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__sel(0, cons(X, Y)) → mark(X)
a__sel(s(X), cons(Y, Z)) → a__sel(mark(X), mark(Z))
mark(from(X)) → a__from(mark(X))
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
a__from(X) → from(X)
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__sel(0, cons(X, Y)) → mark(X)
a__sel(s(X), cons(Y, Z)) → a__sel(mark(X), mark(Z))
mark(from(X)) → a__from(mark(X))
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
a__from(X) → from(X)
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(sel(X1, X2)) → MARK(X1)
MARK(from(X)) → MARK(X)
MARK(sel(X1, X2)) → A__SEL(mark(X1), mark(X2))
A__SEL(s(X), cons(Y, Z)) → MARK(Z)
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)
A__FROM(X) → MARK(X)
MARK(sel(X1, X2)) → MARK(X2)
A__SEL(s(X), cons(Y, Z)) → MARK(X)
A__SEL(0, cons(X, Y)) → MARK(X)

The TRS R consists of the following rules:

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

The TRS R consists of the following rules:

a__from(X) → cons(mark(X), from(s(X)))
a__sel(0, cons(X, Y)) → mark(X)
a__sel(s(X), cons(Y, Z)) → a__sel(mark(X), mark(Z))
mark(from(X)) → a__from(mark(X))
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
a__from(X) → from(X)
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(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(s(x0)), cons(y1, y2)) → A__SEL(s(mark(x0)), 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(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(cons(X1, X2)) → MARK(X1)
A__FROM(X) → MARK(X)
A__SEL(0, cons(X, Y)) → MARK(X)
A__SEL(s(cons(x0, x1)), cons(y1, y2)) → A__SEL(cons(mark(x0), 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(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__sel(0, cons(X, Y)) → mark(X)
a__sel(s(X), cons(Y, Z)) → a__sel(mark(X), mark(Z))
mark(from(X)) → a__from(mark(X))
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
a__from(X) → from(X)
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 1 less node.

↳ 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(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(cons(X1, X2)) → MARK(X1)
A__FROM(X) → MARK(X)
A__SEL(0, cons(X, Y)) → MARK(X)
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(0), cons(y1, y2)) → A__SEL(0, mark(y2))
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__sel(0, cons(X, Y)) → mark(X)
a__sel(s(X), cons(Y, Z)) → a__sel(mark(X), mark(Z))
mark(from(X)) → a__from(mark(X))
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
a__from(X) → from(X)
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(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(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))



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ Narrowing
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ 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(sel(X1, X2)) → MARK(X1)
MARK(from(X)) → MARK(X)
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(from(X)) → A__FROM(mark(X))
A__SEL(s(s(x0)), cons(y1, y2)) → A__SEL(s(mark(x0)), mark(y2))
MARK(s(X)) → MARK(X)
MARK(cons(X1, X2)) → MARK(X1)
MARK(sel(s(x0), y1)) → A__SEL(s(mark(x0)), mark(y1))
A__FROM(X) → MARK(X)
A__SEL(0, cons(X, Y)) → MARK(X)
A__SEL(s(0), cons(y1, y2)) → A__SEL(0, mark(y2))
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(sel(0, y1)) → A__SEL(0, mark(y1))
MARK(sel(sel(x0, x1), y1)) → A__SEL(a__sel(mark(x0), mark(x1)), 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__sel(0, cons(X, Y)) → mark(X)
a__sel(s(X), cons(Y, Z)) → a__sel(mark(X), mark(Z))
mark(from(X)) → a__from(mark(X))
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
a__from(X) → from(X)
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 1 less node.

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



↳ 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:

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

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

The TRS R consists of the following rules:

a__from(X) → cons(mark(X), from(s(X)))
a__sel(0, cons(X, Y)) → mark(X)
a__sel(s(X), cons(Y, Z)) → a__sel(mark(X), mark(Z))
mark(from(X)) → a__from(mark(X))
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
a__from(X) → from(X)
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
                                ↳ 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(sel(sel(y0, y1), sel(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), a__sel(mark(x0), mark(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(x0), y1)) → A__SEL(a__from(mark(x0)), mark(y1))
MARK(sel(s(y0), cons(x0, x1))) → A__SEL(s(mark(y0)), cons(mark(x0), x1))
MARK(s(X)) → MARK(X)
MARK(sel(sel(y0, y1), cons(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), cons(mark(x0), 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)
A__FROM(X) → MARK(X)
A__SEL(0, cons(X, Y)) → MARK(X)
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(X), cons(Y, Z)) → MARK(Z)
MARK(sel(sel(y0, y1), from(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__from(mark(x0)))
MARK(sel(0, y1)) → A__SEL(0, mark(y1))
MARK(sel(s(y0), from(x0))) → A__SEL(s(mark(y0)), a__from(mark(x0)))
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__sel(0, cons(X, Y)) → mark(X)
a__sel(s(X), cons(Y, Z)) → a__sel(mark(X), mark(Z))
mark(from(X)) → a__from(mark(X))
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
a__from(X) → from(X)
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), s(x0))) → A__SEL(a__from(mark(y0)), s(mark(x0)))
MARK(sel(from(y0), 0)) → A__SEL(a__from(mark(y0)), 0)



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

The TRS R consists of the following rules:

a__from(X) → cons(mark(X), from(s(X)))
a__sel(0, cons(X, Y)) → mark(X)
a__sel(s(X), cons(Y, Z)) → a__sel(mark(X), mark(Z))
mark(from(X)) → a__from(mark(X))
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
a__from(X) → from(X)
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
                                ↳ 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(sel(from(y0), cons(x0, x1))) → A__SEL(a__from(mark(y0)), cons(mark(x0), x1))
MARK(sel(sel(y0, y1), sel(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), a__sel(mark(x0), mark(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(sel(s(y0), cons(x0, x1))) → A__SEL(s(mark(y0)), cons(mark(x0), x1))
MARK(s(X)) → MARK(X)
MARK(sel(sel(y0, y1), cons(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), cons(mark(x0), 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)
A__FROM(X) → MARK(X)
A__SEL(0, cons(X, Y)) → MARK(X)
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(X), cons(Y, Z)) → MARK(Z)
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(sel(0, y1)) → A__SEL(0, mark(y1))
MARK(sel(s(y0), from(x0))) → A__SEL(s(mark(y0)), a__from(mark(x0)))
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__sel(0, cons(X, Y)) → mark(X)
a__sel(s(X), cons(Y, Z)) → a__sel(mark(X), mark(Z))
mark(from(X)) → a__from(mark(X))
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
a__from(X) → from(X)
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, s(x0))) → A__SEL(0, s(mark(x0)))
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, from(x0))) → A__SEL(0, a__from(mark(x0)))
MARK(sel(0, 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

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

The TRS R consists of the following rules:

a__from(X) → cons(mark(X), from(s(X)))
a__sel(0, cons(X, Y)) → mark(X)
a__sel(s(X), cons(Y, Z)) → a__sel(mark(X), mark(Z))
mark(from(X)) → a__from(mark(X))
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
a__from(X) → from(X)
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
                                ↳ 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(from(y0), cons(x0, x1))) → A__SEL(a__from(mark(y0)), cons(mark(x0), x1))
MARK(s(X)) → MARK(X)
MARK(sel(s(y0), cons(x0, x1))) → A__SEL(s(mark(y0)), cons(mark(x0), 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(sel(0, from(x0))) → A__SEL(0, a__from(mark(x0)))
A__SEL(0, cons(X, Y)) → MARK(X)
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(sel(s(y0), from(x0))) → A__SEL(s(mark(y0)), a__from(mark(x0)))
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(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(sel(sel(y0, y1), cons(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), cons(mark(x0), x1))
A__FROM(X) → MARK(X)
A__SEL(s(0), cons(y1, y2)) → A__SEL(0, mark(y2))
MARK(sel(0, sel(x0, x1))) → A__SEL(0, 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)))
MARK(sel(0, cons(x0, x1))) → A__SEL(0, cons(mark(x0), 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__sel(0, cons(X, Y)) → mark(X)
a__sel(s(X), cons(Y, Z)) → a__sel(mark(X), mark(Z))
mark(from(X)) → a__from(mark(X))
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
a__from(X) → from(X)
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, 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, 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

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

The TRS R consists of the following rules:

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

The TRS R consists of the following rules:

a__from(X) → cons(mark(X), from(s(X)))
a__sel(0, cons(X, Y)) → mark(X)
a__sel(s(X), cons(Y, Z)) → a__sel(mark(X), mark(Z))
mark(from(X)) → a__from(mark(X))
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
a__from(X) → from(X)
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, 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)
A__SEL(s(0), cons(y0, s(x0))) → A__SEL(0, 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

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(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(0), cons(y0, s(x0))) → A__SEL(0, s(mark(x0)))
MARK(sel(s(y0), sel(x0, x1))) → A__SEL(s(mark(y0)), a__sel(mark(x0), mark(x1)))
MARK(cons(X1, X2)) → MARK(X1)
A__SEL(0, cons(X, Y)) → 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__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(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(X), cons(Y, Z)) → MARK(X)
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(sel(sel(y0, y1), cons(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), cons(mark(x0), x1))
A__SEL(s(0), cons(y0, 0)) → A__SEL(0, 0)
A__FROM(X) → MARK(X)
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, sel(x0, x1))) → A__SEL(0, 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)))
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))
A__SEL(s(0), cons(y0, from(x0))) → A__SEL(0, a__from(mark(x0)))
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__sel(0, cons(X, Y)) → mark(X)
a__sel(s(X), cons(Y, Z)) → a__sel(mark(X), mark(Z))
mark(from(X)) → a__from(mark(X))
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
a__from(X) → from(X)
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
                                ↳ 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(from(y0), cons(x0, x1))) → A__SEL(a__from(mark(y0)), cons(mark(x0), x1))
MARK(s(X)) → MARK(X)
MARK(sel(s(y0), cons(x0, x1))) → A__SEL(s(mark(y0)), cons(mark(x0), 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(sel(0, from(x0))) → A__SEL(0, a__from(mark(x0)))
A__SEL(0, cons(X, Y)) → MARK(X)
A__SEL(s(from(y0)), cons(y1, cons(x0, x1))) → A__SEL(a__from(mark(y0)), 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(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(X), cons(Y, Z)) → MARK(X)
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(sel(sel(y0, y1), cons(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), cons(mark(x0), x1))
A__FROM(X) → MARK(X)
MARK(sel(0, sel(x0, x1))) → A__SEL(0, a__sel(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))
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, from(x0))) → A__SEL(a__from(mark(y0)), a__from(mark(x0)))
A__SEL(s(0), cons(y0, from(x0))) → A__SEL(0, a__from(mark(x0)))
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__sel(0, cons(X, Y)) → mark(X)
a__sel(s(X), cons(Y, Z)) → a__sel(mark(X), mark(Z))
mark(from(X)) → a__from(mark(X))
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
a__from(X) → from(X)
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, 0)) → A__SEL(a__sel(mark(y0), mark(y1)), 0)
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)))



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

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(from(y0), cons(x0, x1))) → A__SEL(a__from(mark(y0)), cons(mark(x0), x1))
MARK(s(X)) → MARK(X)
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(s(y0), sel(x0, x1))) → A__SEL(s(mark(y0)), a__sel(mark(x0), mark(x1)))
MARK(cons(X1, X2)) → MARK(X1)
A__SEL(0, cons(X, Y)) → 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__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)
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(from(y0), from(x0))) → A__SEL(a__from(mark(y0)), a__from(mark(x0)))
MARK(sel(sel(y0, y1), cons(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), cons(mark(x0), 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)
A__SEL(s(s(y0)), cons(y1, 0)) → A__SEL(s(mark(y0)), 0)
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, sel(x0, x1))) → A__SEL(0, 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(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))
A__SEL(s(0), cons(y0, from(x0))) → A__SEL(0, a__from(mark(x0)))
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__sel(0, cons(X, Y)) → mark(X)
a__sel(s(X), cons(Y, Z)) → a__sel(mark(X), mark(Z))
mark(from(X)) → a__from(mark(X))
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
a__from(X) → from(X)
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
                                ↳ QDP
                                  ↳ DependencyGraphProof
                                    ↳ QDP
                                      ↳ 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(from(y0), cons(x0, x1))) → A__SEL(a__from(mark(y0)), cons(mark(x0), x1))
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(s(y0), sel(x0, x1))) → A__SEL(s(mark(y0)), a__sel(mark(x0), mark(x1)))
MARK(cons(X1, X2)) → MARK(X1)
MARK(sel(0, from(x0))) → A__SEL(0, a__from(mark(x0)))
A__SEL(0, cons(X, Y)) → MARK(X)
A__SEL(s(from(y0)), cons(y1, cons(x0, x1))) → A__SEL(a__from(mark(y0)), 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)
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(from(X)) → MARK(X)
MARK(from(X)) → A__FROM(mark(X))
MARK(sel(from(y0), from(x0))) → A__SEL(a__from(mark(y0)), a__from(mark(x0)))
MARK(sel(sel(y0, y1), cons(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), cons(mark(x0), 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(sel(0, sel(x0, x1))) → A__SEL(0, a__sel(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))
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, from(x0))) → A__SEL(a__from(mark(y0)), a__from(mark(x0)))
A__SEL(s(0), cons(y0, from(x0))) → A__SEL(0, a__from(mark(x0)))
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__sel(0, cons(X, Y)) → mark(X)
a__sel(s(X), cons(Y, Z)) → a__sel(mark(X), mark(Z))
mark(from(X)) → a__from(mark(X))
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
a__from(X) → from(X)
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(from(y0), cons(x0, x1))) → A__SEL(a__from(mark(y0)), cons(mark(x0), 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(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)))
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))
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(s(y0), sel(x0, x1))) → A__SEL(s(mark(y0)), a__sel(mark(x0), mark(x1)))
MARK(cons(X1, X2)) → MARK(X1)
MARK(sel(0, from(x0))) → A__SEL(0, a__from(mark(x0)))
A__SEL(0, cons(X, Y)) → MARK(X)
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)
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(from(X)) → MARK(X)
MARK(from(X)) → A__FROM(mark(X))
MARK(sel(sel(y0, y1), cons(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), cons(mark(x0), 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(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(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)))
A__SEL(s(0), cons(y0, sel(x0, x1))) → A__SEL(0, a__sel(mark(x0), mark(x1)))
MARK(sel(X1, X2)) → MARK(X2)
Used ordering: Polynomial interpretation [25]:

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

The following usable rules [17] were oriented:

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

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