Termination w.r.t. Q proof of /tmp/tmptChwh0/Ex4_7_56_Bor03

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__after(0, XS) → mark(XS)
a__after(s(N), cons(X, XS)) → a__after(mark(N), mark(XS))
mark(from(X)) → a__from(mark(X))
mark(after(X1, X2)) → a__after(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__after(X1, X2) → after(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__after(0, XS) → mark(XS)
a__after(s(N), cons(X, XS)) → a__after(mark(N), mark(XS))
mark(from(X)) → a__from(mark(X))
mark(after(X1, X2)) → a__after(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__after(X1, X2) → after(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:

A__AFTER(s(N), cons(X, XS)) → MARK(N)
MARK(from(X)) → MARK(X)
MARK(after(X1, X2)) → A__AFTER(mark(X1), mark(X2))
MARK(from(X)) → A__FROM(mark(X))
MARK(s(X)) → MARK(X)
MARK(after(X1, X2)) → MARK(X1)
A__AFTER(0, XS) → MARK(XS)
MARK(cons(X1, X2)) → MARK(X1)
A__AFTER(s(N), cons(X, XS)) → MARK(XS)
A__AFTER(s(N), cons(X, XS)) → A__AFTER(mark(N), mark(XS))
MARK(after(X1, X2)) → MARK(X2)
A__FROM(X) → MARK(X)

The TRS R consists of the following rules:

a__from(X) → cons(mark(X), from(s(X)))
a__after(0, XS) → mark(XS)
a__after(s(N), cons(X, XS)) → a__after(mark(N), mark(XS))
mark(from(X)) → a__from(mark(X))
mark(after(X1, X2)) → a__after(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__after(X1, X2) → after(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:

A__AFTER(s(N), cons(X, XS)) → MARK(N)
MARK(from(X)) → MARK(X)
MARK(after(X1, X2)) → A__AFTER(mark(X1), mark(X2))
MARK(from(X)) → A__FROM(mark(X))
MARK(s(X)) → MARK(X)
MARK(after(X1, X2)) → MARK(X1)
A__AFTER(0, XS) → MARK(XS)
MARK(cons(X1, X2)) → MARK(X1)
A__AFTER(s(N), cons(X, XS)) → MARK(XS)
A__AFTER(s(N), cons(X, XS)) → A__AFTER(mark(N), mark(XS))
MARK(after(X1, X2)) → MARK(X2)
A__FROM(X) → MARK(X)

The TRS R consists of the following rules:

a__from(X) → cons(mark(X), from(s(X)))
a__after(0, XS) → mark(XS)
a__after(s(N), cons(X, XS)) → a__after(mark(N), mark(XS))
mark(from(X)) → a__from(mark(X))
mark(after(X1, X2)) → a__after(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__after(X1, X2) → after(X1, X2)

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

A__AFTER(s(0), cons(y1, y2)) → A__AFTER(0, mark(y2))
A__AFTER(s(s(x0)), cons(y1, y2)) → A__AFTER(s(mark(x0)), mark(y2))
A__AFTER(s(cons(x0, x1)), cons(y1, y2)) → A__AFTER(cons(mark(x0), x1), mark(y2))
A__AFTER(s(from(x0)), cons(y1, y2)) → A__AFTER(a__from(mark(x0)), mark(y2))
A__AFTER(s(after(x0, x1)), cons(y1, y2)) → A__AFTER(a__after(mark(x0), mark(x1)), mark(y2))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ Narrowing

        ↳ QDP

          ↳ DependencyGraphProof

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

A__AFTER(s(N), cons(X, XS)) → MARK(N)
MARK(from(X)) → MARK(X)
MARK(from(X)) → A__FROM(mark(X))
MARK(after(X1, X2)) → A__AFTER(mark(X1), mark(X2))
MARK(s(X)) → MARK(X)
MARK(after(X1, X2)) → MARK(X1)
A__AFTER(0, XS) → MARK(XS)
MARK(cons(X1, X2)) → MARK(X1)
MARK(after(X1, X2)) → MARK(X2)
A__AFTER(s(cons(x0, x1)), cons(y1, y2)) → A__AFTER(cons(mark(x0), x1), mark(y2))
A__FROM(X) → MARK(X)
A__AFTER(s(N), cons(X, XS)) → MARK(XS)
A__AFTER(s(s(x0)), cons(y1, y2)) → A__AFTER(s(mark(x0)), mark(y2))
A__AFTER(s(0), cons(y1, y2)) → A__AFTER(0, mark(y2))
A__AFTER(s(from(x0)), cons(y1, y2)) → A__AFTER(a__from(mark(x0)), mark(y2))
A__AFTER(s(after(x0, x1)), cons(y1, y2)) → A__AFTER(a__after(mark(x0), mark(x1)), mark(y2))

The TRS R consists of the following rules:

a__from(X) → cons(mark(X), from(s(X)))
a__after(0, XS) → mark(XS)
a__after(s(N), cons(X, XS)) → a__after(mark(N), mark(XS))
mark(from(X)) → a__from(mark(X))
mark(after(X1, X2)) → a__after(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__after(X1, X2) → after(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__AFTER(s(N), cons(X, XS)) → MARK(N)
MARK(from(X)) → MARK(X)
MARK(from(X)) → A__FROM(mark(X))
MARK(after(X1, X2)) → A__AFTER(mark(X1), mark(X2))
MARK(s(X)) → MARK(X)
MARK(after(X1, X2)) → MARK(X1)
A__AFTER(0, XS) → MARK(XS)
MARK(cons(X1, X2)) → MARK(X1)
MARK(after(X1, X2)) → MARK(X2)
A__FROM(X) → MARK(X)
A__AFTER(s(N), cons(X, XS)) → MARK(XS)
A__AFTER(s(0), cons(y1, y2)) → A__AFTER(0, mark(y2))
A__AFTER(s(s(x0)), cons(y1, y2)) → A__AFTER(s(mark(x0)), mark(y2))
A__AFTER(s(from(x0)), cons(y1, y2)) → A__AFTER(a__from(mark(x0)), mark(y2))
A__AFTER(s(after(x0, x1)), cons(y1, y2)) → A__AFTER(a__after(mark(x0), mark(x1)), mark(y2))

The TRS R consists of the following rules:

a__from(X) → cons(mark(X), from(s(X)))
a__after(0, XS) → mark(XS)
a__after(s(N), cons(X, XS)) → a__after(mark(N), mark(XS))
mark(from(X)) → a__from(mark(X))
mark(after(X1, X2)) → a__after(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__after(X1, X2) → after(X1, X2)

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

MARK(after(s(x0), y1)) → A__AFTER(s(mark(x0)), mark(y1))
MARK(after(cons(x0, x1), y1)) → A__AFTER(cons(mark(x0), x1), mark(y1))
MARK(after(from(x0), y1)) → A__AFTER(a__from(mark(x0)), mark(y1))
MARK(after(0, y1)) → A__AFTER(0, mark(y1))
MARK(after(after(x0, x1), y1)) → A__AFTER(a__after(mark(x0), mark(x1)), mark(y1))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ Narrowing

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ QDP

              ↳ Narrowing

                ↳ QDP

                  ↳ DependencyGraphProof

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

A__AFTER(s(N), cons(X, XS)) → MARK(N)
MARK(from(X)) → MARK(X)
MARK(from(X)) → A__FROM(mark(X))
MARK(s(X)) → MARK(X)
MARK(after(0, y1)) → A__AFTER(0, mark(y1))
MARK(after(X1, X2)) → MARK(X1)
A__AFTER(0, XS) → MARK(XS)
MARK(cons(X1, X2)) → MARK(X1)
MARK(after(X1, X2)) → MARK(X2)
MARK(after(from(x0), y1)) → A__AFTER(a__from(mark(x0)), mark(y1))
A__FROM(X) → MARK(X)
MARK(after(after(x0, x1), y1)) → A__AFTER(a__after(mark(x0), mark(x1)), mark(y1))
A__AFTER(s(N), cons(X, XS)) → MARK(XS)
A__AFTER(s(s(x0)), cons(y1, y2)) → A__AFTER(s(mark(x0)), mark(y2))
A__AFTER(s(0), cons(y1, y2)) → A__AFTER(0, mark(y2))
MARK(after(cons(x0, x1), y1)) → A__AFTER(cons(mark(x0), x1), mark(y1))
MARK(after(s(x0), y1)) → A__AFTER(s(mark(x0)), mark(y1))
A__AFTER(s(from(x0)), cons(y1, y2)) → A__AFTER(a__from(mark(x0)), mark(y2))
A__AFTER(s(after(x0, x1)), cons(y1, y2)) → A__AFTER(a__after(mark(x0), mark(x1)), mark(y2))

The TRS R consists of the following rules:

a__from(X) → cons(mark(X), from(s(X)))
a__after(0, XS) → mark(XS)
a__after(s(N), cons(X, XS)) → a__after(mark(N), mark(XS))
mark(from(X)) → a__from(mark(X))
mark(after(X1, X2)) → a__after(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__after(X1, X2) → after(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__AFTER(s(N), cons(X, XS)) → MARK(N)
MARK(from(X)) → MARK(X)
MARK(from(X)) → A__FROM(mark(X))
MARK(s(X)) → MARK(X)
MARK(after(0, y1)) → A__AFTER(0, mark(y1))
MARK(after(X1, X2)) → MARK(X1)
A__AFTER(0, XS) → MARK(XS)
MARK(cons(X1, X2)) → MARK(X1)
MARK(after(X1, X2)) → MARK(X2)
A__FROM(X) → MARK(X)
MARK(after(from(x0), y1)) → A__AFTER(a__from(mark(x0)), mark(y1))
MARK(after(after(x0, x1), y1)) → A__AFTER(a__after(mark(x0), mark(x1)), mark(y1))
A__AFTER(s(N), cons(X, XS)) → MARK(XS)
A__AFTER(s(s(x0)), cons(y1, y2)) → A__AFTER(s(mark(x0)), mark(y2))
A__AFTER(s(0), cons(y1, y2)) → A__AFTER(0, mark(y2))
A__AFTER(s(from(x0)), cons(y1, y2)) → A__AFTER(a__from(mark(x0)), mark(y2))
MARK(after(s(x0), y1)) → A__AFTER(s(mark(x0)), mark(y1))
A__AFTER(s(after(x0, x1)), cons(y1, y2)) → A__AFTER(a__after(mark(x0), mark(x1)), mark(y2))

The TRS R consists of the following rules:

a__from(X) → cons(mark(X), from(s(X)))
a__after(0, XS) → mark(XS)
a__after(s(N), cons(X, XS)) → a__after(mark(N), mark(XS))
mark(from(X)) → a__from(mark(X))
mark(after(X1, X2)) → a__after(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__after(X1, X2) → after(X1, X2)

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

MARK(after(s(y0), cons(x0, x1))) → A__AFTER(s(mark(y0)), cons(mark(x0), x1))
MARK(after(s(y0), after(x0, x1))) → A__AFTER(s(mark(y0)), a__after(mark(x0), mark(x1)))
MARK(after(s(y0), 0)) → A__AFTER(s(mark(y0)), 0)
MARK(after(s(y0), from(x0))) → A__AFTER(s(mark(y0)), a__from(mark(x0)))
MARK(after(s(y0), s(x0))) → A__AFTER(s(mark(y0)), 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:

MARK(after(s(y0), cons(x0, x1))) → A__AFTER(s(mark(y0)), cons(mark(x0), x1))
A__AFTER(s(N), cons(X, XS)) → MARK(N)
MARK(from(X)) → MARK(X)
MARK(from(X)) → A__FROM(mark(X))
MARK(s(X)) → MARK(X)
MARK(after(s(y0), from(x0))) → A__AFTER(s(mark(y0)), a__from(mark(x0)))
MARK(after(0, y1)) → A__AFTER(0, mark(y1))
MARK(after(X1, X2)) → MARK(X1)
A__AFTER(0, XS) → MARK(XS)
MARK(cons(X1, X2)) → MARK(X1)
MARK(after(s(y0), after(x0, x1))) → A__AFTER(s(mark(y0)), a__after(mark(x0), mark(x1)))
MARK(after(X1, X2)) → MARK(X2)
MARK(after(from(x0), y1)) → A__AFTER(a__from(mark(x0)), mark(y1))
A__FROM(X) → MARK(X)
MARK(after(s(y0), 0)) → A__AFTER(s(mark(y0)), 0)
MARK(after(after(x0, x1), y1)) → A__AFTER(a__after(mark(x0), mark(x1)), mark(y1))
A__AFTER(s(N), cons(X, XS)) → MARK(XS)
A__AFTER(s(0), cons(y1, y2)) → A__AFTER(0, mark(y2))
A__AFTER(s(s(x0)), cons(y1, y2)) → A__AFTER(s(mark(x0)), mark(y2))
A__AFTER(s(from(x0)), cons(y1, y2)) → A__AFTER(a__from(mark(x0)), mark(y2))
MARK(after(s(y0), s(x0))) → A__AFTER(s(mark(y0)), s(mark(x0)))
A__AFTER(s(after(x0, x1)), cons(y1, y2)) → A__AFTER(a__after(mark(x0), mark(x1)), mark(y2))

The TRS R consists of the following rules:

a__from(X) → cons(mark(X), from(s(X)))
a__after(0, XS) → mark(XS)
a__after(s(N), cons(X, XS)) → a__after(mark(N), mark(XS))
mark(from(X)) → a__from(mark(X))
mark(after(X1, X2)) → a__after(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__after(X1, X2) → after(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__AFTER(s(N), cons(X, XS)) → MARK(N)
MARK(after(s(y0), cons(x0, x1))) → A__AFTER(s(mark(y0)), cons(mark(x0), x1))
MARK(from(X)) → MARK(X)
MARK(from(X)) → A__FROM(mark(X))
MARK(s(X)) → MARK(X)
MARK(after(s(y0), from(x0))) → A__AFTER(s(mark(y0)), a__from(mark(x0)))
MARK(after(0, y1)) → A__AFTER(0, mark(y1))
MARK(after(X1, X2)) → MARK(X1)
A__AFTER(0, XS) → MARK(XS)
MARK(cons(X1, X2)) → MARK(X1)
MARK(after(s(y0), after(x0, x1))) → A__AFTER(s(mark(y0)), a__after(mark(x0), mark(x1)))
MARK(after(X1, X2)) → MARK(X2)
A__FROM(X) → MARK(X)
MARK(after(from(x0), y1)) → A__AFTER(a__from(mark(x0)), mark(y1))
MARK(after(after(x0, x1), y1)) → A__AFTER(a__after(mark(x0), mark(x1)), mark(y1))
A__AFTER(s(N), cons(X, XS)) → MARK(XS)
A__AFTER(s(s(x0)), cons(y1, y2)) → A__AFTER(s(mark(x0)), mark(y2))
A__AFTER(s(0), cons(y1, y2)) → A__AFTER(0, mark(y2))
A__AFTER(s(from(x0)), cons(y1, y2)) → A__AFTER(a__from(mark(x0)), mark(y2))
A__AFTER(s(after(x0, x1)), cons(y1, y2)) → A__AFTER(a__after(mark(x0), mark(x1)), mark(y2))

The TRS R consists of the following rules:

a__from(X) → cons(mark(X), from(s(X)))
a__after(0, XS) → mark(XS)
a__after(s(N), cons(X, XS)) → a__after(mark(N), mark(XS))
mark(from(X)) → a__from(mark(X))
mark(after(X1, X2)) → a__after(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__after(X1, X2) → after(X1, X2)

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

A__AFTER(s(s(y0)), cons(y1, s(x0))) → A__AFTER(s(mark(y0)), s(mark(x0)))
A__AFTER(s(s(y0)), cons(y1, cons(x0, x1))) → A__AFTER(s(mark(y0)), cons(mark(x0), x1))
A__AFTER(s(s(y0)), cons(y1, after(x0, x1))) → A__AFTER(s(mark(y0)), a__after(mark(x0), mark(x1)))
A__AFTER(s(s(y0)), cons(y1, from(x0))) → A__AFTER(s(mark(y0)), a__from(mark(x0)))
A__AFTER(s(s(y0)), cons(y1, 0)) → A__AFTER(s(mark(y0)), 0)

↳ 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(after(s(y0), cons(x0, x1))) → A__AFTER(s(mark(y0)), cons(mark(x0), x1))
MARK(s(X)) → MARK(X)
MARK(after(s(y0), from(x0))) → A__AFTER(s(mark(y0)), a__from(mark(x0)))
MARK(cons(X1, X2)) → MARK(X1)
A__AFTER(s(s(y0)), cons(y1, cons(x0, x1))) → A__AFTER(s(mark(y0)), cons(mark(x0), x1))
MARK(after(s(y0), after(x0, x1))) → A__AFTER(s(mark(y0)), a__after(mark(x0), mark(x1)))
MARK(after(X1, X2)) → MARK(X2)
MARK(after(from(x0), y1)) → A__AFTER(a__from(mark(x0)), mark(y1))
A__AFTER(s(s(y0)), cons(y1, after(x0, x1))) → A__AFTER(s(mark(y0)), a__after(mark(x0), mark(x1)))
MARK(after(after(x0, x1), y1)) → A__AFTER(a__after(mark(x0), mark(x1)), mark(y1))
A__AFTER(s(N), cons(X, XS)) → MARK(XS)
A__AFTER(s(s(y0)), cons(y1, 0)) → A__AFTER(s(mark(y0)), 0)
A__AFTER(s(N), cons(X, XS)) → MARK(N)
MARK(from(X)) → MARK(X)
MARK(from(X)) → A__FROM(mark(X))
MARK(after(0, y1)) → A__AFTER(0, mark(y1))
MARK(after(X1, X2)) → MARK(X1)
A__AFTER(0, XS) → MARK(XS)
A__AFTER(s(s(y0)), cons(y1, s(x0))) → A__AFTER(s(mark(y0)), s(mark(x0)))
A__FROM(X) → MARK(X)
A__AFTER(s(s(y0)), cons(y1, from(x0))) → A__AFTER(s(mark(y0)), a__from(mark(x0)))
A__AFTER(s(0), cons(y1, y2)) → A__AFTER(0, mark(y2))
A__AFTER(s(from(x0)), cons(y1, y2)) → A__AFTER(a__from(mark(x0)), mark(y2))
A__AFTER(s(after(x0, x1)), cons(y1, y2)) → A__AFTER(a__after(mark(x0), mark(x1)), mark(y2))

The TRS R consists of the following rules:

a__from(X) → cons(mark(X), from(s(X)))
a__after(0, XS) → mark(XS)
a__after(s(N), cons(X, XS)) → a__after(mark(N), mark(XS))
mark(from(X)) → a__from(mark(X))
mark(after(X1, X2)) → a__after(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__after(X1, X2) → after(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

                                      ↳ QDPOrderProof

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

A__AFTER(s(N), cons(X, XS)) → MARK(N)
MARK(after(s(y0), cons(x0, x1))) → A__AFTER(s(mark(y0)), cons(mark(x0), x1))
MARK(from(X)) → MARK(X)
MARK(from(X)) → A__FROM(mark(X))
MARK(s(X)) → MARK(X)
MARK(after(s(y0), from(x0))) → A__AFTER(s(mark(y0)), a__from(mark(x0)))
MARK(after(0, y1)) → A__AFTER(0, mark(y1))
MARK(after(X1, X2)) → MARK(X1)
A__AFTER(0, XS) → MARK(XS)
A__AFTER(s(s(y0)), cons(y1, cons(x0, x1))) → A__AFTER(s(mark(y0)), cons(mark(x0), x1))
MARK(cons(X1, X2)) → MARK(X1)
MARK(after(s(y0), after(x0, x1))) → A__AFTER(s(mark(y0)), a__after(mark(x0), mark(x1)))
MARK(after(X1, X2)) → MARK(X2)
A__FROM(X) → MARK(X)
MARK(after(from(x0), y1)) → A__AFTER(a__from(mark(x0)), mark(y1))
A__AFTER(s(s(y0)), cons(y1, after(x0, x1))) → A__AFTER(s(mark(y0)), a__after(mark(x0), mark(x1)))
A__AFTER(s(s(y0)), cons(y1, from(x0))) → A__AFTER(s(mark(y0)), a__from(mark(x0)))
MARK(after(after(x0, x1), y1)) → A__AFTER(a__after(mark(x0), mark(x1)), mark(y1))
A__AFTER(s(N), cons(X, XS)) → MARK(XS)
A__AFTER(s(0), cons(y1, y2)) → A__AFTER(0, mark(y2))
A__AFTER(s(from(x0)), cons(y1, y2)) → A__AFTER(a__from(mark(x0)), mark(y2))
A__AFTER(s(after(x0, x1)), cons(y1, y2)) → A__AFTER(a__after(mark(x0), mark(x1)), mark(y2))

The TRS R consists of the following rules:

a__from(X) → cons(mark(X), from(s(X)))
a__after(0, XS) → mark(XS)
a__after(s(N), cons(X, XS)) → a__after(mark(N), mark(XS))
mark(from(X)) → a__from(mark(X))
mark(after(X1, X2)) → a__after(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__after(X1, X2) → after(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(after(from(x0), y1)) → A__AFTER(a__from(mark(x0)), mark(y1))
A__AFTER(s(from(x0)), cons(y1, y2)) → A__AFTER(a__from(mark(x0)), mark(y2))

The remaining pairs can at least be oriented weakly.

A__AFTER(s(N), cons(X, XS)) → MARK(N)
MARK(after(s(y0), cons(x0, x1))) → A__AFTER(s(mark(y0)), cons(mark(x0), x1))
MARK(from(X)) → MARK(X)
MARK(from(X)) → A__FROM(mark(X))
MARK(s(X)) → MARK(X)
MARK(after(s(y0), from(x0))) → A__AFTER(s(mark(y0)), a__from(mark(x0)))
MARK(after(0, y1)) → A__AFTER(0, mark(y1))
MARK(after(X1, X2)) → MARK(X1)
A__AFTER(0, XS) → MARK(XS)
A__AFTER(s(s(y0)), cons(y1, cons(x0, x1))) → A__AFTER(s(mark(y0)), cons(mark(x0), x1))
MARK(cons(X1, X2)) → MARK(X1)
MARK(after(s(y0), after(x0, x1))) → A__AFTER(s(mark(y0)), a__after(mark(x0), mark(x1)))
MARK(after(X1, X2)) → MARK(X2)
A__FROM(X) → MARK(X)
A__AFTER(s(s(y0)), cons(y1, after(x0, x1))) → A__AFTER(s(mark(y0)), a__after(mark(x0), mark(x1)))
A__AFTER(s(s(y0)), cons(y1, from(x0))) → A__AFTER(s(mark(y0)), a__from(mark(x0)))
MARK(after(after(x0, x1), y1)) → A__AFTER(a__after(mark(x0), mark(x1)), mark(y1))
A__AFTER(s(N), cons(X, XS)) → MARK(XS)
A__AFTER(s(0), cons(y1, y2)) → A__AFTER(0, mark(y2))
A__AFTER(s(after(x0, x1)), cons(y1, y2)) → A__AFTER(a__after(mark(x0), mark(x1)), mark(y2))

Used ordering: Polynomial interpretation [25]:

POL(0) = 1
POL(A__AFTER(x₁, x₂)) = x₁
POL(A__FROM(x₁)) = 1
POL(MARK(x₁)) = 1
POL(a__after(x₁, x₂)) = x₁
POL(a__from(x₁)) = 0
POL(after(x₁, x₂)) = 0
POL(cons(x₁, x₂)) = 0
POL(from(x₁)) = 0
POL(mark(x₁)) = 1
POL(s(x₁)) = 1

The following usable rules [17] were oriented:

a__after(s(N), cons(X, XS)) → a__after(mark(N), mark(XS))
mark(after(X1, X2)) → a__after(mark(X1), mark(X2))
a__after(0, XS) → mark(XS)
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(from(X)) → a__from(mark(X))
a__from(X) → from(X)
a__after(X1, X2) → after(X1, X2)
mark(s(X)) → s(mark(X))
mark(0) → 0
a__from(X) → cons(mark(X), from(s(X)))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ 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:

A__AFTER(s(N), cons(X, XS)) → MARK(N)
MARK(after(s(y0), cons(x0, x1))) → A__AFTER(s(mark(y0)), cons(mark(x0), x1))
MARK(from(X)) → MARK(X)
MARK(from(X)) → A__FROM(mark(X))
MARK(s(X)) → MARK(X)
MARK(after(s(y0), from(x0))) → A__AFTER(s(mark(y0)), a__from(mark(x0)))
MARK(after(0, y1)) → A__AFTER(0, mark(y1))
MARK(after(X1, X2)) → MARK(X1)
A__AFTER(0, XS) → MARK(XS)
A__AFTER(s(s(y0)), cons(y1, cons(x0, x1))) → A__AFTER(s(mark(y0)), cons(mark(x0), x1))
MARK(cons(X1, X2)) → MARK(X1)
MARK(after(s(y0), after(x0, x1))) → A__AFTER(s(mark(y0)), a__after(mark(x0), mark(x1)))
MARK(after(X1, X2)) → MARK(X2)
A__FROM(X) → MARK(X)
A__AFTER(s(s(y0)), cons(y1, after(x0, x1))) → A__AFTER(s(mark(y0)), a__after(mark(x0), mark(x1)))
A__AFTER(s(s(y0)), cons(y1, from(x0))) → A__AFTER(s(mark(y0)), a__from(mark(x0)))
MARK(after(after(x0, x1), y1)) → A__AFTER(a__after(mark(x0), mark(x1)), mark(y1))
A__AFTER(s(N), cons(X, XS)) → MARK(XS)
A__AFTER(s(0), cons(y1, y2)) → A__AFTER(0, mark(y2))
A__AFTER(s(after(x0, x1)), cons(y1, y2)) → A__AFTER(a__after(mark(x0), mark(x1)), mark(y2))

The TRS R consists of the following rules:

a__from(X) → cons(mark(X), from(s(X)))
a__after(0, XS) → mark(XS)
a__after(s(N), cons(X, XS)) → a__after(mark(N), mark(XS))
mark(from(X)) → a__from(mark(X))
mark(after(X1, X2)) → a__after(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__after(X1, X2) → after(X1, X2)

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