A__SEL(a__first(mark(y0), mark(y1)), a__sel(mark(x0), mark(x1)))
MARK(first(s(y0), cons(x0, x1))) → A__FIRST(s(mark(y0)), cons(mark(x0), x1))
MARK(sel(0, from(x0))) → A__SEL(0, a__from(mark(x0)))
A__SEL(0, cons(X, Z)) → MARK(X)
A__FIRST(s(X), cons(Y, Z)) → MARK(Y)
A__SEL(s(from(y0)), cons(y1, cons(x0, x1))) → A__SEL(a__from(mark(y0)), cons(mark(x0), x1))
MARK(first(sel(y0, y1), cons(x0, x1))) → A__FIRST(a__sel(mark(y0), mark(y1)), cons(mark(x0), x1))
A__SEL(s(0), cons(y0, cons(x0, x1))) → A__SEL(0, cons(mark(x0), x1))
A__SEL(s(s(y0)), cons(y1, from(x0))) → A__SEL(s(mark(y0)), a__from(mark(x0)))
A__SEL(s(X), cons(Y, Z)) → MARK(X)
MARK(first(X1, X2)) → MARK(X1)
MARK(from(X)) → MARK(X)
MARK(sel(s(y0), first(x0, x1))) → A__SEL(s(mark(y0)), a__first(mark(x0), mark(x1)))
MARK(from(X)) → A__FROM(mark(X))
MARK(first(s(y0), first(x0, x1))) → A__FIRST(s(mark(y0)), a__first(mark(x0), mark(x1)))
MARK(sel(from(y0), from(x0))) → A__SEL(a__from(mark(y0)), a__from(mark(x0)))
MARK(first(s(y0), sel(x0, x1))) → A__FIRST(s(mark(y0)), a__sel(mark(x0), mark(x1)))
MARK(sel(0, first(x0, x1))) → A__SEL(0, a__first(mark(x0), mark(x1)))
MARK(sel(0, sel(x0, x1))) → A__SEL(0, a__sel(mark(x0), mark(x1)))
A__SEL(s(first(y0, y1)), cons(y2, sel(x0, x1))) → A__SEL(a__first(mark(y0), mark(y1)), a__sel(mark(x0), mark(x1)))
A__SEL(s(sel(y0, y1)), cons(y2, first(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), a__first(mark(x0), mark(x1)))
A__SEL(s(0), cons(y0, from(x0))) → A__SEL(0, a__from(mark(x0)))
MARK(sel(X1, X2)) → MARK(X2)
MARK(sel(first(y0, y1), from(x0))) → A__SEL(a__first(mark(y0), mark(y1)), a__from(mark(x0)))
MARK(sel(from(y0), first(x0, x1))) → A__SEL(a__from(mark(y0)), a__first(mark(x0), mark(x1)))
A__SEL(s(first(y0, y1)), cons(y2, first(x0, x1))) → A__SEL(a__first(mark(y0), mark(y1)), a__first(mark(x0), mark(x1)))

Used ordering: Polynomial interpretation [25]:

POL(0) = 0
POL(A__FIRST(x₁, x₂)) = x₁
POL(A__FROM(x₁)) = 1
POL(A__SEL(x₁, x₂)) = 1
POL(MARK(x₁)) = 1
POL(a__first(x₁, x₂)) = 0
POL(a__from(x₁)) = 0
POL(a__sel(x₁, x₂)) = 1
POL(cons(x₁, x₂)) = 0
POL(first(x₁, x₂)) = 0
POL(from(x₁)) = 0
POL(mark(x₁)) = 1
POL(nil) = 0
POL(s(x₁)) = 1
POL(sel(x₁, x₂)) = 0

The following usable rules [17] were oriented:

a__first(s(X), cons(Y, Z)) → cons(mark(Y), first(X, Z))
a__from(X) → cons(mark(X), from(s(X)))
a__first(0, Z) → nil
mark(first(X1, X2)) → a__first(mark(X1), mark(X2))
a__sel(0, cons(X, Z)) → mark(X)
a__sel(s(X), cons(Y, Z)) → a__sel(mark(X), mark(Z))
mark(sel(X1, X2)) → a__sel(mark(X1), mark(X2))
mark(from(X)) → a__from(mark(X))
mark(nil) → nil
mark(0) → 0
mark(s(X)) → s(mark(X))
mark(cons(X1, X2)) → cons(mark(X1), X2)
a__sel(X1, X2) → sel(X1, X2)
a__first(X1, X2) → first(X1, X2)
a__from(X) → from(X)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ Narrowing

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ QDP

              ↳ Narrowing

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ QDP

                      ↳ Narrowing

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ QDP

                              ↳ Narrowing

                                ↳ QDP

                                  ↳ DependencyGraphProof

                                    ↳ QDP

                                      ↳ Narrowing

                                        ↳ QDP

                                          ↳ DependencyGraphProof

                                            ↳ QDP

                                              ↳ Narrowing

                                                ↳ QDP

                                                  ↳ DependencyGraphProof

                                                    ↳ QDP

                                                      ↳ Narrowing

                                                        ↳ QDP

                                                          ↳ DependencyGraphProof

                                                            ↳ QDP

                                                              ↳ Narrowing

                                                                ↳ QDP

                                                                  ↳ DependencyGraphProof

                                                                    ↳ QDP

                                                                      ↳ Narrowing

                                                                        ↳ QDP

                                                                          ↳ DependencyGraphProof

                                                                            ↳ QDP

                                                                              ↳ Narrowing

                                                                                ↳ QDP

                                                                                  ↳ DependencyGraphProof

                                                                                    ↳ QDP

                                                                                      ↳ Narrowing

                                                                                        ↳ QDP

                                                                                          ↳ DependencyGraphProof

                                                                                            ↳ QDP

                                                                                              ↳ 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

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

MARK(sel(sel(y0, y1), sel(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), a__sel(mark(x0), mark(x1)))
MARK(sel(s(y0), cons(x0, x1))) → A__SEL(s(mark(y0)), cons(mark(x0), x1))
A__SEL(s(sel(y0, y1)), cons(y2, from(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__from(mark(x0)))
MARK(sel(first(y0, y1), first(x0, x1))) → A__SEL(a__first(mark(y0), mark(y1)), a__first(mark(x0), mark(x1)))
MARK(cons(X1, X2)) → MARK(X1)
MARK(first(sel(y0, y1), from(x0))) → A__FIRST(a__sel(mark(y0), mark(y1)), a__from(mark(x0)))
A__SEL(s(0), cons(y0, first(x0, x1))) → A__SEL(0, a__first(mark(x0), mark(x1)))
A__SEL(s(X), cons(Y, Z)) → MARK(Z)
MARK(sel(s(y0), from(x0))) → A__SEL(s(mark(y0)), a__from(mark(x0)))
MARK(first(sel(y0, y1), sel(x0, x1))) → A__FIRST(a__sel(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(s(y0)), cons(y1, sel(x0, x1))) → A__SEL(s(mark(y0)), a__sel(mark(x0), mark(x1)))
MARK(sel(X1, X2)) → MARK(X1)
MARK(sel(sel(y0, y1), cons(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), cons(mark(x0), x1))
A__SEL(s(from(y0)), cons(y1, first(x0, x1))) → A__SEL(a__from(mark(y0)), a__first(mark(x0), mark(x1)))
MARK(first(X1, X2)) → MARK(X2)
A__SEL(s(s(y0)), cons(y1, first(x0, x1))) → A__SEL(s(mark(y0)), a__first(mark(x0), mark(x1)))
A__SEL(s(sel(y0, y1)), cons(y2, cons(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), cons(mark(x0), x1))
A__SEL(s(s(y0)), cons(y1, cons(x0, x1))) → A__SEL(s(mark(y0)), cons(mark(x0), x1))
A__FROM(X) → MARK(X)
MARK(first(s(y0), from(x0))) → A__FIRST(s(mark(y0)), a__from(mark(x0)))
A__SEL(s(first(y0, y1)), cons(y2, cons(x0, x1))) → A__SEL(a__first(mark(y0), mark(y1)), cons(mark(x0), x1))
A__SEL(s(from(y0)), cons(y1, sel(x0, x1))) → A__SEL(a__from(mark(y0)), a__sel(mark(x0), mark(x1)))
MARK(sel(0, cons(x0, x1))) → A__SEL(0, cons(mark(x0), x1))
MARK(sel(sel(y0, y1), from(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__from(mark(x0)))
MARK(sel(from(y0), sel(x0, x1))) → A__SEL(a__from(mark(y0)), a__sel(mark(x0), mark(x1)))
A__SEL(s(from(y0)), cons(y1, from(x0))) → A__SEL(a__from(mark(y0)), a__from(mark(x0)))
A__SEL(s(first(y0, y1)), cons(y2, from(x0))) → A__SEL(a__first(mark(y0), mark(y1)), a__from(mark(x0)))
MARK(first(sel(y0, y1), first(x0, x1))) → A__FIRST(a__sel(mark(y0), mark(y1)), a__first(mark(x0), mark(x1)))
MARK(sel(first(y0, y1), cons(x0, x1))) → A__SEL(a__first(mark(y0), mark(y1)), cons(mark(x0), x1))
A__SEL(s(0), cons(y0, sel(x0, x1))) → A__SEL(0, a__sel(mark(x0), mark(x1)))
MARK(sel(from(y0), cons(x0, x1))) → A__SEL(a__from(mark(y0)), cons(mark(x0), x1))
MARK(s(X)) → MARK(X)
MARK(sel(s(y0), sel(x0, x1))) → A__SEL(s(mark(y0)), a__sel(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), first(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), a__first(mark(x0), mark(x1)))
MARK(sel(first(y0, y1), sel(x0, x1))) → A__SEL(a__first(mark(y0), mark(y1)), a__sel(mark(x0), mark(x1)))
MARK(first(s(y0), cons(x0, x1))) → A__FIRST(s(mark(y0)), cons(mark(x0), x1))
MARK(sel(0, from(x0))) → A__SEL(0, a__from(mark(x0)))
A__SEL(0, cons(X, Z)) → MARK(X)
A__FIRST(s(X), cons(Y, Z)) → MARK(Y)
A__SEL(s(from(y0)), cons(y1, cons(x0, x1))) → A__SEL(a__from(mark(y0)), cons(mark(x0), x1))
MARK(first(sel(y0, y1), cons(x0, x1))) → A__FIRST(a__sel(mark(y0), mark(y1)), cons(mark(x0), x1))
A__SEL(s(0), cons(y0, cons(x0, x1))) → A__SEL(0, cons(mark(x0), x1))
A__SEL(s(s(y0)), cons(y1, from(x0))) → A__SEL(s(mark(y0)), a__from(mark(x0)))
A__SEL(s(X), cons(Y, Z)) → MARK(X)
MARK(first(X1, X2)) → MARK(X1)
MARK(from(X)) → MARK(X)
MARK(sel(s(y0), first(x0, x1))) → A__SEL(s(mark(y0)), a__first(mark(x0), mark(x1)))
MARK(from(X)) → A__FROM(mark(X))
MARK(sel(from(y0), from(x0))) → A__SEL(a__from(mark(y0)), a__from(mark(x0)))
MARK(first(s(y0), first(x0, x1))) → A__FIRST(s(mark(y0)), a__first(mark(x0), mark(x1)))
MARK(first(s(y0), sel(x0, x1))) → A__FIRST(s(mark(y0)), a__sel(mark(x0), mark(x1)))
MARK(sel(0, first(x0, x1))) → A__SEL(0, a__first(mark(x0), mark(x1)))
MARK(sel(0, sel(x0, x1))) → A__SEL(0, a__sel(mark(x0), mark(x1)))
A__SEL(s(sel(y0, y1)), cons(y2, first(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), a__first(mark(x0), mark(x1)))
A__SEL(s(first(y0, y1)), cons(y2, sel(x0, x1))) → A__SEL(a__first(mark(y0), mark(y1)), a__sel(mark(x0), mark(x1)))
A__SEL(s(0), cons(y0, from(x0))) → A__SEL(0, a__from(mark(x0)))
MARK(sel(X1, X2)) → MARK(X2)
A__SEL(s(first(y0, y1)), cons(y2, first(x0, x1))) → A__SEL(a__first(mark(y0), mark(y1)), a__first(mark(x0), mark(x1)))
MARK(sel(from(y0), first(x0, x1))) → A__SEL(a__from(mark(y0)), a__first(mark(x0), mark(x1)))
MARK(sel(first(y0, y1), from(x0))) → A__SEL(a__first(mark(y0), mark(y1)), a__from(mark(x0)))

The TRS R consists of the following rules:

a__from(X) → cons(mark(X), from(s(X)))
a__first(0, Z) → nil
a__first(s(X), cons(Y, Z)) → cons(mark(Y), first(X, Z))
a__sel(0, cons(X, Z)) → mark(X)
a__sel(s(X), cons(Y, Z)) → a__sel(mark(X), mark(Z))
mark(from(X)) → a__from(mark(X))
mark(first(X1, X2)) → a__first(mark(X1), mark(X2))
mark(sel(X1, X2)) → a__sel(mark(X1), mark(X2))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(s(X)) → s(mark(X))
mark(0) → 0
mark(nil) → nil
a__from(X) → from(X)
a__first(X1, X2) → first(X1, X2)
a__sel(X1, X2) → sel(X1, X2)

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(first(y0, y1), first(x0, x1))) → A__SEL(a__first(mark(y0), mark(y1)), a__first(mark(x0), mark(x1)))
A__SEL(s(from(y0)), cons(y1, first(x0, x1))) → A__SEL(a__from(mark(y0)), a__first(mark(x0), mark(x1)))
A__SEL(s(first(y0, y1)), cons(y2, cons(x0, x1))) → A__SEL(a__first(mark(y0), mark(y1)), cons(mark(x0), x1))
A__SEL(s(from(y0)), cons(y1, sel(x0, x1))) → A__SEL(a__from(mark(y0)), a__sel(mark(x0), mark(x1)))
MARK(sel(from(y0), sel(x0, x1))) → A__SEL(a__from(mark(y0)), a__sel(mark(x0), mark(x1)))
A__SEL(s(from(y0)), cons(y1, from(x0))) → A__SEL(a__from(mark(y0)), a__from(mark(x0)))
A__SEL(s(first(y0, y1)), cons(y2, from(x0))) → A__SEL(a__first(mark(y0), mark(y1)), a__from(mark(x0)))
MARK(sel(first(y0, y1), cons(x0, x1))) → A__SEL(a__first(mark(y0), mark(y1)), cons(mark(x0), x1))
MARK(sel(from(y0), cons(x0, x1))) → A__SEL(a__from(mark(y0)), cons(mark(x0), x1))
MARK(sel(first(y0, y1), sel(x0, x1))) → A__SEL(a__first(mark(y0), mark(y1)), a__sel(mark(x0), mark(x1)))
A__SEL(s(from(y0)), cons(y1, cons(x0, x1))) → A__SEL(a__from(mark(y0)), cons(mark(x0), x1))
MARK(sel(from(y0), from(x0))) → A__SEL(a__from(mark(y0)), a__from(mark(x0)))
A__SEL(s(first(y0, y1)), cons(y2, sel(x0, x1))) → A__SEL(a__first(mark(y0), mark(y1)), a__sel(mark(x0), mark(x1)))
A__SEL(s(first(y0, y1)), cons(y2, first(x0, x1))) → A__SEL(a__first(mark(y0), mark(y1)), a__first(mark(x0), mark(x1)))
MARK(sel(from(y0), first(x0, x1))) → A__SEL(a__from(mark(y0)), a__first(mark(x0), mark(x1)))
MARK(sel(first(y0, y1), from(x0))) → A__SEL(a__first(mark(y0), mark(y1)), a__from(mark(x0)))

The remaining pairs can at least be oriented weakly.

MARK(sel(sel(y0, y1), sel(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), a__sel(mark(x0), mark(x1)))
MARK(sel(s(y0), cons(x0, x1))) → A__SEL(s(mark(y0)), cons(mark(x0), x1))
A__SEL(s(sel(y0, y1)), cons(y2, from(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__from(mark(x0)))
MARK(cons(X1, X2)) → MARK(X1)
MARK(first(sel(y0, y1), from(x0))) → A__FIRST(a__sel(mark(y0), mark(y1)), a__from(mark(x0)))
A__SEL(s(0), cons(y0, first(x0, x1))) → A__SEL(0, a__first(mark(x0), mark(x1)))
A__SEL(s(X), cons(Y, Z)) → MARK(Z)
MARK(sel(s(y0), from(x0))) → A__SEL(s(mark(y0)), a__from(mark(x0)))
MARK(first(sel(y0, y1), sel(x0, x1))) → A__FIRST(a__sel(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(s(y0)), cons(y1, sel(x0, x1))) → A__SEL(s(mark(y0)), a__sel(mark(x0), mark(x1)))
MARK(sel(X1, X2)) → MARK(X1)
MARK(sel(sel(y0, y1), cons(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), cons(mark(x0), x1))
MARK(first(X1, X2)) → MARK(X2)
A__SEL(s(s(y0)), cons(y1, first(x0, x1))) → A__SEL(s(mark(y0)), a__first(mark(x0), mark(x1)))
A__SEL(s(sel(y0, y1)), cons(y2, cons(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), cons(mark(x0), x1))
A__SEL(s(s(y0)), cons(y1, cons(x0, x1))) → A__SEL(s(mark(y0)), cons(mark(x0), x1))
A__FROM(X) → MARK(X)
MARK(first(s(y0), from(x0))) → A__FIRST(s(mark(y0)), a__from(mark(x0)))
MARK(sel(0, cons(x0, x1))) → A__SEL(0, cons(mark(x0), x1))
MARK(sel(sel(y0, y1), from(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__from(mark(x0)))
MARK(first(sel(y0, y1), first(x0, x1))) → A__FIRST(a__sel(mark(y0), mark(y1)), a__first(mark(x0), mark(x1)))
A__SEL(s(0), cons(y0, sel(x0, x1))) → A__SEL(0, a__sel(mark(x0), mark(x1)))
MARK(s(X)) → MARK(X)
MARK(sel(s(y0), sel(x0, x1))) → A__SEL(s(mark(y0)), a__sel(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), first(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), a__first(mark(x0), mark(x1)))
MARK(first(s(y0), cons(x0, x1))) → A__FIRST(s(mark(y0)), cons(mark(x0), x1))
MARK(sel(0, from(x0))) → A__SEL(0, a__from(mark(x0)))
A__SEL(0, cons(X, Z)) → MARK(X)
A__FIRST(s(X), cons(Y, Z)) → MARK(Y)
MARK(first(sel(y0, y1), cons(x0, x1))) → A__FIRST(a__sel(mark(y0), mark(y1)), cons(mark(x0), x1))
A__SEL(s(0), cons(y0, cons(x0, x1))) → A__SEL(0, cons(mark(x0), x1))
A__SEL(s(s(y0)), cons(y1, from(x0))) → A__SEL(s(mark(y0)), a__from(mark(x0)))
A__SEL(s(X), cons(Y, Z)) → MARK(X)
MARK(first(X1, X2)) → MARK(X1)
MARK(from(X)) → MARK(X)
MARK(sel(s(y0), first(x0, x1))) → A__SEL(s(mark(y0)), a__first(mark(x0), mark(x1)))
MARK(from(X)) → A__FROM(mark(X))
MARK(first(s(y0), first(x0, x1))) → A__FIRST(s(mark(y0)), a__first(mark(x0), mark(x1)))
MARK(first(s(y0), sel(x0, x1))) → A__FIRST(s(mark(y0)), a__sel(mark(x0), mark(x1)))
MARK(sel(0, first(x0, x1))) → A__SEL(0, a__first(mark(x0), mark(x1)))
MARK(sel(0, sel(x0, x1))) → A__SEL(0, a__sel(mark(x0), mark(x1)))
A__SEL(s(sel(y0, y1)), cons(y2, first(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), a__first(mark(x0), mark(x1)))
A__SEL(s(0), cons(y0, from(x0))) → A__SEL(0, a__from(mark(x0)))
MARK(sel(X1, X2)) → MARK(X2)

Used ordering: Polynomial interpretation [25]:

POL(0) = 1
POL(A__FIRST(x₁, x₂)) = x₁
POL(A__FROM(x₁)) = 1
POL(A__SEL(x₁, x₂)) = x₁
POL(MARK(x₁)) = 1
POL(a__first(x₁, x₂)) = 0
POL(a__from(x₁)) = 0
POL(a__sel(x₁, x₂)) = x₁
POL(cons(x₁, x₂)) = 0
POL(first(x₁, x₂)) = 0
POL(from(x₁)) = 0
POL(mark(x₁)) = 1
POL(nil) = 0
POL(s(x₁)) = 1
POL(sel(x₁, x₂)) = 0

The following usable rules [17] were oriented:

a__first(s(X), cons(Y, Z)) → cons(mark(Y), first(X, Z))
a__from(X) → cons(mark(X), from(s(X)))
a__first(0, Z) → nil
mark(first(X1, X2)) → a__first(mark(X1), mark(X2))
a__sel(0, cons(X, Z)) → mark(X)
a__sel(s(X), cons(Y, Z)) → a__sel(mark(X), mark(Z))
mark(sel(X1, X2)) → a__sel(mark(X1), mark(X2))
mark(from(X)) → a__from(mark(X))
mark(nil) → nil
mark(0) → 0
mark(s(X)) → s(mark(X))
mark(cons(X1, X2)) → cons(mark(X1), X2)
a__sel(X1, X2) → sel(X1, X2)
a__first(X1, X2) → first(X1, X2)
a__from(X) → from(X)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ Narrowing

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ QDP

              ↳ Narrowing

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ QDP

                      ↳ Narrowing

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ QDP

                              ↳ Narrowing

                                ↳ QDP

                                  ↳ DependencyGraphProof

                                    ↳ QDP

                                      ↳ Narrowing

                                        ↳ QDP

                                          ↳ DependencyGraphProof

                                            ↳ QDP

                                              ↳ Narrowing

                                                ↳ QDP

                                                  ↳ DependencyGraphProof

                                                    ↳ QDP

                                                      ↳ Narrowing

                                                        ↳ QDP

                                                          ↳ DependencyGraphProof

                                                            ↳ QDP

                                                              ↳ Narrowing

                                                                ↳ QDP

                                                                  ↳ DependencyGraphProof

                                                                    ↳ QDP

                                                                      ↳ Narrowing

                                                                        ↳ QDP

                                                                          ↳ DependencyGraphProof

                                                                            ↳ QDP

                                                                              ↳ Narrowing

                                                                                ↳ QDP

                                                                                  ↳ DependencyGraphProof

                                                                                    ↳ QDP

                                                                                      ↳ Narrowing

                                                                                        ↳ QDP

                                                                                          ↳ DependencyGraphProof

                                                                                            ↳ QDP

                                                                                              ↳ 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

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

MARK(sel(sel(y0, y1), sel(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), a__sel(mark(x0), mark(x1)))
MARK(sel(s(y0), cons(x0, x1))) → A__SEL(s(mark(y0)), cons(mark(x0), x1))
MARK(s(X)) → MARK(X)
A__SEL(s(sel(y0, y1)), cons(y2, from(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__from(mark(x0)))
MARK(sel(sel(y0, y1), first(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), a__first(mark(x0), mark(x1)))
MARK(sel(s(y0), sel(x0, x1))) → A__SEL(s(mark(y0)), a__sel(mark(x0), mark(x1)))
MARK(cons(X1, X2)) → MARK(X1)
MARK(first(s(y0), cons(x0, x1))) → A__FIRST(s(mark(y0)), cons(mark(x0), x1))
MARK(first(sel(y0, y1), from(x0))) → A__FIRST(a__sel(mark(y0), mark(y1)), a__from(mark(x0)))
A__SEL(0, cons(X, Z)) → MARK(X)
MARK(sel(0, from(x0))) → A__SEL(0, a__from(mark(x0)))
A__FIRST(s(X), cons(Y, Z)) → MARK(Y)
A__SEL(s(0), cons(y0, first(x0, x1))) → A__SEL(0, a__first(mark(x0), mark(x1)))
MARK(first(sel(y0, y1), cons(x0, x1))) → A__FIRST(a__sel(mark(y0), mark(y1)), cons(mark(x0), x1))
A__SEL(s(X), cons(Y, Z)) → MARK(Z)
MARK(sel(s(y0), from(x0))) → A__SEL(s(mark(y0)), a__from(mark(x0)))
A__SEL(s(0), cons(y0, cons(x0, x1))) → A__SEL(0, cons(mark(x0), x1))
A__SEL(s(s(y0)), cons(y1, from(x0))) → A__SEL(s(mark(y0)), a__from(mark(x0)))
A__SEL(s(X), cons(Y, Z)) → MARK(X)
MARK(first(sel(y0, y1), sel(x0, x1))) → A__FIRST(a__sel(mark(y0), mark(y1)), a__sel(mark(x0), mark(x1)))
MARK(first(X1, X2)) → 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, sel(x0, x1))) → A__SEL(s(mark(y0)), a__sel(mark(x0), mark(x1)))
MARK(from(X)) → MARK(X)
MARK(sel(X1, X2)) → MARK(X1)
MARK(from(X)) → A__FROM(mark(X))
MARK(sel(s(y0), first(x0, x1))) → A__SEL(s(mark(y0)), a__first(mark(x0), mark(x1)))
MARK(first(s(y0), first(x0, x1))) → A__FIRST(s(mark(y0)), a__first(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), cons(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), cons(mark(x0), x1))
MARK(first(s(y0), sel(x0, x1))) → A__FIRST(s(mark(y0)), a__sel(mark(x0), mark(x1)))
MARK(first(X1, X2)) → MARK(X2)
A__SEL(s(s(y0)), cons(y1, first(x0, x1))) → A__SEL(s(mark(y0)), a__first(mark(x0), mark(x1)))
A__SEL(s(sel(y0, y1)), cons(y2, cons(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), cons(mark(x0), x1))
MARK(sel(0, first(x0, x1))) → A__SEL(0, a__first(mark(x0), mark(x1)))
A__SEL(s(s(y0)), cons(y1, cons(x0, x1))) → A__SEL(s(mark(y0)), cons(mark(x0), x1))
A__FROM(X) → MARK(X)
MARK(first(s(y0), from(x0))) → A__FIRST(s(mark(y0)), a__from(mark(x0)))
MARK(sel(0, sel(x0, x1))) → A__SEL(0, a__sel(mark(x0), mark(x1)))
A__SEL(s(sel(y0, y1)), cons(y2, first(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), a__first(mark(x0), mark(x1)))
MARK(sel(0, cons(x0, x1))) → A__SEL(0, cons(mark(x0), x1))
MARK(sel(sel(y0, y1), from(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__from(mark(x0)))
A__SEL(s(0), cons(y0, from(x0))) → A__SEL(0, a__from(mark(x0)))
MARK(first(sel(y0, y1), first(x0, x1))) → A__FIRST(a__sel(mark(y0), mark(y1)), a__first(mark(x0), mark(x1)))
A__SEL(s(0), cons(y0, sel(x0, x1))) → A__SEL(0, a__sel(mark(x0), mark(x1)))
MARK(sel(X1, X2)) → MARK(X2)

The TRS R consists of the following rules:

a__from(X) → cons(mark(X), from(s(X)))
a__first(0, Z) → nil
a__first(s(X), cons(Y, Z)) → cons(mark(Y), first(X, Z))
a__sel(0, cons(X, Z)) → mark(X)
a__sel(s(X), cons(Y, Z)) → a__sel(mark(X), mark(Z))
mark(from(X)) → a__from(mark(X))
mark(first(X1, X2)) → a__first(mark(X1), mark(X2))
mark(sel(X1, X2)) → a__sel(mark(X1), mark(X2))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(s(X)) → s(mark(X))
mark(0) → 0
mark(nil) → nil
a__from(X) → from(X)
a__first(X1, X2) → first(X1, X2)
a__sel(X1, X2) → sel(X1, X2)

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