Termination w.r.t. Q proof of /tmp/tmp07Sf0J/Ex4_Zan97

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__f(X) → cons(mark(X), f(g(X)))
a__g(0) → s(0)
a__g(s(X)) → s(s(a__g(mark(X))))
a__sel(0, cons(X, Y)) → mark(X)
a__sel(s(X), cons(Y, Z)) → a__sel(mark(X), mark(Z))
mark(f(X)) → a__f(mark(X))
mark(g(X)) → a__g(mark(X))
mark(sel(X1, X2)) → a__sel(mark(X1), mark(X2))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0) → 0
mark(s(X)) → s(mark(X))
a__f(X) → f(X)
a__g(X) → g(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__f(X) → cons(mark(X), f(g(X)))
a__g(0) → s(0)
a__g(s(X)) → s(s(a__g(mark(X))))
a__sel(0, cons(X, Y)) → mark(X)
a__sel(s(X), cons(Y, Z)) → a__sel(mark(X), mark(Z))
mark(f(X)) → a__f(mark(X))
mark(g(X)) → a__g(mark(X))
mark(sel(X1, X2)) → a__sel(mark(X1), mark(X2))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0) → 0
mark(s(X)) → s(mark(X))
a__f(X) → f(X)
a__g(X) → g(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(g(X)) → A__G(mark(X))
MARK(sel(X1, X2)) → MARK(X1)
MARK(sel(X1, X2)) → A__SEL(mark(X1), mark(X2))
A__G(s(X)) → A__G(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__F(X) → MARK(X)
A__SEL(0, cons(X, Y)) → MARK(X)
MARK(g(X)) → MARK(X)
A__SEL(s(X), cons(Y, Z)) → MARK(Z)
A__G(s(X)) → MARK(X)
MARK(f(X)) → A__F(mark(X))
MARK(sel(X1, X2)) → MARK(X2)
A__SEL(s(X), cons(Y, Z)) → MARK(X)
MARK(f(X)) → MARK(X)

The TRS R consists of the following rules:

a__f(X) → cons(mark(X), f(g(X)))
a__g(0) → s(0)
a__g(s(X)) → s(s(a__g(mark(X))))
a__sel(0, cons(X, Y)) → mark(X)
a__sel(s(X), cons(Y, Z)) → a__sel(mark(X), mark(Z))
mark(f(X)) → a__f(mark(X))
mark(g(X)) → a__g(mark(X))
mark(sel(X1, X2)) → a__sel(mark(X1), mark(X2))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0) → 0
mark(s(X)) → s(mark(X))
a__f(X) → f(X)
a__g(X) → g(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(g(X)) → A__G(mark(X))
MARK(sel(X1, X2)) → MARK(X1)
MARK(sel(X1, X2)) → A__SEL(mark(X1), mark(X2))
A__G(s(X)) → A__G(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__F(X) → MARK(X)
A__SEL(0, cons(X, Y)) → MARK(X)
MARK(g(X)) → MARK(X)
A__SEL(s(X), cons(Y, Z)) → MARK(Z)
A__G(s(X)) → MARK(X)
MARK(f(X)) → A__F(mark(X))
MARK(sel(X1, X2)) → MARK(X2)
A__SEL(s(X), cons(Y, Z)) → MARK(X)
MARK(f(X)) → MARK(X)

The TRS R consists of the following rules:

a__f(X) → cons(mark(X), f(g(X)))
a__g(0) → s(0)
a__g(s(X)) → s(s(a__g(mark(X))))
a__sel(0, cons(X, Y)) → mark(X)
a__sel(s(X), cons(Y, Z)) → a__sel(mark(X), mark(Z))
mark(f(X)) → a__f(mark(X))
mark(g(X)) → a__g(mark(X))
mark(sel(X1, X2)) → a__sel(mark(X1), mark(X2))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0) → 0
mark(s(X)) → s(mark(X))
a__f(X) → f(X)
a__g(X) → g(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__G(s(X)) → A__G(mark(X)) at position [0] we obtained the following new rules:

A__G(s(cons(x0, x1))) → A__G(cons(mark(x0), x1))
A__G(s(f(x0))) → A__G(a__f(mark(x0)))
A__G(s(0)) → A__G(0)
A__G(s(g(x0))) → A__G(a__g(mark(x0)))
A__G(s(sel(x0, x1))) → A__G(a__sel(mark(x0), mark(x1)))
A__G(s(s(x0))) → A__G(s(mark(x0)))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ Narrowing

        ↳ QDP

          ↳ DependencyGraphProof

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

MARK(g(X)) → A__G(mark(X))
A__G(s(f(x0))) → A__G(a__f(mark(x0)))
MARK(sel(X1, X2)) → MARK(X1)
A__G(s(0)) → A__G(0)
MARK(sel(X1, X2)) → A__SEL(mark(X1), mark(X2))
MARK(s(X)) → MARK(X)
A__SEL(s(X), cons(Y, Z)) → A__SEL(mark(X), mark(Z))
A__G(s(cons(x0, x1))) → A__G(cons(mark(x0), x1))
MARK(cons(X1, X2)) → MARK(X1)
A__F(X) → MARK(X)
A__SEL(0, cons(X, Y)) → MARK(X)
MARK(g(X)) → MARK(X)
A__SEL(s(X), cons(Y, Z)) → MARK(Z)
A__G(s(sel(x0, x1))) → A__G(a__sel(mark(x0), mark(x1)))
A__G(s(X)) → MARK(X)
MARK(f(X)) → A__F(mark(X))
A__G(s(g(x0))) → A__G(a__g(mark(x0)))
A__SEL(s(X), cons(Y, Z)) → MARK(X)
MARK(sel(X1, X2)) → MARK(X2)
A__G(s(s(x0))) → A__G(s(mark(x0)))
MARK(f(X)) → MARK(X)

The TRS R consists of the following rules:

a__f(X) → cons(mark(X), f(g(X)))
a__g(0) → s(0)
a__g(s(X)) → s(s(a__g(mark(X))))
a__sel(0, cons(X, Y)) → mark(X)
a__sel(s(X), cons(Y, Z)) → a__sel(mark(X), mark(Z))
mark(f(X)) → a__f(mark(X))
mark(g(X)) → a__g(mark(X))
mark(sel(X1, X2)) → a__sel(mark(X1), mark(X2))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0) → 0
mark(s(X)) → s(mark(X))
a__f(X) → f(X)
a__g(X) → g(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

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

MARK(g(X)) → A__G(mark(X))
A__G(s(f(x0))) → A__G(a__f(mark(x0)))
MARK(sel(X1, X2)) → MARK(X1)
MARK(sel(X1, X2)) → A__SEL(mark(X1), mark(X2))
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__F(X) → MARK(X)
A__SEL(0, cons(X, Y)) → MARK(X)
MARK(g(X)) → MARK(X)
A__SEL(s(X), cons(Y, Z)) → MARK(Z)
A__G(s(sel(x0, x1))) → A__G(a__sel(mark(x0), mark(x1)))
A__G(s(X)) → MARK(X)
MARK(f(X)) → A__F(mark(X))
A__G(s(g(x0))) → A__G(a__g(mark(x0)))
A__SEL(s(X), cons(Y, Z)) → MARK(X)
MARK(sel(X1, X2)) → MARK(X2)
A__G(s(s(x0))) → A__G(s(mark(x0)))
MARK(f(X)) → MARK(X)

The TRS R consists of the following rules:

a__f(X) → cons(mark(X), f(g(X)))
a__g(0) → s(0)
a__g(s(X)) → s(s(a__g(mark(X))))
a__sel(0, cons(X, Y)) → mark(X)
a__sel(s(X), cons(Y, Z)) → a__sel(mark(X), mark(Z))
mark(f(X)) → a__f(mark(X))
mark(g(X)) → a__g(mark(X))
mark(sel(X1, X2)) → a__sel(mark(X1), mark(X2))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0) → 0
mark(s(X)) → s(mark(X))
a__f(X) → f(X)
a__g(X) → g(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(g(X)) → A__G(mark(X)) at position [0] we obtained the following new rules:

MARK(g(cons(x0, x1))) → A__G(cons(mark(x0), x1))
MARK(g(sel(x0, x1))) → A__G(a__sel(mark(x0), mark(x1)))
MARK(g(g(x0))) → A__G(a__g(mark(x0)))
MARK(g(f(x0))) → A__G(a__f(mark(x0)))
MARK(g(s(x0))) → A__G(s(mark(x0)))
MARK(g(0)) → A__G(0)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ Narrowing

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ QDP

              ↳ Narrowing

                ↳ QDP

                  ↳ DependencyGraphProof

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

MARK(s(X)) → MARK(X)
MARK(cons(X1, X2)) → MARK(X1)
A__F(X) → MARK(X)
A__SEL(0, cons(X, Y)) → MARK(X)
MARK(g(X)) → MARK(X)
A__SEL(s(X), cons(Y, Z)) → MARK(Z)
MARK(g(f(x0))) → A__G(a__f(mark(x0)))
A__G(s(sel(x0, x1))) → A__G(a__sel(mark(x0), mark(x1)))
MARK(g(cons(x0, x1))) → A__G(cons(mark(x0), x1))
MARK(g(g(x0))) → A__G(a__g(mark(x0)))
A__G(s(g(x0))) → A__G(a__g(mark(x0)))
A__SEL(s(X), cons(Y, Z)) → MARK(X)
A__G(s(s(x0))) → A__G(s(mark(x0)))
MARK(f(X)) → MARK(X)
MARK(g(0)) → A__G(0)
A__G(s(f(x0))) → A__G(a__f(mark(x0)))
MARK(sel(X1, X2)) → MARK(X1)
MARK(sel(X1, X2)) → A__SEL(mark(X1), mark(X2))
MARK(g(sel(x0, x1))) → A__G(a__sel(mark(x0), mark(x1)))
A__SEL(s(X), cons(Y, Z)) → A__SEL(mark(X), mark(Z))
MARK(g(s(x0))) → A__G(s(mark(x0)))
A__G(s(X)) → MARK(X)
MARK(f(X)) → A__F(mark(X))
MARK(sel(X1, X2)) → MARK(X2)

The TRS R consists of the following rules:

a__f(X) → cons(mark(X), f(g(X)))
a__g(0) → s(0)
a__g(s(X)) → s(s(a__g(mark(X))))
a__sel(0, cons(X, Y)) → mark(X)
a__sel(s(X), cons(Y, Z)) → a__sel(mark(X), mark(Z))
mark(f(X)) → a__f(mark(X))
mark(g(X)) → a__g(mark(X))
mark(sel(X1, X2)) → a__sel(mark(X1), mark(X2))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0) → 0
mark(s(X)) → s(mark(X))
a__f(X) → f(X)
a__g(X) → g(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

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

A__G(s(f(x0))) → A__G(a__f(mark(x0)))
MARK(sel(X1, X2)) → MARK(X1)
MARK(sel(X1, X2)) → A__SEL(mark(X1), mark(X2))
MARK(g(sel(x0, x1))) → A__G(a__sel(mark(x0), mark(x1)))
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__F(X) → MARK(X)
A__SEL(0, cons(X, Y)) → MARK(X)
MARK(g(X)) → MARK(X)
A__SEL(s(X), cons(Y, Z)) → MARK(Z)
MARK(g(f(x0))) → A__G(a__f(mark(x0)))
MARK(g(s(x0))) → A__G(s(mark(x0)))
A__G(s(sel(x0, x1))) → A__G(a__sel(mark(x0), mark(x1)))
A__G(s(X)) → MARK(X)
MARK(f(X)) → A__F(mark(X))
MARK(g(g(x0))) → A__G(a__g(mark(x0)))
A__G(s(g(x0))) → A__G(a__g(mark(x0)))
A__SEL(s(X), cons(Y, Z)) → MARK(X)
MARK(sel(X1, X2)) → MARK(X2)
A__G(s(s(x0))) → A__G(s(mark(x0)))
MARK(f(X)) → MARK(X)

The TRS R consists of the following rules:

a__f(X) → cons(mark(X), f(g(X)))
a__g(0) → s(0)
a__g(s(X)) → s(s(a__g(mark(X))))
a__sel(0, cons(X, Y)) → mark(X)
a__sel(s(X), cons(Y, Z)) → a__sel(mark(X), mark(Z))
mark(f(X)) → a__f(mark(X))
mark(g(X)) → a__g(mark(X))
mark(sel(X1, X2)) → a__sel(mark(X1), mark(X2))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0) → 0
mark(s(X)) → s(mark(X))
a__f(X) → f(X)
a__g(X) → g(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(g(x0), y1)) → A__SEL(a__g(mark(x0)), mark(y1))
MARK(sel(f(x0), y1)) → A__SEL(a__f(mark(x0)), mark(y1))
MARK(sel(0, y1)) → A__SEL(0, mark(y1))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ Narrowing

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ QDP

              ↳ Narrowing

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ QDP

                      ↳ Narrowing

                        ↳ QDP

                          ↳ DependencyGraphProof

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

MARK(sel(cons(x0, x1), y1)) → A__SEL(cons(mark(x0), x1), mark(y1))
MARK(s(X)) → MARK(X)
MARK(cons(X1, X2)) → MARK(X1)
A__F(X) → MARK(X)
A__SEL(0, cons(X, Y)) → MARK(X)
MARK(g(X)) → MARK(X)
A__SEL(s(X), cons(Y, Z)) → MARK(Z)
MARK(g(f(x0))) → A__G(a__f(mark(x0)))
MARK(sel(0, y1)) → A__SEL(0, mark(y1))
A__G(s(sel(x0, x1))) → A__G(a__sel(mark(x0), mark(x1)))
MARK(g(g(x0))) → A__G(a__g(mark(x0)))
A__G(s(g(x0))) → A__G(a__g(mark(x0)))
A__SEL(s(X), cons(Y, Z)) → MARK(X)
A__G(s(s(x0))) → A__G(s(mark(x0)))
MARK(f(X)) → MARK(X)
A__G(s(f(x0))) → A__G(a__f(mark(x0)))
MARK(sel(X1, X2)) → MARK(X1)
MARK(g(sel(x0, x1))) → A__G(a__sel(mark(x0), mark(x1)))
A__SEL(s(X), cons(Y, Z)) → A__SEL(mark(X), mark(Z))
MARK(sel(s(x0), y1)) → A__SEL(s(mark(x0)), mark(y1))
MARK(sel(f(x0), y1)) → A__SEL(a__f(mark(x0)), mark(y1))
MARK(g(s(x0))) → A__G(s(mark(x0)))
A__G(s(X)) → MARK(X)
MARK(sel(sel(x0, x1), y1)) → A__SEL(a__sel(mark(x0), mark(x1)), mark(y1))
MARK(f(X)) → A__F(mark(X))
MARK(sel(g(x0), y1)) → A__SEL(a__g(mark(x0)), mark(y1))
MARK(sel(X1, X2)) → MARK(X2)

The TRS R consists of the following rules:

a__f(X) → cons(mark(X), f(g(X)))
a__g(0) → s(0)
a__g(s(X)) → s(s(a__g(mark(X))))
a__sel(0, cons(X, Y)) → mark(X)
a__sel(s(X), cons(Y, Z)) → a__sel(mark(X), mark(Z))
mark(f(X)) → a__f(mark(X))
mark(g(X)) → a__g(mark(X))
mark(sel(X1, X2)) → a__sel(mark(X1), mark(X2))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0) → 0
mark(s(X)) → s(mark(X))
a__f(X) → f(X)
a__g(X) → g(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

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ QDP

                              ↳ Narrowing

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

MARK(s(X)) → MARK(X)
MARK(cons(X1, X2)) → MARK(X1)
A__F(X) → MARK(X)
A__SEL(0, cons(X, Y)) → MARK(X)
MARK(g(X)) → MARK(X)
A__SEL(s(X), cons(Y, Z)) → MARK(Z)
MARK(g(f(x0))) → A__G(a__f(mark(x0)))
MARK(sel(0, y1)) → A__SEL(0, mark(y1))
A__G(s(sel(x0, x1))) → A__G(a__sel(mark(x0), mark(x1)))
MARK(g(g(x0))) → A__G(a__g(mark(x0)))
A__G(s(g(x0))) → A__G(a__g(mark(x0)))
A__SEL(s(X), cons(Y, Z)) → MARK(X)
MARK(f(X)) → MARK(X)
A__G(s(s(x0))) → A__G(s(mark(x0)))
A__G(s(f(x0))) → A__G(a__f(mark(x0)))
MARK(sel(X1, X2)) → MARK(X1)
MARK(g(sel(x0, x1))) → A__G(a__sel(mark(x0), mark(x1)))
A__SEL(s(X), cons(Y, Z)) → A__SEL(mark(X), mark(Z))
MARK(sel(s(x0), y1)) → A__SEL(s(mark(x0)), mark(y1))
MARK(sel(f(x0), y1)) → A__SEL(a__f(mark(x0)), mark(y1))
MARK(g(s(x0))) → A__G(s(mark(x0)))
A__G(s(X)) → MARK(X)
MARK(sel(sel(x0, x1), y1)) → A__SEL(a__sel(mark(x0), mark(x1)), mark(y1))
MARK(f(X)) → A__F(mark(X))
MARK(sel(g(x0), y1)) → A__SEL(a__g(mark(x0)), mark(y1))
MARK(sel(X1, X2)) → MARK(X2)

The TRS R consists of the following rules:

a__f(X) → cons(mark(X), f(g(X)))
a__g(0) → s(0)
a__g(s(X)) → s(s(a__g(mark(X))))
a__sel(0, cons(X, Y)) → mark(X)
a__sel(s(X), cons(Y, Z)) → a__sel(mark(X), mark(Z))
mark(f(X)) → a__f(mark(X))
mark(g(X)) → a__g(mark(X))
mark(sel(X1, X2)) → a__sel(mark(X1), mark(X2))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0) → 0
mark(s(X)) → s(mark(X))
a__f(X) → f(X)
a__g(X) → g(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), cons(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), cons(mark(x0), x1))
MARK(sel(sel(y0, y1), s(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), s(mark(x0)))
MARK(sel(sel(y0, y1), g(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__g(mark(x0)))
MARK(sel(sel(y0, y1), f(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__f(mark(x0)))
MARK(sel(sel(y0, y1), 0)) → A__SEL(a__sel(mark(y0), mark(y1)), 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(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(sel(y0, y1), g(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__g(mark(x0)))
MARK(sel(sel(y0, y1), s(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), s(mark(x0)))
MARK(cons(X1, X2)) → MARK(X1)
A__F(X) → MARK(X)
A__SEL(0, cons(X, Y)) → MARK(X)
MARK(g(X)) → MARK(X)
A__SEL(s(X), cons(Y, Z)) → MARK(Z)
MARK(g(f(x0))) → A__G(a__f(mark(x0)))
A__G(s(sel(x0, x1))) → A__G(a__sel(mark(x0), mark(x1)))
MARK(sel(0, y1)) → A__SEL(0, mark(y1))
MARK(g(g(x0))) → A__G(a__g(mark(x0)))
A__G(s(g(x0))) → A__G(a__g(mark(x0)))
A__SEL(s(X), cons(Y, Z)) → MARK(X)
A__G(s(s(x0))) → A__G(s(mark(x0)))
MARK(f(X)) → MARK(X)
MARK(sel(X1, X2)) → MARK(X1)
A__G(s(f(x0))) → A__G(a__f(mark(x0)))
MARK(g(sel(x0, x1))) → A__G(a__sel(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), cons(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), cons(mark(x0), x1))
A__SEL(s(X), cons(Y, Z)) → A__SEL(mark(X), mark(Z))
MARK(sel(s(x0), y1)) → A__SEL(s(mark(x0)), mark(y1))
MARK(sel(sel(y0, y1), f(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__f(mark(x0)))
MARK(sel(f(x0), y1)) → A__SEL(a__f(mark(x0)), mark(y1))
MARK(sel(sel(y0, y1), 0)) → A__SEL(a__sel(mark(y0), mark(y1)), 0)
MARK(g(s(x0))) → A__G(s(mark(x0)))
A__G(s(X)) → MARK(X)
MARK(f(X)) → A__F(mark(X))
MARK(sel(g(x0), y1)) → A__SEL(a__g(mark(x0)), mark(y1))
MARK(sel(X1, X2)) → MARK(X2)

The TRS R consists of the following rules:

a__f(X) → cons(mark(X), f(g(X)))
a__g(0) → s(0)
a__g(s(X)) → s(s(a__g(mark(X))))
a__sel(0, cons(X, Y)) → mark(X)
a__sel(s(X), cons(Y, Z)) → a__sel(mark(X), mark(Z))
mark(f(X)) → a__f(mark(X))
mark(g(X)) → a__g(mark(X))
mark(sel(X1, X2)) → a__sel(mark(X1), mark(X2))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0) → 0
mark(s(X)) → s(mark(X))
a__f(X) → f(X)
a__g(X) → g(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:

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(sel(y0, y1), g(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__g(mark(x0)))
MARK(cons(X1, X2)) → MARK(X1)
A__F(X) → MARK(X)
A__SEL(0, cons(X, Y)) → MARK(X)
MARK(g(X)) → MARK(X)
A__SEL(s(X), cons(Y, Z)) → MARK(Z)
MARK(g(f(x0))) → A__G(a__f(mark(x0)))
A__G(s(sel(x0, x1))) → A__G(a__sel(mark(x0), mark(x1)))
MARK(sel(0, y1)) → A__SEL(0, mark(y1))
MARK(g(g(x0))) → A__G(a__g(mark(x0)))
A__G(s(g(x0))) → A__G(a__g(mark(x0)))
A__SEL(s(X), cons(Y, Z)) → MARK(X)
MARK(f(X)) → MARK(X)
A__G(s(s(x0))) → A__G(s(mark(x0)))
A__G(s(f(x0))) → A__G(a__f(mark(x0)))
MARK(sel(X1, X2)) → MARK(X1)
MARK(g(sel(x0, x1))) → A__G(a__sel(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), cons(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), cons(mark(x0), x1))
A__SEL(s(X), cons(Y, Z)) → A__SEL(mark(X), mark(Z))
MARK(sel(s(x0), y1)) → A__SEL(s(mark(x0)), mark(y1))
MARK(sel(sel(y0, y1), f(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__f(mark(x0)))
MARK(sel(f(x0), y1)) → A__SEL(a__f(mark(x0)), mark(y1))
MARK(g(s(x0))) → A__G(s(mark(x0)))
A__G(s(X)) → MARK(X)
MARK(f(X)) → A__F(mark(X))
MARK(sel(g(x0), y1)) → A__SEL(a__g(mark(x0)), mark(y1))
MARK(sel(X1, X2)) → MARK(X2)

The TRS R consists of the following rules:

a__f(X) → cons(mark(X), f(g(X)))
a__g(0) → s(0)
a__g(s(X)) → s(s(a__g(mark(X))))
a__sel(0, cons(X, Y)) → mark(X)
a__sel(s(X), cons(Y, Z)) → a__sel(mark(X), mark(Z))
mark(f(X)) → a__f(mark(X))
mark(g(X)) → a__g(mark(X))
mark(sel(X1, X2)) → a__sel(mark(X1), mark(X2))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0) → 0
mark(s(X)) → s(mark(X))
a__f(X) → f(X)
a__g(X) → g(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), f(x0))) → A__SEL(s(mark(y0)), a__f(mark(x0)))
MARK(sel(s(y0), s(x0))) → A__SEL(s(mark(y0)), s(mark(x0)))
MARK(sel(s(y0), g(x0))) → A__SEL(s(mark(y0)), a__g(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

                                    ↳ QDP

                                      ↳ Narrowing

                                        ↳ QDP

                                          ↳ DependencyGraphProof

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

MARK(sel(sel(y0, y1), sel(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), a__sel(mark(x0), mark(x1)))
MARK(sel(s(y0), cons(x0, x1))) → A__SEL(s(mark(y0)), cons(mark(x0), x1))
MARK(s(X)) → MARK(X)
MARK(sel(s(y0), sel(x0, x1))) → A__SEL(s(mark(y0)), a__sel(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), g(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__g(mark(x0)))
MARK(cons(X1, X2)) → MARK(X1)
A__F(X) → MARK(X)
A__SEL(0, cons(X, Y)) → MARK(X)
MARK(g(X)) → MARK(X)
A__SEL(s(X), cons(Y, Z)) → MARK(Z)
MARK(g(f(x0))) → A__G(a__f(mark(x0)))
MARK(sel(0, y1)) → A__SEL(0, mark(y1))
A__G(s(sel(x0, x1))) → A__G(a__sel(mark(x0), mark(x1)))
MARK(sel(s(y0), s(x0))) → A__SEL(s(mark(y0)), s(mark(x0)))
MARK(g(g(x0))) → A__G(a__g(mark(x0)))
MARK(sel(s(y0), g(x0))) → A__SEL(s(mark(y0)), a__g(mark(x0)))
A__G(s(g(x0))) → A__G(a__g(mark(x0)))
A__SEL(s(X), cons(Y, Z)) → MARK(X)
A__G(s(s(x0))) → A__G(s(mark(x0)))
MARK(f(X)) → MARK(X)
MARK(sel(X1, X2)) → MARK(X1)
A__G(s(f(x0))) → A__G(a__f(mark(x0)))
MARK(g(sel(x0, x1))) → A__G(a__sel(mark(x0), mark(x1)))
A__SEL(s(X), cons(Y, Z)) → A__SEL(mark(X), mark(Z))
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)
MARK(sel(sel(y0, y1), f(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__f(mark(x0)))
MARK(sel(s(y0), f(x0))) → A__SEL(s(mark(y0)), a__f(mark(x0)))
MARK(sel(f(x0), y1)) → A__SEL(a__f(mark(x0)), mark(y1))
MARK(g(s(x0))) → A__G(s(mark(x0)))
A__G(s(X)) → MARK(X)
MARK(f(X)) → A__F(mark(X))
MARK(sel(g(x0), y1)) → A__SEL(a__g(mark(x0)), mark(y1))
MARK(sel(X1, X2)) → MARK(X2)

The TRS R consists of the following rules:

a__f(X) → cons(mark(X), f(g(X)))
a__g(0) → s(0)
a__g(s(X)) → s(s(a__g(mark(X))))
a__sel(0, cons(X, Y)) → mark(X)
a__sel(s(X), cons(Y, Z)) → a__sel(mark(X), mark(Z))
mark(f(X)) → a__f(mark(X))
mark(g(X)) → a__g(mark(X))
mark(sel(X1, X2)) → a__sel(mark(X1), mark(X2))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0) → 0
mark(s(X)) → s(mark(X))
a__f(X) → f(X)
a__g(X) → g(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:

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(sel(sel(y0, y1), g(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__g(mark(x0)))
MARK(cons(X1, X2)) → MARK(X1)
A__F(X) → MARK(X)
A__SEL(0, cons(X, Y)) → MARK(X)
MARK(g(X)) → MARK(X)
A__SEL(s(X), cons(Y, Z)) → MARK(Z)
MARK(g(f(x0))) → A__G(a__f(mark(x0)))
A__G(s(sel(x0, x1))) → A__G(a__sel(mark(x0), mark(x1)))
MARK(sel(0, y1)) → A__SEL(0, mark(y1))
MARK(g(g(x0))) → A__G(a__g(mark(x0)))
MARK(sel(s(y0), g(x0))) → A__SEL(s(mark(y0)), a__g(mark(x0)))
A__G(s(g(x0))) → A__G(a__g(mark(x0)))
A__SEL(s(X), cons(Y, Z)) → MARK(X)
MARK(f(X)) → MARK(X)
A__G(s(s(x0))) → A__G(s(mark(x0)))
A__G(s(f(x0))) → A__G(a__f(mark(x0)))
MARK(sel(X1, X2)) → MARK(X1)
MARK(g(sel(x0, x1))) → A__G(a__sel(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), cons(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), cons(mark(x0), x1))
A__SEL(s(X), cons(Y, Z)) → A__SEL(mark(X), mark(Z))
MARK(sel(sel(y0, y1), f(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__f(mark(x0)))
MARK(sel(s(y0), f(x0))) → A__SEL(s(mark(y0)), a__f(mark(x0)))
MARK(sel(f(x0), y1)) → A__SEL(a__f(mark(x0)), mark(y1))
MARK(g(s(x0))) → A__G(s(mark(x0)))
A__G(s(X)) → MARK(X)
MARK(f(X)) → A__F(mark(X))
MARK(sel(g(x0), y1)) → A__SEL(a__g(mark(x0)), mark(y1))
MARK(sel(X1, X2)) → MARK(X2)

The TRS R consists of the following rules:

a__f(X) → cons(mark(X), f(g(X)))
a__g(0) → s(0)
a__g(s(X)) → s(s(a__g(mark(X))))
a__sel(0, cons(X, Y)) → mark(X)
a__sel(s(X), cons(Y, Z)) → a__sel(mark(X), mark(Z))
mark(f(X)) → a__f(mark(X))
mark(g(X)) → a__g(mark(X))
mark(sel(X1, X2)) → a__sel(mark(X1), mark(X2))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0) → 0
mark(s(X)) → s(mark(X))
a__f(X) → f(X)
a__g(X) → g(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(g(x0), y1)) → A__SEL(a__g(mark(x0)), mark(y1)) at position [1] we obtained the following new rules:

MARK(sel(g(y0), cons(x0, x1))) → A__SEL(a__g(mark(y0)), cons(mark(x0), x1))
MARK(sel(g(y0), 0)) → A__SEL(a__g(mark(y0)), 0)
MARK(sel(g(y0), g(x0))) → A__SEL(a__g(mark(y0)), a__g(mark(x0)))
MARK(sel(g(y0), f(x0))) → A__SEL(a__g(mark(y0)), a__f(mark(x0)))
MARK(sel(g(y0), sel(x0, x1))) → A__SEL(a__g(mark(y0)), a__sel(mark(x0), mark(x1)))
MARK(sel(g(y0), s(x0))) → A__SEL(a__g(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

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

MARK(sel(g(y0), cons(x0, x1))) → A__SEL(a__g(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(s(X)) → MARK(X)
MARK(sel(s(y0), cons(x0, x1))) → A__SEL(s(mark(y0)), cons(mark(x0), x1))
MARK(sel(g(y0), g(x0))) → A__SEL(a__g(mark(y0)), a__g(mark(x0)))
MARK(sel(g(y0), sel(x0, x1))) → A__SEL(a__g(mark(y0)), a__sel(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), g(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__g(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__F(X) → MARK(X)
A__SEL(0, cons(X, Y)) → MARK(X)
MARK(g(X)) → MARK(X)
A__SEL(s(X), cons(Y, Z)) → MARK(Z)
MARK(g(f(x0))) → A__G(a__f(mark(x0)))
MARK(sel(g(y0), 0)) → A__SEL(a__g(mark(y0)), 0)
MARK(sel(0, y1)) → A__SEL(0, mark(y1))
A__G(s(sel(x0, x1))) → A__G(a__sel(mark(x0), mark(x1)))
MARK(g(g(x0))) → A__G(a__g(mark(x0)))
MARK(sel(g(y0), s(x0))) → A__SEL(a__g(mark(y0)), s(mark(x0)))
A__G(s(g(x0))) → A__G(a__g(mark(x0)))
MARK(sel(s(y0), g(x0))) → A__SEL(s(mark(y0)), a__g(mark(x0)))
A__SEL(s(X), cons(Y, Z)) → MARK(X)
A__G(s(s(x0))) → A__G(s(mark(x0)))
MARK(f(X)) → MARK(X)
MARK(sel(X1, X2)) → MARK(X1)
A__G(s(f(x0))) → A__G(a__f(mark(x0)))
MARK(g(sel(x0, x1))) → A__G(a__sel(mark(x0), mark(x1)))
A__SEL(s(X), cons(Y, Z)) → A__SEL(mark(X), mark(Z))
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), f(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__f(mark(x0)))
MARK(sel(s(y0), f(x0))) → A__SEL(s(mark(y0)), a__f(mark(x0)))
MARK(sel(f(x0), y1)) → A__SEL(a__f(mark(x0)), mark(y1))
MARK(g(s(x0))) → A__G(s(mark(x0)))
MARK(sel(g(y0), f(x0))) → A__SEL(a__g(mark(y0)), a__f(mark(x0)))
A__G(s(X)) → MARK(X)
MARK(f(X)) → A__F(mark(X))
MARK(sel(X1, X2)) → MARK(X2)

The TRS R consists of the following rules:

a__f(X) → cons(mark(X), f(g(X)))
a__g(0) → s(0)
a__g(s(X)) → s(s(a__g(mark(X))))
a__sel(0, cons(X, Y)) → mark(X)
a__sel(s(X), cons(Y, Z)) → a__sel(mark(X), mark(Z))
mark(f(X)) → a__f(mark(X))
mark(g(X)) → a__g(mark(X))
mark(sel(X1, X2)) → a__sel(mark(X1), mark(X2))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0) → 0
mark(s(X)) → s(mark(X))
a__f(X) → f(X)
a__g(X) → g(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(g(y0), cons(x0, x1))) → A__SEL(a__g(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(g(y0), g(x0))) → A__SEL(a__g(mark(y0)), a__g(mark(x0)))
MARK(sel(g(y0), sel(x0, x1))) → A__SEL(a__g(mark(y0)), a__sel(mark(x0), mark(x1)))
MARK(sel(s(y0), sel(x0, x1))) → A__SEL(s(mark(y0)), a__sel(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), g(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__g(mark(x0)))
MARK(cons(X1, X2)) → MARK(X1)
A__F(X) → MARK(X)
A__SEL(0, cons(X, Y)) → MARK(X)
MARK(g(X)) → MARK(X)
A__SEL(s(X), cons(Y, Z)) → MARK(Z)
MARK(g(f(x0))) → A__G(a__f(mark(x0)))
A__G(s(sel(x0, x1))) → A__G(a__sel(mark(x0), mark(x1)))
MARK(sel(0, y1)) → A__SEL(0, mark(y1))
MARK(g(g(x0))) → A__G(a__g(mark(x0)))
MARK(sel(s(y0), g(x0))) → A__SEL(s(mark(y0)), a__g(mark(x0)))
A__G(s(g(x0))) → A__G(a__g(mark(x0)))
A__SEL(s(X), cons(Y, Z)) → MARK(X)
MARK(f(X)) → MARK(X)
A__G(s(s(x0))) → A__G(s(mark(x0)))
A__G(s(f(x0))) → A__G(a__f(mark(x0)))
MARK(sel(X1, X2)) → MARK(X1)
MARK(g(sel(x0, x1))) → A__G(a__sel(mark(x0), mark(x1)))
A__SEL(s(X), cons(Y, Z)) → A__SEL(mark(X), mark(Z))
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), f(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__f(mark(x0)))
MARK(sel(s(y0), f(x0))) → A__SEL(s(mark(y0)), a__f(mark(x0)))
MARK(sel(f(x0), y1)) → A__SEL(a__f(mark(x0)), mark(y1))
MARK(g(s(x0))) → A__G(s(mark(x0)))
MARK(sel(g(y0), f(x0))) → A__SEL(a__g(mark(y0)), a__f(mark(x0)))
A__G(s(X)) → MARK(X)
MARK(f(X)) → A__F(mark(X))
MARK(sel(X1, X2)) → MARK(X2)

The TRS R consists of the following rules:

a__f(X) → cons(mark(X), f(g(X)))
a__g(0) → s(0)
a__g(s(X)) → s(s(a__g(mark(X))))
a__sel(0, cons(X, Y)) → mark(X)
a__sel(s(X), cons(Y, Z)) → a__sel(mark(X), mark(Z))
mark(f(X)) → a__f(mark(X))
mark(g(X)) → a__g(mark(X))
mark(sel(X1, X2)) → a__sel(mark(X1), mark(X2))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0) → 0
mark(s(X)) → s(mark(X))
a__f(X) → f(X)
a__g(X) → g(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(f(x0), y1)) → A__SEL(a__f(mark(x0)), mark(y1)) at position [1] we obtained the following new rules:

MARK(sel(f(y0), s(x0))) → A__SEL(a__f(mark(y0)), s(mark(x0)))
MARK(sel(f(y0), f(x0))) → A__SEL(a__f(mark(y0)), a__f(mark(x0)))
MARK(sel(f(y0), 0)) → A__SEL(a__f(mark(y0)), 0)
MARK(sel(f(y0), cons(x0, x1))) → A__SEL(a__f(mark(y0)), cons(mark(x0), x1))
MARK(sel(f(y0), g(x0))) → A__SEL(a__f(mark(y0)), a__g(mark(x0)))
MARK(sel(f(y0), sel(x0, x1))) → A__SEL(a__f(mark(y0)), a__sel(mark(x0), mark(x1)))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ Narrowing

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ QDP

              ↳ Narrowing

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ QDP

                      ↳ Narrowing

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ QDP

                              ↳ Narrowing

                                ↳ QDP

                                  ↳ DependencyGraphProof

                                    ↳ QDP

                                      ↳ Narrowing

                                        ↳ QDP

                                          ↳ DependencyGraphProof

                                            ↳ QDP

                                              ↳ Narrowing

                                                ↳ QDP

                                                  ↳ DependencyGraphProof

                                                    ↳ QDP

                                                      ↳ Narrowing

                                                        ↳ QDP

                                                          ↳ DependencyGraphProof

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

MARK(sel(sel(y0, y1), sel(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), a__sel(mark(x0), mark(x1)))
MARK(sel(g(y0), cons(x0, x1))) → A__SEL(a__g(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(g(y0), g(x0))) → A__SEL(a__g(mark(y0)), a__g(mark(x0)))
MARK(sel(f(y0), cons(x0, x1))) → A__SEL(a__f(mark(y0)), cons(mark(x0), x1))
MARK(sel(sel(y0, y1), g(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__g(mark(x0)))
MARK(sel(s(y0), sel(x0, x1))) → A__SEL(s(mark(y0)), a__sel(mark(x0), mark(x1)))
MARK(sel(g(y0), sel(x0, x1))) → A__SEL(a__g(mark(y0)), a__sel(mark(x0), mark(x1)))
MARK(cons(X1, X2)) → MARK(X1)
A__F(X) → MARK(X)
A__SEL(0, cons(X, Y)) → MARK(X)
MARK(g(X)) → MARK(X)
MARK(sel(f(y0), 0)) → A__SEL(a__f(mark(y0)), 0)
A__SEL(s(X), cons(Y, Z)) → MARK(Z)
MARK(g(f(x0))) → A__G(a__f(mark(x0)))
MARK(sel(0, y1)) → A__SEL(0, mark(y1))
A__G(s(sel(x0, x1))) → A__G(a__sel(mark(x0), mark(x1)))
MARK(g(g(x0))) → A__G(a__g(mark(x0)))
A__G(s(g(x0))) → A__G(a__g(mark(x0)))
MARK(sel(s(y0), g(x0))) → A__SEL(s(mark(y0)), a__g(mark(x0)))
A__SEL(s(X), cons(Y, Z)) → MARK(X)
A__G(s(s(x0))) → A__G(s(mark(x0)))
MARK(f(X)) → MARK(X)
MARK(sel(X1, X2)) → MARK(X1)
A__G(s(f(x0))) → A__G(a__f(mark(x0)))
MARK(g(sel(x0, x1))) → A__G(a__sel(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), cons(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), cons(mark(x0), x1))
A__SEL(s(X), cons(Y, Z)) → A__SEL(mark(X), mark(Z))
MARK(sel(f(y0), sel(x0, x1))) → A__SEL(a__f(mark(y0)), a__sel(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), f(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__f(mark(x0)))
MARK(sel(f(y0), f(x0))) → A__SEL(a__f(mark(y0)), a__f(mark(x0)))
MARK(sel(f(y0), s(x0))) → A__SEL(a__f(mark(y0)), s(mark(x0)))
MARK(sel(s(y0), f(x0))) → A__SEL(s(mark(y0)), a__f(mark(x0)))
MARK(g(s(x0))) → A__G(s(mark(x0)))
MARK(sel(f(y0), g(x0))) → A__SEL(a__f(mark(y0)), a__g(mark(x0)))
MARK(sel(g(y0), f(x0))) → A__SEL(a__g(mark(y0)), a__f(mark(x0)))
A__G(s(X)) → MARK(X)
MARK(f(X)) → A__F(mark(X))
MARK(sel(X1, X2)) → MARK(X2)

The TRS R consists of the following rules:

a__f(X) → cons(mark(X), f(g(X)))
a__g(0) → s(0)
a__g(s(X)) → s(s(a__g(mark(X))))
a__sel(0, cons(X, Y)) → mark(X)
a__sel(s(X), cons(Y, Z)) → a__sel(mark(X), mark(Z))
mark(f(X)) → a__f(mark(X))
mark(g(X)) → a__g(mark(X))
mark(sel(X1, X2)) → a__sel(mark(X1), mark(X2))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0) → 0
mark(s(X)) → s(mark(X))
a__f(X) → f(X)
a__g(X) → g(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(g(y0), cons(x0, x1))) → A__SEL(a__g(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(g(y0), g(x0))) → A__SEL(a__g(mark(y0)), a__g(mark(x0)))
MARK(sel(f(y0), cons(x0, x1))) → A__SEL(a__f(mark(y0)), cons(mark(x0), x1))
MARK(sel(g(y0), sel(x0, x1))) → A__SEL(a__g(mark(y0)), a__sel(mark(x0), mark(x1)))
MARK(sel(s(y0), sel(x0, x1))) → A__SEL(s(mark(y0)), a__sel(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), g(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__g(mark(x0)))
MARK(cons(X1, X2)) → MARK(X1)
A__F(X) → MARK(X)
A__SEL(0, cons(X, Y)) → MARK(X)
MARK(g(X)) → MARK(X)
A__SEL(s(X), cons(Y, Z)) → MARK(Z)
MARK(g(f(x0))) → A__G(a__f(mark(x0)))
A__G(s(sel(x0, x1))) → A__G(a__sel(mark(x0), mark(x1)))
MARK(sel(0, y1)) → A__SEL(0, mark(y1))
MARK(g(g(x0))) → A__G(a__g(mark(x0)))
MARK(sel(s(y0), g(x0))) → A__SEL(s(mark(y0)), a__g(mark(x0)))
A__G(s(g(x0))) → A__G(a__g(mark(x0)))
A__SEL(s(X), cons(Y, Z)) → MARK(X)
MARK(f(X)) → MARK(X)
A__G(s(s(x0))) → A__G(s(mark(x0)))
A__G(s(f(x0))) → A__G(a__f(mark(x0)))
MARK(sel(X1, X2)) → MARK(X1)
MARK(g(sel(x0, x1))) → A__G(a__sel(mark(x0), mark(x1)))
A__SEL(s(X), cons(Y, Z)) → A__SEL(mark(X), mark(Z))
MARK(sel(sel(y0, y1), cons(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), cons(mark(x0), x1))
MARK(sel(f(y0), sel(x0, x1))) → A__SEL(a__f(mark(y0)), a__sel(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), f(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__f(mark(x0)))
MARK(sel(f(y0), f(x0))) → A__SEL(a__f(mark(y0)), a__f(mark(x0)))
MARK(sel(s(y0), f(x0))) → A__SEL(s(mark(y0)), a__f(mark(x0)))
MARK(g(s(x0))) → A__G(s(mark(x0)))
MARK(sel(f(y0), g(x0))) → A__SEL(a__f(mark(y0)), a__g(mark(x0)))
MARK(sel(g(y0), f(x0))) → A__SEL(a__g(mark(y0)), a__f(mark(x0)))
A__G(s(X)) → MARK(X)
MARK(f(X)) → A__F(mark(X))
MARK(sel(X1, X2)) → MARK(X2)

The TRS R consists of the following rules:

a__f(X) → cons(mark(X), f(g(X)))
a__g(0) → s(0)
a__g(s(X)) → s(s(a__g(mark(X))))
a__sel(0, cons(X, Y)) → mark(X)
a__sel(s(X), cons(Y, Z)) → a__sel(mark(X), mark(Z))
mark(f(X)) → a__f(mark(X))
mark(g(X)) → a__g(mark(X))
mark(sel(X1, X2)) → a__sel(mark(X1), mark(X2))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0) → 0
mark(s(X)) → s(mark(X))
a__f(X) → f(X)
a__g(X) → g(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, f(x0))) → A__SEL(0, a__f(mark(x0)))
MARK(sel(0, sel(x0, x1))) → A__SEL(0, a__sel(mark(x0), mark(x1)))
MARK(sel(0, g(x0))) → A__SEL(0, a__g(mark(x0)))
MARK(sel(0, cons(x0, x1))) → A__SEL(0, cons(mark(x0), x1))
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

                                                    ↳ 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(g(y0), cons(x0, x1))) → A__SEL(a__g(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(g(y0), g(x0))) → A__SEL(a__g(mark(y0)), a__g(mark(x0)))
MARK(sel(f(y0), cons(x0, x1))) → A__SEL(a__f(mark(y0)), cons(mark(x0), x1))
MARK(sel(sel(y0, y1), g(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__g(mark(x0)))
MARK(sel(s(y0), sel(x0, x1))) → A__SEL(s(mark(y0)), a__sel(mark(x0), mark(x1)))
MARK(sel(g(y0), sel(x0, x1))) → A__SEL(a__g(mark(y0)), a__sel(mark(x0), mark(x1)))
MARK(cons(X1, X2)) → MARK(X1)
A__F(X) → MARK(X)
A__SEL(0, cons(X, Y)) → MARK(X)
MARK(g(X)) → MARK(X)
MARK(sel(0, f(x0))) → A__SEL(0, a__f(mark(x0)))
A__SEL(s(X), cons(Y, Z)) → MARK(Z)
MARK(g(f(x0))) → A__G(a__f(mark(x0)))
MARK(sel(0, g(x0))) → A__SEL(0, a__g(mark(x0)))
A__G(s(sel(x0, x1))) → A__G(a__sel(mark(x0), mark(x1)))
MARK(g(g(x0))) → A__G(a__g(mark(x0)))
A__G(s(g(x0))) → A__G(a__g(mark(x0)))
MARK(sel(s(y0), g(x0))) → A__SEL(s(mark(y0)), a__g(mark(x0)))
A__SEL(s(X), cons(Y, Z)) → MARK(X)
A__G(s(s(x0))) → A__G(s(mark(x0)))
MARK(f(X)) → MARK(X)
MARK(sel(X1, X2)) → MARK(X1)
A__G(s(f(x0))) → A__G(a__f(mark(x0)))
MARK(g(sel(x0, x1))) → A__G(a__sel(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), cons(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), cons(mark(x0), x1))
A__SEL(s(X), cons(Y, Z)) → A__SEL(mark(X), mark(Z))
MARK(sel(0, s(x0))) → A__SEL(0, s(mark(x0)))
MARK(sel(sel(y0, y1), f(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__f(mark(x0)))
MARK(sel(f(y0), sel(x0, x1))) → A__SEL(a__f(mark(y0)), a__sel(mark(x0), mark(x1)))
MARK(sel(s(y0), f(x0))) → A__SEL(s(mark(y0)), a__f(mark(x0)))
MARK(sel(f(y0), f(x0))) → A__SEL(a__f(mark(y0)), a__f(mark(x0)))
MARK(sel(0, sel(x0, x1))) → A__SEL(0, a__sel(mark(x0), mark(x1)))
MARK(g(s(x0))) → A__G(s(mark(x0)))
MARK(sel(0, cons(x0, x1))) → A__SEL(0, cons(mark(x0), x1))
MARK(sel(0, 0)) → A__SEL(0, 0)
MARK(sel(g(y0), f(x0))) → A__SEL(a__g(mark(y0)), a__f(mark(x0)))
MARK(sel(f(y0), g(x0))) → A__SEL(a__f(mark(y0)), a__g(mark(x0)))
A__G(s(X)) → MARK(X)
MARK(f(X)) → A__F(mark(X))
MARK(sel(X1, X2)) → MARK(X2)

The TRS R consists of the following rules:

a__f(X) → cons(mark(X), f(g(X)))
a__g(0) → s(0)
a__g(s(X)) → s(s(a__g(mark(X))))
a__sel(0, cons(X, Y)) → mark(X)
a__sel(s(X), cons(Y, Z)) → a__sel(mark(X), mark(Z))
mark(f(X)) → a__f(mark(X))
mark(g(X)) → a__g(mark(X))
mark(sel(X1, X2)) → a__sel(mark(X1), mark(X2))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0) → 0
mark(s(X)) → s(mark(X))
a__f(X) → f(X)
a__g(X) → g(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(g(y0), cons(x0, x1))) → A__SEL(a__g(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(g(y0), g(x0))) → A__SEL(a__g(mark(y0)), a__g(mark(x0)))
MARK(sel(f(y0), cons(x0, x1))) → A__SEL(a__f(mark(y0)), cons(mark(x0), x1))
MARK(sel(g(y0), sel(x0, x1))) → A__SEL(a__g(mark(y0)), a__sel(mark(x0), mark(x1)))
MARK(sel(s(y0), sel(x0, x1))) → A__SEL(s(mark(y0)), a__sel(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), g(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__g(mark(x0)))
MARK(cons(X1, X2)) → MARK(X1)
A__F(X) → MARK(X)
A__SEL(0, cons(X, Y)) → MARK(X)
MARK(g(X)) → MARK(X)
MARK(sel(0, f(x0))) → A__SEL(0, a__f(mark(x0)))
A__SEL(s(X), cons(Y, Z)) → MARK(Z)
MARK(g(f(x0))) → A__G(a__f(mark(x0)))
MARK(sel(0, g(x0))) → A__SEL(0, a__g(mark(x0)))
A__G(s(sel(x0, x1))) → A__G(a__sel(mark(x0), mark(x1)))
MARK(g(g(x0))) → A__G(a__g(mark(x0)))
MARK(sel(s(y0), g(x0))) → A__SEL(s(mark(y0)), a__g(mark(x0)))
A__G(s(g(x0))) → A__G(a__g(mark(x0)))
A__SEL(s(X), cons(Y, Z)) → MARK(X)
MARK(f(X)) → MARK(X)
A__G(s(s(x0))) → A__G(s(mark(x0)))
A__G(s(f(x0))) → A__G(a__f(mark(x0)))
MARK(sel(X1, X2)) → MARK(X1)
MARK(g(sel(x0, x1))) → A__G(a__sel(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), cons(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), cons(mark(x0), x1))
A__SEL(s(X), cons(Y, Z)) → A__SEL(mark(X), mark(Z))
MARK(sel(f(y0), sel(x0, x1))) → A__SEL(a__f(mark(y0)), a__sel(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), f(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__f(mark(x0)))
MARK(sel(f(y0), f(x0))) → A__SEL(a__f(mark(y0)), a__f(mark(x0)))
MARK(sel(s(y0), f(x0))) → A__SEL(s(mark(y0)), a__f(mark(x0)))
MARK(sel(0, sel(x0, x1))) → A__SEL(0, a__sel(mark(x0), mark(x1)))
MARK(g(s(x0))) → A__G(s(mark(x0)))
MARK(sel(0, cons(x0, x1))) → A__SEL(0, cons(mark(x0), x1))
MARK(sel(f(y0), g(x0))) → A__SEL(a__f(mark(y0)), a__g(mark(x0)))
MARK(sel(g(y0), f(x0))) → A__SEL(a__g(mark(y0)), a__f(mark(x0)))
A__G(s(X)) → MARK(X)
MARK(f(X)) → A__F(mark(X))
MARK(sel(X1, X2)) → MARK(X2)

The TRS R consists of the following rules:

a__f(X) → cons(mark(X), f(g(X)))
a__g(0) → s(0)
a__g(s(X)) → s(s(a__g(mark(X))))
a__sel(0, cons(X, Y)) → mark(X)
a__sel(s(X), cons(Y, Z)) → a__sel(mark(X), mark(Z))
mark(f(X)) → a__f(mark(X))
mark(g(X)) → a__g(mark(X))
mark(sel(X1, X2)) → a__sel(mark(X1), mark(X2))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0) → 0
mark(s(X)) → s(mark(X))
a__f(X) → f(X)
a__g(X) → g(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(f(x0)), cons(y1, y2)) → A__SEL(a__f(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(g(x0)), cons(y1, y2)) → A__SEL(a__g(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

            ↳ QDP

              ↳ Narrowing

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ QDP

                      ↳ Narrowing

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ 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(g(y0), cons(x0, x1))) → A__SEL(a__g(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(g(y0), g(x0))) → A__SEL(a__g(mark(y0)), a__g(mark(x0)))
MARK(sel(f(y0), cons(x0, x1))) → A__SEL(a__f(mark(y0)), cons(mark(x0), x1))
MARK(sel(sel(y0, y1), g(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__g(mark(x0)))
MARK(sel(s(y0), sel(x0, x1))) → A__SEL(s(mark(y0)), a__sel(mark(x0), mark(x1)))
MARK(sel(g(y0), sel(x0, x1))) → A__SEL(a__g(mark(y0)), a__sel(mark(x0), mark(x1)))
A__SEL(s(f(x0)), cons(y1, y2)) → A__SEL(a__f(mark(x0)), mark(y2))
MARK(cons(X1, X2)) → MARK(X1)
A__F(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))
MARK(g(X)) → 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, f(x0))) → A__SEL(0, a__f(mark(x0)))
MARK(g(f(x0))) → A__G(a__f(mark(x0)))
MARK(sel(0, g(x0))) → A__SEL(0, a__g(mark(x0)))
A__G(s(sel(x0, x1))) → A__G(a__sel(mark(x0), mark(x1)))
MARK(g(g(x0))) → A__G(a__g(mark(x0)))
A__G(s(g(x0))) → A__G(a__g(mark(x0)))
MARK(sel(s(y0), g(x0))) → A__SEL(s(mark(y0)), a__g(mark(x0)))
A__SEL(s(X), cons(Y, Z)) → MARK(X)
A__G(s(s(x0))) → A__G(s(mark(x0)))
MARK(f(X)) → MARK(X)
MARK(sel(X1, X2)) → MARK(X1)
A__G(s(f(x0))) → A__G(a__f(mark(x0)))
MARK(g(sel(x0, x1))) → A__G(a__sel(mark(x0), mark(x1)))
A__SEL(s(s(x0)), cons(y1, y2)) → A__SEL(s(mark(x0)), mark(y2))
A__SEL(s(g(x0)), cons(y1, y2)) → A__SEL(a__g(mark(x0)), mark(y2))
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), f(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__f(mark(x0)))
MARK(sel(f(y0), sel(x0, x1))) → A__SEL(a__f(mark(y0)), a__sel(mark(x0), mark(x1)))
MARK(sel(s(y0), f(x0))) → A__SEL(s(mark(y0)), a__f(mark(x0)))
MARK(sel(f(y0), f(x0))) → A__SEL(a__f(mark(y0)), a__f(mark(x0)))
A__SEL(s(0), cons(y1, y2)) → A__SEL(0, mark(y2))
MARK(g(s(x0))) → A__G(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(g(y0), f(x0))) → A__SEL(a__g(mark(y0)), a__f(mark(x0)))
MARK(sel(f(y0), g(x0))) → A__SEL(a__f(mark(y0)), a__g(mark(x0)))
A__G(s(X)) → MARK(X)
MARK(f(X)) → A__F(mark(X))
MARK(sel(X1, X2)) → MARK(X2)

The TRS R consists of the following rules:

a__f(X) → cons(mark(X), f(g(X)))
a__g(0) → s(0)
a__g(s(X)) → s(s(a__g(mark(X))))
a__sel(0, cons(X, Y)) → mark(X)
a__sel(s(X), cons(Y, Z)) → a__sel(mark(X), mark(Z))
mark(f(X)) → a__f(mark(X))
mark(g(X)) → a__g(mark(X))
mark(sel(X1, X2)) → a__sel(mark(X1), mark(X2))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0) → 0
mark(s(X)) → s(mark(X))
a__f(X) → f(X)
a__g(X) → g(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

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ QDP

                              ↳ Narrowing

                                ↳ 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(g(y0), cons(x0, x1))) → A__SEL(a__g(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(g(y0), g(x0))) → A__SEL(a__g(mark(y0)), a__g(mark(x0)))
MARK(sel(f(y0), cons(x0, x1))) → A__SEL(a__f(mark(y0)), cons(mark(x0), x1))
MARK(sel(g(y0), sel(x0, x1))) → A__SEL(a__g(mark(y0)), a__sel(mark(x0), mark(x1)))
MARK(sel(s(y0), sel(x0, x1))) → A__SEL(s(mark(y0)), a__sel(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), g(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__g(mark(x0)))
MARK(cons(X1, X2)) → MARK(X1)
A__SEL(s(f(x0)), cons(y1, y2)) → A__SEL(a__f(mark(x0)), mark(y2))
A__F(X) → MARK(X)
A__SEL(0, cons(X, Y)) → MARK(X)
MARK(g(X)) → MARK(X)
MARK(sel(0, f(x0))) → A__SEL(0, a__f(mark(x0)))
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(g(f(x0))) → A__G(a__f(mark(x0)))
MARK(sel(0, g(x0))) → A__SEL(0, a__g(mark(x0)))
A__G(s(sel(x0, x1))) → A__G(a__sel(mark(x0), mark(x1)))
MARK(g(g(x0))) → A__G(a__g(mark(x0)))
MARK(sel(s(y0), g(x0))) → A__SEL(s(mark(y0)), a__g(mark(x0)))
A__G(s(g(x0))) → A__G(a__g(mark(x0)))
A__SEL(s(X), cons(Y, Z)) → MARK(X)
MARK(f(X)) → MARK(X)
A__G(s(s(x0))) → A__G(s(mark(x0)))
A__G(s(f(x0))) → A__G(a__f(mark(x0)))
MARK(sel(X1, X2)) → MARK(X1)
MARK(g(sel(x0, x1))) → A__G(a__sel(mark(x0), mark(x1)))
A__SEL(s(s(x0)), cons(y1, y2)) → A__SEL(s(mark(x0)), mark(y2))
A__SEL(s(g(x0)), cons(y1, y2)) → A__SEL(a__g(mark(x0)), mark(y2))
MARK(sel(sel(y0, y1), cons(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), cons(mark(x0), x1))
MARK(sel(f(y0), sel(x0, x1))) → A__SEL(a__f(mark(y0)), a__sel(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), f(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__f(mark(x0)))
MARK(sel(f(y0), f(x0))) → A__SEL(a__f(mark(y0)), a__f(mark(x0)))
MARK(sel(s(y0), f(x0))) → A__SEL(s(mark(y0)), a__f(mark(x0)))
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(g(s(x0))) → A__G(s(mark(x0)))
MARK(sel(0, cons(x0, x1))) → A__SEL(0, cons(mark(x0), x1))
MARK(sel(f(y0), g(x0))) → A__SEL(a__f(mark(y0)), a__g(mark(x0)))
MARK(sel(g(y0), f(x0))) → A__SEL(a__g(mark(y0)), a__f(mark(x0)))
A__G(s(X)) → MARK(X)
MARK(f(X)) → A__F(mark(X))
MARK(sel(X1, X2)) → MARK(X2)

The TRS R consists of the following rules:

a__f(X) → cons(mark(X), f(g(X)))
a__g(0) → s(0)
a__g(s(X)) → s(s(a__g(mark(X))))
a__sel(0, cons(X, Y)) → mark(X)
a__sel(s(X), cons(Y, Z)) → a__sel(mark(X), mark(Z))
mark(f(X)) → a__f(mark(X))
mark(g(X)) → a__g(mark(X))
mark(sel(X1, X2)) → a__sel(mark(X1), mark(X2))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0) → 0
mark(s(X)) → s(mark(X))
a__f(X) → f(X)
a__g(X) → g(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(f(x0)), cons(y1, y2)) → A__SEL(a__f(mark(x0)), mark(y2)) at position [1] we obtained the following new rules:

A__SEL(s(f(y0)), cons(y1, g(x0))) → A__SEL(a__f(mark(y0)), a__g(mark(x0)))
A__SEL(s(f(y0)), cons(y1, cons(x0, x1))) → A__SEL(a__f(mark(y0)), cons(mark(x0), x1))
A__SEL(s(f(y0)), cons(y1, sel(x0, x1))) → A__SEL(a__f(mark(y0)), a__sel(mark(x0), mark(x1)))
A__SEL(s(f(y0)), cons(y1, 0)) → A__SEL(a__f(mark(y0)), 0)
A__SEL(s(f(y0)), cons(y1, f(x0))) → A__SEL(a__f(mark(y0)), a__f(mark(x0)))
A__SEL(s(f(y0)), cons(y1, s(x0))) → A__SEL(a__f(mark(y0)), s(mark(x0)))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ Narrowing

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ QDP

              ↳ Narrowing

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ QDP

                      ↳ Narrowing

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ QDP

                              ↳ Narrowing

                                ↳ QDP

                                  ↳ DependencyGraphProof

                                    ↳ QDP

                                      ↳ Narrowing

                                        ↳ QDP

                                          ↳ DependencyGraphProof

                                            ↳ QDP

                                              ↳ Narrowing

                                                ↳ QDP

                                                  ↳ DependencyGraphProof

                                                    ↳ QDP

                                                      ↳ Narrowing

                                                        ↳ QDP

                                                          ↳ DependencyGraphProof

                                                            ↳ QDP

                                                              ↳ Narrowing

                                                                ↳ QDP

                                                                  ↳ DependencyGraphProof

                                                                    ↳ QDP

                                                                      ↳ Narrowing

                                                                        ↳ QDP

                                                                          ↳ DependencyGraphProof

                                                                            ↳ QDP

                                                                              ↳ Narrowing

                                                                                ↳ QDP

                                                                                  ↳ DependencyGraphProof

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

MARK(sel(sel(y0, y1), sel(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), a__sel(mark(x0), mark(x1)))
MARK(sel(s(y0), cons(x0, x1))) → A__SEL(s(mark(y0)), cons(mark(x0), x1))
MARK(sel(g(y0), g(x0))) → A__SEL(a__g(mark(y0)), a__g(mark(x0)))
MARK(sel(f(y0), cons(x0, x1))) → A__SEL(a__f(mark(y0)), cons(mark(x0), x1))
MARK(cons(X1, X2)) → MARK(X1)
A__F(X) → MARK(X)
MARK(g(X)) → 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(g(f(x0))) → A__G(a__f(mark(x0)))
MARK(sel(0, g(x0))) → A__SEL(0, a__g(mark(x0)))
A__SEL(s(f(y0)), cons(y1, 0)) → A__SEL(a__f(mark(y0)), 0)
MARK(g(g(x0))) → A__G(a__g(mark(x0)))
MARK(sel(s(y0), g(x0))) → A__SEL(s(mark(y0)), a__g(mark(x0)))
MARK(f(X)) → MARK(X)
A__G(s(f(x0))) → A__G(a__f(mark(x0)))
MARK(sel(X1, X2)) → MARK(X1)
MARK(g(sel(x0, x1))) → A__G(a__sel(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), cons(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), cons(mark(x0), x1))
MARK(sel(s(y0), f(x0))) → A__SEL(s(mark(y0)), a__f(mark(x0)))
A__SEL(s(0), cons(y1, y2)) → A__SEL(0, mark(y2))
MARK(sel(0, cons(x0, x1))) → A__SEL(0, cons(mark(x0), x1))
A__SEL(s(f(y0)), cons(y1, cons(x0, x1))) → A__SEL(a__f(mark(y0)), cons(mark(x0), x1))
MARK(sel(g(y0), cons(x0, x1))) → A__SEL(a__g(mark(y0)), cons(mark(x0), x1))
MARK(s(X)) → MARK(X)
MARK(sel(g(y0), sel(x0, x1))) → A__SEL(a__g(mark(y0)), a__sel(mark(x0), mark(x1)))
MARK(sel(s(y0), sel(x0, x1))) → A__SEL(s(mark(y0)), a__sel(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), g(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__g(mark(x0)))
A__SEL(0, cons(X, Y)) → MARK(X)
MARK(sel(0, f(x0))) → A__SEL(0, a__f(mark(x0)))
A__G(s(sel(x0, x1))) → A__G(a__sel(mark(x0), mark(x1)))
A__SEL(s(f(y0)), cons(y1, s(x0))) → A__SEL(a__f(mark(y0)), s(mark(x0)))
A__G(s(g(x0))) → A__G(a__g(mark(x0)))
A__SEL(s(X), cons(Y, Z)) → MARK(X)
A__G(s(s(x0))) → A__G(s(mark(x0)))
A__SEL(s(s(x0)), cons(y1, y2)) → A__SEL(s(mark(x0)), mark(y2))
A__SEL(s(f(y0)), cons(y1, g(x0))) → A__SEL(a__f(mark(y0)), a__g(mark(x0)))
A__SEL(s(g(x0)), cons(y1, y2)) → A__SEL(a__g(mark(x0)), mark(y2))
A__SEL(s(f(y0)), cons(y1, f(x0))) → A__SEL(a__f(mark(y0)), a__f(mark(x0)))
MARK(sel(f(y0), sel(x0, x1))) → A__SEL(a__f(mark(y0)), a__sel(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), f(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__f(mark(x0)))
MARK(sel(f(y0), f(x0))) → A__SEL(a__f(mark(y0)), a__f(mark(x0)))
MARK(sel(0, sel(x0, x1))) → A__SEL(0, a__sel(mark(x0), mark(x1)))
MARK(g(s(x0))) → A__G(s(mark(x0)))
MARK(sel(f(y0), g(x0))) → A__SEL(a__f(mark(y0)), a__g(mark(x0)))
MARK(sel(g(y0), f(x0))) → A__SEL(a__g(mark(y0)), a__f(mark(x0)))
A__SEL(s(f(y0)), cons(y1, sel(x0, x1))) → A__SEL(a__f(mark(y0)), a__sel(mark(x0), mark(x1)))
A__G(s(X)) → MARK(X)
MARK(f(X)) → A__F(mark(X))
MARK(sel(X1, X2)) → MARK(X2)

The TRS R consists of the following rules:

a__f(X) → cons(mark(X), f(g(X)))
a__g(0) → s(0)
a__g(s(X)) → s(s(a__g(mark(X))))
a__sel(0, cons(X, Y)) → mark(X)
a__sel(s(X), cons(Y, Z)) → a__sel(mark(X), mark(Z))
mark(f(X)) → a__f(mark(X))
mark(g(X)) → a__g(mark(X))
mark(sel(X1, X2)) → a__sel(mark(X1), mark(X2))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0) → 0
mark(s(X)) → s(mark(X))
a__f(X) → f(X)
a__g(X) → g(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

                                                                                      ↳ Narrowing

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

MARK(sel(g(y0), cons(x0, x1))) → A__SEL(a__g(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(g(y0), g(x0))) → A__SEL(a__g(mark(y0)), a__g(mark(x0)))
MARK(sel(f(y0), cons(x0, x1))) → A__SEL(a__f(mark(y0)), cons(mark(x0), x1))
MARK(sel(g(y0), sel(x0, x1))) → A__SEL(a__g(mark(y0)), a__sel(mark(x0), mark(x1)))
MARK(sel(s(y0), sel(x0, x1))) → A__SEL(s(mark(y0)), a__sel(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), g(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__g(mark(x0)))
MARK(cons(X1, X2)) → MARK(X1)
A__F(X) → MARK(X)
A__SEL(0, cons(X, Y)) → MARK(X)
MARK(g(X)) → MARK(X)
MARK(sel(0, f(x0))) → A__SEL(0, a__f(mark(x0)))
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(g(f(x0))) → A__G(a__f(mark(x0)))
MARK(sel(0, g(x0))) → A__SEL(0, a__g(mark(x0)))
A__G(s(sel(x0, x1))) → A__G(a__sel(mark(x0), mark(x1)))
MARK(g(g(x0))) → A__G(a__g(mark(x0)))
MARK(sel(s(y0), g(x0))) → A__SEL(s(mark(y0)), a__g(mark(x0)))
A__G(s(g(x0))) → A__G(a__g(mark(x0)))
A__SEL(s(X), cons(Y, Z)) → MARK(X)
MARK(f(X)) → MARK(X)
A__G(s(s(x0))) → A__G(s(mark(x0)))
A__G(s(f(x0))) → A__G(a__f(mark(x0)))
MARK(sel(X1, X2)) → MARK(X1)
MARK(g(sel(x0, x1))) → A__G(a__sel(mark(x0), mark(x1)))
A__SEL(s(f(y0)), cons(y1, g(x0))) → A__SEL(a__f(mark(y0)), a__g(mark(x0)))
A__SEL(s(s(x0)), cons(y1, y2)) → A__SEL(s(mark(x0)), mark(y2))
A__SEL(s(g(x0)), cons(y1, y2)) → A__SEL(a__g(mark(x0)), mark(y2))
MARK(sel(sel(y0, y1), cons(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), cons(mark(x0), x1))
A__SEL(s(f(y0)), cons(y1, f(x0))) → A__SEL(a__f(mark(y0)), a__f(mark(x0)))
MARK(sel(f(y0), sel(x0, x1))) → A__SEL(a__f(mark(y0)), a__sel(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), f(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__f(mark(x0)))
MARK(sel(f(y0), f(x0))) → A__SEL(a__f(mark(y0)), a__f(mark(x0)))
MARK(sel(s(y0), f(x0))) → A__SEL(s(mark(y0)), a__f(mark(x0)))
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(g(s(x0))) → A__G(s(mark(x0)))
MARK(sel(0, cons(x0, x1))) → A__SEL(0, cons(mark(x0), x1))
MARK(sel(f(y0), g(x0))) → A__SEL(a__f(mark(y0)), a__g(mark(x0)))
MARK(sel(g(y0), f(x0))) → A__SEL(a__g(mark(y0)), a__f(mark(x0)))
A__SEL(s(f(y0)), cons(y1, sel(x0, x1))) → A__SEL(a__f(mark(y0)), a__sel(mark(x0), mark(x1)))
A__SEL(s(f(y0)), cons(y1, cons(x0, x1))) → A__SEL(a__f(mark(y0)), cons(mark(x0), x1))
A__G(s(X)) → MARK(X)
MARK(f(X)) → A__F(mark(X))
MARK(sel(X1, X2)) → MARK(X2)

The TRS R consists of the following rules:

a__f(X) → cons(mark(X), f(g(X)))
a__g(0) → s(0)
a__g(s(X)) → s(s(a__g(mark(X))))
a__sel(0, cons(X, Y)) → mark(X)
a__sel(s(X), cons(Y, Z)) → a__sel(mark(X), mark(Z))
mark(f(X)) → a__f(mark(X))
mark(g(X)) → a__g(mark(X))
mark(sel(X1, X2)) → a__sel(mark(X1), mark(X2))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0) → 0
mark(s(X)) → s(mark(X))
a__f(X) → f(X)
a__g(X) → g(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, 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, g(x0))) → A__SEL(0, a__g(mark(x0)))
A__SEL(s(0), cons(y0, f(x0))) → A__SEL(0, a__f(mark(x0)))
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

                                                                    ↳ QDP

                                                                      ↳ Narrowing

                                                                        ↳ QDP

                                                                          ↳ DependencyGraphProof

                                                                            ↳ QDP

                                                                              ↳ Narrowing

                                                                                ↳ QDP

                                                                                  ↳ DependencyGraphProof

                                                                                    ↳ QDP

                                                                                      ↳ Narrowing

                                                                                        ↳ QDP

                                                                                          ↳ DependencyGraphProof

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

MARK(sel(sel(y0, y1), sel(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), a__sel(mark(x0), mark(x1)))
MARK(sel(s(y0), cons(x0, x1))) → A__SEL(s(mark(y0)), cons(mark(x0), x1))
MARK(sel(g(y0), g(x0))) → A__SEL(a__g(mark(y0)), a__g(mark(x0)))
A__SEL(s(0), cons(y0, s(x0))) → A__SEL(0, s(mark(x0)))
MARK(sel(f(y0), cons(x0, x1))) → A__SEL(a__f(mark(y0)), cons(mark(x0), x1))
MARK(cons(X1, X2)) → MARK(X1)
A__F(X) → MARK(X)
MARK(g(X)) → 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(g(f(x0))) → A__G(a__f(mark(x0)))
A__SEL(s(0), cons(y0, f(x0))) → A__SEL(0, a__f(mark(x0)))
MARK(sel(0, g(x0))) → A__SEL(0, a__g(mark(x0)))
MARK(g(g(x0))) → A__G(a__g(mark(x0)))
MARK(sel(s(y0), g(x0))) → A__SEL(s(mark(y0)), a__g(mark(x0)))
A__SEL(s(0), cons(y0, g(x0))) → A__SEL(0, a__g(mark(x0)))
MARK(f(X)) → MARK(X)
MARK(sel(X1, X2)) → MARK(X1)
A__G(s(f(x0))) → A__G(a__f(mark(x0)))
MARK(g(sel(x0, x1))) → A__G(a__sel(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), cons(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), cons(mark(x0), x1))
MARK(sel(s(y0), f(x0))) → A__SEL(s(mark(y0)), a__f(mark(x0)))
MARK(sel(0, cons(x0, x1))) → A__SEL(0, cons(mark(x0), x1))
A__SEL(s(f(y0)), cons(y1, cons(x0, x1))) → A__SEL(a__f(mark(y0)), cons(mark(x0), x1))
A__SEL(s(0), cons(y0, sel(x0, x1))) → A__SEL(0, a__sel(mark(x0), mark(x1)))
MARK(sel(g(y0), cons(x0, x1))) → A__SEL(a__g(mark(y0)), cons(mark(x0), x1))
MARK(s(X)) → MARK(X)
MARK(sel(sel(y0, y1), g(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__g(mark(x0)))
MARK(sel(s(y0), sel(x0, x1))) → A__SEL(s(mark(y0)), a__sel(mark(x0), mark(x1)))
MARK(sel(g(y0), sel(x0, x1))) → A__SEL(a__g(mark(y0)), a__sel(mark(x0), mark(x1)))
A__SEL(0, cons(X, Y)) → MARK(X)
MARK(sel(0, f(x0))) → A__SEL(0, a__f(mark(x0)))
A__G(s(sel(x0, x1))) → A__G(a__sel(mark(x0), mark(x1)))
A__SEL(s(0), cons(y0, cons(x0, x1))) → A__SEL(0, cons(mark(x0), x1))
A__G(s(g(x0))) → A__G(a__g(mark(x0)))
A__SEL(s(X), cons(Y, Z)) → MARK(X)
A__G(s(s(x0))) → A__G(s(mark(x0)))
A__SEL(s(s(x0)), cons(y1, y2)) → A__SEL(s(mark(x0)), mark(y2))
A__SEL(s(f(y0)), cons(y1, g(x0))) → A__SEL(a__f(mark(y0)), a__g(mark(x0)))
A__SEL(s(g(x0)), cons(y1, y2)) → A__SEL(a__g(mark(x0)), mark(y2))
A__SEL(s(f(y0)), cons(y1, f(x0))) → A__SEL(a__f(mark(y0)), a__f(mark(x0)))
A__SEL(s(0), cons(y0, 0)) → A__SEL(0, 0)
MARK(sel(sel(y0, y1), f(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__f(mark(x0)))
MARK(sel(f(y0), sel(x0, x1))) → A__SEL(a__f(mark(y0)), a__sel(mark(x0), mark(x1)))
MARK(sel(f(y0), f(x0))) → A__SEL(a__f(mark(y0)), a__f(mark(x0)))
MARK(g(s(x0))) → A__G(s(mark(x0)))
MARK(sel(0, sel(x0, x1))) → A__SEL(0, a__sel(mark(x0), mark(x1)))
A__SEL(s(f(y0)), cons(y1, sel(x0, x1))) → A__SEL(a__f(mark(y0)), a__sel(mark(x0), mark(x1)))
MARK(sel(g(y0), f(x0))) → A__SEL(a__g(mark(y0)), a__f(mark(x0)))
MARK(sel(f(y0), g(x0))) → A__SEL(a__f(mark(y0)), a__g(mark(x0)))
A__G(s(X)) → MARK(X)
MARK(f(X)) → A__F(mark(X))
MARK(sel(X1, X2)) → MARK(X2)

The TRS R consists of the following rules:

a__f(X) → cons(mark(X), f(g(X)))
a__g(0) → s(0)
a__g(s(X)) → s(s(a__g(mark(X))))
a__sel(0, cons(X, Y)) → mark(X)
a__sel(s(X), cons(Y, Z)) → a__sel(mark(X), mark(Z))
mark(f(X)) → a__f(mark(X))
mark(g(X)) → a__g(mark(X))
mark(sel(X1, X2)) → a__sel(mark(X1), mark(X2))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0) → 0
mark(s(X)) → s(mark(X))
a__f(X) → f(X)
a__g(X) → g(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

                                                                                      ↳ Narrowing

                                                                                        ↳ QDP

                                                                                          ↳ DependencyGraphProof

                                                                                            ↳ QDP

                                                                                              ↳ Narrowing

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

MARK(sel(sel(y0, y1), sel(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), a__sel(mark(x0), mark(x1)))
MARK(sel(s(y0), cons(x0, x1))) → A__SEL(s(mark(y0)), cons(mark(x0), x1))
MARK(sel(g(y0), g(x0))) → A__SEL(a__g(mark(y0)), a__g(mark(x0)))
MARK(sel(f(y0), cons(x0, x1))) → A__SEL(a__f(mark(y0)), cons(mark(x0), x1))
MARK(cons(X1, X2)) → MARK(X1)
A__F(X) → MARK(X)
MARK(g(X)) → 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(g(f(x0))) → A__G(a__f(mark(x0)))
MARK(sel(0, g(x0))) → A__SEL(0, a__g(mark(x0)))
A__SEL(s(0), cons(y0, f(x0))) → A__SEL(0, a__f(mark(x0)))
MARK(g(g(x0))) → A__G(a__g(mark(x0)))
MARK(sel(s(y0), g(x0))) → A__SEL(s(mark(y0)), a__g(mark(x0)))
A__SEL(s(0), cons(y0, g(x0))) → A__SEL(0, a__g(mark(x0)))
MARK(f(X)) → MARK(X)
MARK(sel(X1, X2)) → MARK(X1)
A__G(s(f(x0))) → A__G(a__f(mark(x0)))
MARK(g(sel(x0, x1))) → A__G(a__sel(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), cons(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), cons(mark(x0), x1))
MARK(sel(s(y0), f(x0))) → A__SEL(s(mark(y0)), a__f(mark(x0)))
MARK(sel(0, cons(x0, x1))) → A__SEL(0, cons(mark(x0), x1))
A__SEL(s(f(y0)), cons(y1, cons(x0, x1))) → A__SEL(a__f(mark(y0)), cons(mark(x0), x1))
A__SEL(s(0), cons(y0, sel(x0, x1))) → A__SEL(0, a__sel(mark(x0), mark(x1)))
MARK(sel(g(y0), cons(x0, x1))) → A__SEL(a__g(mark(y0)), cons(mark(x0), x1))
MARK(s(X)) → MARK(X)
MARK(sel(sel(y0, y1), g(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__g(mark(x0)))
MARK(sel(s(y0), sel(x0, x1))) → A__SEL(s(mark(y0)), a__sel(mark(x0), mark(x1)))
MARK(sel(g(y0), sel(x0, x1))) → A__SEL(a__g(mark(y0)), a__sel(mark(x0), mark(x1)))
A__SEL(0, cons(X, Y)) → MARK(X)
MARK(sel(0, f(x0))) → A__SEL(0, a__f(mark(x0)))
A__G(s(sel(x0, x1))) → A__G(a__sel(mark(x0), mark(x1)))
A__SEL(s(0), cons(y0, cons(x0, x1))) → A__SEL(0, cons(mark(x0), x1))
A__G(s(g(x0))) → A__G(a__g(mark(x0)))
A__SEL(s(X), cons(Y, Z)) → MARK(X)
A__G(s(s(x0))) → A__G(s(mark(x0)))
A__SEL(s(s(x0)), cons(y1, y2)) → A__SEL(s(mark(x0)), mark(y2))
A__SEL(s(f(y0)), cons(y1, g(x0))) → A__SEL(a__f(mark(y0)), a__g(mark(x0)))
A__SEL(s(g(x0)), cons(y1, y2)) → A__SEL(a__g(mark(x0)), mark(y2))
A__SEL(s(f(y0)), cons(y1, f(x0))) → A__SEL(a__f(mark(y0)), a__f(mark(x0)))
MARK(sel(sel(y0, y1), f(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__f(mark(x0)))
MARK(sel(f(y0), sel(x0, x1))) → A__SEL(a__f(mark(y0)), a__sel(mark(x0), mark(x1)))
MARK(sel(f(y0), f(x0))) → A__SEL(a__f(mark(y0)), a__f(mark(x0)))
MARK(g(s(x0))) → A__G(s(mark(x0)))
MARK(sel(0, sel(x0, x1))) → A__SEL(0, a__sel(mark(x0), mark(x1)))
A__SEL(s(f(y0)), cons(y1, sel(x0, x1))) → A__SEL(a__f(mark(y0)), a__sel(mark(x0), mark(x1)))
MARK(sel(g(y0), f(x0))) → A__SEL(a__g(mark(y0)), a__f(mark(x0)))
MARK(sel(f(y0), g(x0))) → A__SEL(a__f(mark(y0)), a__g(mark(x0)))
A__G(s(X)) → MARK(X)
MARK(f(X)) → A__F(mark(X))
MARK(sel(X1, X2)) → MARK(X2)

The TRS R consists of the following rules:

a__f(X) → cons(mark(X), f(g(X)))
a__g(0) → s(0)
a__g(s(X)) → s(s(a__g(mark(X))))
a__sel(0, cons(X, Y)) → mark(X)
a__sel(s(X), cons(Y, Z)) → a__sel(mark(X), mark(Z))
mark(f(X)) → a__f(mark(X))
mark(g(X)) → a__g(mark(X))
mark(sel(X1, X2)) → a__sel(mark(X1), mark(X2))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0) → 0
mark(s(X)) → s(mark(X))
a__f(X) → f(X)
a__g(X) → g(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, 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, cons(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), cons(mark(x0), x1))
A__SEL(s(sel(y0, y1)), cons(y2, f(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__f(mark(x0)))
A__SEL(s(sel(y0, y1)), cons(y2, s(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), s(mark(x0)))
A__SEL(s(sel(y0, y1)), cons(y2, 0)) → A__SEL(a__sel(mark(y0), mark(y1)), 0)
A__SEL(s(sel(y0, y1)), cons(y2, g(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__g(mark(x0)))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ Narrowing

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ QDP

              ↳ Narrowing

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ QDP

                      ↳ Narrowing

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ QDP

                              ↳ Narrowing

                                ↳ QDP

                                  ↳ DependencyGraphProof

                                    ↳ QDP

                                      ↳ Narrowing

                                        ↳ QDP

                                          ↳ DependencyGraphProof

                                            ↳ QDP

                                              ↳ Narrowing

                                                ↳ QDP

                                                  ↳ DependencyGraphProof

                                                    ↳ QDP

                                                      ↳ Narrowing

                                                        ↳ QDP

                                                          ↳ DependencyGraphProof

                                                            ↳ QDP

                                                              ↳ Narrowing

                                                                ↳ QDP

                                                                  ↳ DependencyGraphProof

                                                                    ↳ QDP

                                                                      ↳ Narrowing

                                                                        ↳ QDP

                                                                          ↳ DependencyGraphProof

                                                                            ↳ QDP

                                                                              ↳ Narrowing

                                                                                ↳ QDP

                                                                                  ↳ DependencyGraphProof

                                                                                    ↳ QDP

                                                                                      ↳ Narrowing

                                                                                        ↳ QDP

                                                                                          ↳ DependencyGraphProof

                                                                                            ↳ QDP

                                                                                              ↳ Narrowing

                                                                                                ↳ QDP

                                                                                                  ↳ DependencyGraphProof

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

MARK(sel(sel(y0, y1), sel(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), a__sel(mark(x0), mark(x1)))
MARK(sel(s(y0), cons(x0, x1))) → A__SEL(s(mark(y0)), cons(mark(x0), x1))
MARK(sel(g(y0), g(x0))) → A__SEL(a__g(mark(y0)), a__g(mark(x0)))
MARK(sel(f(y0), cons(x0, x1))) → A__SEL(a__f(mark(y0)), cons(mark(x0), x1))
MARK(cons(X1, X2)) → MARK(X1)
A__F(X) → MARK(X)
MARK(g(X)) → MARK(X)
A__SEL(s(sel(y0, y1)), cons(y2, 0)) → A__SEL(a__sel(mark(y0), mark(y1)), 0)
A__SEL(s(X), cons(Y, Z)) → MARK(Z)
MARK(g(f(x0))) → A__G(a__f(mark(x0)))
A__SEL(s(0), cons(y0, f(x0))) → A__SEL(0, a__f(mark(x0)))
MARK(sel(0, g(x0))) → A__SEL(0, a__g(mark(x0)))
MARK(g(g(x0))) → A__G(a__g(mark(x0)))
MARK(sel(s(y0), g(x0))) → A__SEL(s(mark(y0)), a__g(mark(x0)))
A__SEL(s(0), cons(y0, g(x0))) → A__SEL(0, a__g(mark(x0)))
MARK(f(X)) → 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__G(s(f(x0))) → A__G(a__f(mark(x0)))
MARK(sel(X1, X2)) → MARK(X1)
MARK(g(sel(x0, x1))) → A__G(a__sel(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), cons(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), cons(mark(x0), x1))
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, f(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__f(mark(x0)))
MARK(sel(s(y0), f(x0))) → A__SEL(s(mark(y0)), a__f(mark(x0)))
MARK(sel(0, cons(x0, x1))) → A__SEL(0, cons(mark(x0), x1))
A__SEL(s(f(y0)), cons(y1, cons(x0, x1))) → A__SEL(a__f(mark(y0)), cons(mark(x0), x1))
A__SEL(s(0), cons(y0, sel(x0, x1))) → A__SEL(0, a__sel(mark(x0), mark(x1)))
MARK(sel(g(y0), cons(x0, x1))) → A__SEL(a__g(mark(y0)), cons(mark(x0), x1))
MARK(s(X)) → MARK(X)
MARK(sel(g(y0), sel(x0, x1))) → A__SEL(a__g(mark(y0)), a__sel(mark(x0), mark(x1)))
MARK(sel(s(y0), sel(x0, x1))) → A__SEL(s(mark(y0)), a__sel(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), g(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__g(mark(x0)))
A__SEL(0, cons(X, Y)) → MARK(X)
MARK(sel(0, f(x0))) → A__SEL(0, a__f(mark(x0)))
A__G(s(sel(x0, x1))) → A__G(a__sel(mark(x0), mark(x1)))
A__SEL(s(0), cons(y0, cons(x0, x1))) → A__SEL(0, cons(mark(x0), x1))
A__SEL(s(sel(y0, y1)), cons(y2, g(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__g(mark(x0)))
A__G(s(g(x0))) → A__G(a__g(mark(x0)))
A__SEL(s(X), cons(Y, Z)) → MARK(X)
A__G(s(s(x0))) → A__G(s(mark(x0)))
A__SEL(s(f(y0)), cons(y1, g(x0))) → A__SEL(a__f(mark(y0)), a__g(mark(x0)))
A__SEL(s(s(x0)), cons(y1, y2)) → A__SEL(s(mark(x0)), mark(y2))
A__SEL(s(g(x0)), cons(y1, y2)) → A__SEL(a__g(mark(x0)), mark(y2))
A__SEL(s(f(y0)), cons(y1, f(x0))) → A__SEL(a__f(mark(y0)), a__f(mark(x0)))
A__SEL(s(sel(y0, y1)), cons(y2, s(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), s(mark(x0)))
MARK(sel(f(y0), sel(x0, x1))) → A__SEL(a__f(mark(y0)), a__sel(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), f(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__f(mark(x0)))
MARK(sel(f(y0), f(x0))) → A__SEL(a__f(mark(y0)), a__f(mark(x0)))
MARK(sel(0, sel(x0, x1))) → A__SEL(0, a__sel(mark(x0), mark(x1)))
MARK(g(s(x0))) → A__G(s(mark(x0)))
MARK(sel(f(y0), g(x0))) → A__SEL(a__f(mark(y0)), a__g(mark(x0)))
MARK(sel(g(y0), f(x0))) → A__SEL(a__g(mark(y0)), a__f(mark(x0)))
A__SEL(s(f(y0)), cons(y1, sel(x0, x1))) → A__SEL(a__f(mark(y0)), a__sel(mark(x0), mark(x1)))
A__G(s(X)) → MARK(X)
MARK(f(X)) → A__F(mark(X))
MARK(sel(X1, X2)) → MARK(X2)

The TRS R consists of the following rules:

a__f(X) → cons(mark(X), f(g(X)))
a__g(0) → s(0)
a__g(s(X)) → s(s(a__g(mark(X))))
a__sel(0, cons(X, Y)) → mark(X)
a__sel(s(X), cons(Y, Z)) → a__sel(mark(X), mark(Z))
mark(f(X)) → a__f(mark(X))
mark(g(X)) → a__g(mark(X))
mark(sel(X1, X2)) → a__sel(mark(X1), mark(X2))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0) → 0
mark(s(X)) → s(mark(X))
a__f(X) → f(X)
a__g(X) → g(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

                                                                                      ↳ Narrowing

                                                                                        ↳ QDP

                                                                                          ↳ DependencyGraphProof

                                                                                            ↳ QDP

                                                                                              ↳ Narrowing

                                                                                                ↳ QDP

                                                                                                  ↳ DependencyGraphProof

                                                                                                    ↳ QDP

                                                                                                      ↳ Narrowing

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

MARK(sel(sel(y0, y1), sel(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), a__sel(mark(x0), mark(x1)))
MARK(sel(s(y0), cons(x0, x1))) → A__SEL(s(mark(y0)), cons(mark(x0), x1))
MARK(sel(g(y0), g(x0))) → A__SEL(a__g(mark(y0)), a__g(mark(x0)))
MARK(sel(f(y0), cons(x0, x1))) → A__SEL(a__f(mark(y0)), cons(mark(x0), x1))
MARK(cons(X1, X2)) → MARK(X1)
A__F(X) → MARK(X)
MARK(g(X)) → MARK(X)
A__SEL(s(X), cons(Y, Z)) → MARK(Z)
MARK(g(f(x0))) → A__G(a__f(mark(x0)))
MARK(sel(0, g(x0))) → A__SEL(0, a__g(mark(x0)))
A__SEL(s(0), cons(y0, f(x0))) → A__SEL(0, a__f(mark(x0)))
MARK(g(g(x0))) → A__G(a__g(mark(x0)))
MARK(sel(s(y0), g(x0))) → A__SEL(s(mark(y0)), a__g(mark(x0)))
A__SEL(s(0), cons(y0, g(x0))) → A__SEL(0, a__g(mark(x0)))
MARK(f(X)) → 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)
A__G(s(f(x0))) → A__G(a__f(mark(x0)))
MARK(g(sel(x0, x1))) → A__G(a__sel(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), cons(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), cons(mark(x0), x1))
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, f(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__f(mark(x0)))
MARK(sel(s(y0), f(x0))) → A__SEL(s(mark(y0)), a__f(mark(x0)))
MARK(sel(0, cons(x0, x1))) → A__SEL(0, cons(mark(x0), x1))
A__SEL(s(f(y0)), cons(y1, cons(x0, x1))) → A__SEL(a__f(mark(y0)), cons(mark(x0), x1))
A__SEL(s(0), cons(y0, sel(x0, x1))) → A__SEL(0, a__sel(mark(x0), mark(x1)))
MARK(sel(g(y0), cons(x0, x1))) → A__SEL(a__g(mark(y0)), cons(mark(x0), x1))
MARK(s(X)) → MARK(X)
MARK(sel(sel(y0, y1), g(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__g(mark(x0)))
MARK(sel(s(y0), sel(x0, x1))) → A__SEL(s(mark(y0)), a__sel(mark(x0), mark(x1)))
MARK(sel(g(y0), sel(x0, x1))) → A__SEL(a__g(mark(y0)), a__sel(mark(x0), mark(x1)))
A__SEL(0, cons(X, Y)) → MARK(X)
MARK(sel(0, f(x0))) → A__SEL(0, a__f(mark(x0)))
A__G(s(sel(x0, x1))) → A__G(a__sel(mark(x0), mark(x1)))
A__SEL(s(0), cons(y0, cons(x0, x1))) → A__SEL(0, cons(mark(x0), x1))
A__SEL(s(sel(y0, y1)), cons(y2, g(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__g(mark(x0)))
A__G(s(g(x0))) → A__G(a__g(mark(x0)))
A__SEL(s(X), cons(Y, Z)) → MARK(X)
A__G(s(s(x0))) → A__G(s(mark(x0)))
A__SEL(s(s(x0)), cons(y1, y2)) → A__SEL(s(mark(x0)), mark(y2))
A__SEL(s(f(y0)), cons(y1, g(x0))) → A__SEL(a__f(mark(y0)), a__g(mark(x0)))
A__SEL(s(g(x0)), cons(y1, y2)) → A__SEL(a__g(mark(x0)), mark(y2))
A__SEL(s(f(y0)), cons(y1, f(x0))) → A__SEL(a__f(mark(y0)), a__f(mark(x0)))
MARK(sel(f(y0), sel(x0, x1))) → A__SEL(a__f(mark(y0)), a__sel(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), f(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__f(mark(x0)))
MARK(sel(f(y0), f(x0))) → A__SEL(a__f(mark(y0)), a__f(mark(x0)))
MARK(sel(0, sel(x0, x1))) → A__SEL(0, a__sel(mark(x0), mark(x1)))
MARK(g(s(x0))) → A__G(s(mark(x0)))
A__SEL(s(f(y0)), cons(y1, sel(x0, x1))) → A__SEL(a__f(mark(y0)), a__sel(mark(x0), mark(x1)))
MARK(sel(g(y0), f(x0))) → A__SEL(a__g(mark(y0)), a__f(mark(x0)))
MARK(sel(f(y0), g(x0))) → A__SEL(a__f(mark(y0)), a__g(mark(x0)))
A__G(s(X)) → MARK(X)
MARK(f(X)) → A__F(mark(X))
MARK(sel(X1, X2)) → MARK(X2)

The TRS R consists of the following rules:

a__f(X) → cons(mark(X), f(g(X)))
a__g(0) → s(0)
a__g(s(X)) → s(s(a__g(mark(X))))
a__sel(0, cons(X, Y)) → mark(X)
a__sel(s(X), cons(Y, Z)) → a__sel(mark(X), mark(Z))
mark(f(X)) → a__f(mark(X))
mark(g(X)) → a__g(mark(X))
mark(sel(X1, X2)) → a__sel(mark(X1), mark(X2))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0) → 0
mark(s(X)) → s(mark(X))
a__f(X) → f(X)
a__g(X) → g(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, f(x0))) → A__SEL(s(mark(y0)), a__f(mark(x0)))
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, g(x0))) → A__SEL(s(mark(y0)), a__g(mark(x0)))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ Narrowing

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ QDP

              ↳ Narrowing

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ QDP

                      ↳ Narrowing

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ QDP

                              ↳ Narrowing

                                ↳ QDP

                                  ↳ DependencyGraphProof

                                    ↳ QDP

                                      ↳ Narrowing

                                        ↳ QDP

                                          ↳ DependencyGraphProof

                                            ↳ QDP

                                              ↳ Narrowing

                                                ↳ QDP

                                                  ↳ DependencyGraphProof

                                                    ↳ QDP

                                                      ↳ Narrowing

                                                        ↳ QDP

                                                          ↳ DependencyGraphProof

                                                            ↳ QDP

                                                              ↳ Narrowing

                                                                ↳ QDP

                                                                  ↳ DependencyGraphProof

                                                                    ↳ QDP

                                                                      ↳ Narrowing

                                                                        ↳ QDP

                                                                          ↳ DependencyGraphProof

                                                                            ↳ QDP

                                                                              ↳ Narrowing

                                                                                ↳ QDP

                                                                                  ↳ DependencyGraphProof

                                                                                    ↳ QDP

                                                                                      ↳ Narrowing

                                                                                        ↳ QDP

                                                                                          ↳ DependencyGraphProof

                                                                                            ↳ QDP

                                                                                              ↳ Narrowing

                                                                                                ↳ QDP

                                                                                                  ↳ DependencyGraphProof

                                                                                                    ↳ QDP

                                                                                                      ↳ Narrowing

                                                                                                        ↳ QDP

                                                                                                          ↳ DependencyGraphProof

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

A__SEL(s(s(y0)), cons(y1, s(x0))) → A__SEL(s(mark(y0)), s(mark(x0)))
MARK(sel(sel(y0, y1), sel(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), a__sel(mark(x0), mark(x1)))
MARK(sel(s(y0), cons(x0, x1))) → A__SEL(s(mark(y0)), cons(mark(x0), x1))
MARK(sel(g(y0), g(x0))) → A__SEL(a__g(mark(y0)), a__g(mark(x0)))
MARK(sel(f(y0), cons(x0, x1))) → A__SEL(a__f(mark(y0)), cons(mark(x0), x1))
MARK(cons(X1, X2)) → MARK(X1)
A__F(X) → MARK(X)
MARK(g(X)) → MARK(X)
A__SEL(s(X), cons(Y, Z)) → MARK(Z)
MARK(g(f(x0))) → A__G(a__f(mark(x0)))
A__SEL(s(0), cons(y0, f(x0))) → A__SEL(0, a__f(mark(x0)))
MARK(sel(0, g(x0))) → A__SEL(0, a__g(mark(x0)))
MARK(g(g(x0))) → A__G(a__g(mark(x0)))
MARK(sel(s(y0), g(x0))) → A__SEL(s(mark(y0)), a__g(mark(x0)))
A__SEL(s(0), cons(y0, g(x0))) → A__SEL(0, a__g(mark(x0)))
MARK(f(X)) → 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)))
A__G(s(f(x0))) → A__G(a__f(mark(x0)))
MARK(sel(X1, X2)) → MARK(X1)
MARK(g(sel(x0, x1))) → A__G(a__sel(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), cons(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), cons(mark(x0), x1))
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__SEL(s(sel(y0, y1)), cons(y2, f(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__f(mark(x0)))
MARK(sel(s(y0), f(x0))) → A__SEL(s(mark(y0)), a__f(mark(x0)))
A__SEL(s(s(y0)), cons(y1, 0)) → A__SEL(s(mark(y0)), 0)
MARK(sel(0, cons(x0, x1))) → A__SEL(0, cons(mark(x0), x1))
A__SEL(s(f(y0)), cons(y1, cons(x0, x1))) → A__SEL(a__f(mark(y0)), cons(mark(x0), x1))
A__SEL(s(0), cons(y0, sel(x0, x1))) → A__SEL(0, a__sel(mark(x0), mark(x1)))
MARK(sel(g(y0), cons(x0, x1))) → A__SEL(a__g(mark(y0)), cons(mark(x0), x1))
MARK(s(X)) → MARK(X)
A__SEL(s(s(y0)), cons(y1, g(x0))) → A__SEL(s(mark(y0)), a__g(mark(x0)))
MARK(sel(g(y0), sel(x0, x1))) → A__SEL(a__g(mark(y0)), a__sel(mark(x0), mark(x1)))
MARK(sel(s(y0), sel(x0, x1))) → A__SEL(s(mark(y0)), a__sel(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), g(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__g(mark(x0)))
A__SEL(0, cons(X, Y)) → MARK(X)
MARK(sel(0, f(x0))) → A__SEL(0, a__f(mark(x0)))
A__G(s(sel(x0, x1))) → A__G(a__sel(mark(x0), mark(x1)))
A__SEL(s(0), cons(y0, cons(x0, x1))) → A__SEL(0, cons(mark(x0), x1))
A__SEL(s(sel(y0, y1)), cons(y2, g(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__g(mark(x0)))
A__SEL(s(s(y0)), cons(y1, f(x0))) → A__SEL(s(mark(y0)), a__f(mark(x0)))
A__G(s(g(x0))) → A__G(a__g(mark(x0)))
A__SEL(s(X), cons(Y, Z)) → MARK(X)
A__G(s(s(x0))) → A__G(s(mark(x0)))
A__SEL(s(f(y0)), cons(y1, g(x0))) → A__SEL(a__f(mark(y0)), a__g(mark(x0)))
A__SEL(s(g(x0)), cons(y1, y2)) → A__SEL(a__g(mark(x0)), mark(y2))
A__SEL(s(f(y0)), cons(y1, f(x0))) → A__SEL(a__f(mark(y0)), a__f(mark(x0)))
MARK(sel(sel(y0, y1), f(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__f(mark(x0)))
MARK(sel(f(y0), sel(x0, x1))) → A__SEL(a__f(mark(y0)), a__sel(mark(x0), mark(x1)))
MARK(sel(f(y0), f(x0))) → A__SEL(a__f(mark(y0)), a__f(mark(x0)))
MARK(g(s(x0))) → A__G(s(mark(x0)))
MARK(sel(0, sel(x0, x1))) → A__SEL(0, a__sel(mark(x0), mark(x1)))
MARK(sel(f(y0), g(x0))) → A__SEL(a__f(mark(y0)), a__g(mark(x0)))
MARK(sel(g(y0), f(x0))) → A__SEL(a__g(mark(y0)), a__f(mark(x0)))
A__SEL(s(f(y0)), cons(y1, sel(x0, x1))) → A__SEL(a__f(mark(y0)), a__sel(mark(x0), mark(x1)))
A__G(s(X)) → MARK(X)
MARK(f(X)) → A__F(mark(X))
MARK(sel(X1, X2)) → MARK(X2)

The TRS R consists of the following rules:

a__f(X) → cons(mark(X), f(g(X)))
a__g(0) → s(0)
a__g(s(X)) → s(s(a__g(mark(X))))
a__sel(0, cons(X, Y)) → mark(X)
a__sel(s(X), cons(Y, Z)) → a__sel(mark(X), mark(Z))
mark(f(X)) → a__f(mark(X))
mark(g(X)) → a__g(mark(X))
mark(sel(X1, X2)) → a__sel(mark(X1), mark(X2))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0) → 0
mark(s(X)) → s(mark(X))
a__f(X) → f(X)
a__g(X) → g(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

                                                                                      ↳ Narrowing

                                                                                        ↳ QDP

                                                                                          ↳ DependencyGraphProof

                                                                                            ↳ QDP

                                                                                              ↳ Narrowing

                                                                                                ↳ QDP

                                                                                                  ↳ DependencyGraphProof

                                                                                                    ↳ QDP

                                                                                                      ↳ Narrowing

                                                                                                        ↳ QDP

                                                                                                          ↳ DependencyGraphProof

                                                                                                            ↳ QDP

                                                                                                              ↳ Narrowing

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

MARK(sel(sel(y0, y1), sel(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), a__sel(mark(x0), mark(x1)))
MARK(sel(s(y0), cons(x0, x1))) → A__SEL(s(mark(y0)), cons(mark(x0), x1))
MARK(sel(g(y0), g(x0))) → A__SEL(a__g(mark(y0)), a__g(mark(x0)))
MARK(sel(f(y0), cons(x0, x1))) → A__SEL(a__f(mark(y0)), cons(mark(x0), x1))
MARK(cons(X1, X2)) → MARK(X1)
A__F(X) → MARK(X)
MARK(g(X)) → MARK(X)
A__SEL(s(X), cons(Y, Z)) → MARK(Z)
MARK(g(f(x0))) → A__G(a__f(mark(x0)))
MARK(sel(0, g(x0))) → A__SEL(0, a__g(mark(x0)))
A__SEL(s(0), cons(y0, f(x0))) → A__SEL(0, a__f(mark(x0)))
MARK(g(g(x0))) → A__G(a__g(mark(x0)))
MARK(sel(s(y0), g(x0))) → A__SEL(s(mark(y0)), a__g(mark(x0)))
A__SEL(s(0), cons(y0, g(x0))) → A__SEL(0, a__g(mark(x0)))
MARK(f(X)) → 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)
A__G(s(f(x0))) → A__G(a__f(mark(x0)))
MARK(g(sel(x0, x1))) → A__G(a__sel(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), cons(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), cons(mark(x0), x1))
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, f(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__f(mark(x0)))
A__SEL(s(s(y0)), cons(y1, cons(x0, x1))) → A__SEL(s(mark(y0)), cons(mark(x0), x1))
MARK(sel(s(y0), f(x0))) → A__SEL(s(mark(y0)), a__f(mark(x0)))
MARK(sel(0, cons(x0, x1))) → A__SEL(0, cons(mark(x0), x1))
A__SEL(s(f(y0)), cons(y1, cons(x0, x1))) → A__SEL(a__f(mark(y0)), cons(mark(x0), x1))
A__SEL(s(0), cons(y0, sel(x0, x1))) → A__SEL(0, a__sel(mark(x0), mark(x1)))
MARK(sel(g(y0), cons(x0, x1))) → A__SEL(a__g(mark(y0)), cons(mark(x0), x1))
MARK(s(X)) → MARK(X)
A__SEL(s(s(y0)), cons(y1, g(x0))) → A__SEL(s(mark(y0)), a__g(mark(x0)))
MARK(sel(sel(y0, y1), g(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__g(mark(x0)))
MARK(sel(s(y0), sel(x0, x1))) → A__SEL(s(mark(y0)), a__sel(mark(x0), mark(x1)))
MARK(sel(g(y0), sel(x0, x1))) → A__SEL(a__g(mark(y0)), a__sel(mark(x0), mark(x1)))
A__SEL(0, cons(X, Y)) → MARK(X)
MARK(sel(0, f(x0))) → A__SEL(0, a__f(mark(x0)))
A__G(s(sel(x0, x1))) → A__G(a__sel(mark(x0), mark(x1)))
A__SEL(s(0), cons(y0, cons(x0, x1))) → A__SEL(0, cons(mark(x0), x1))
A__SEL(s(sel(y0, y1)), cons(y2, g(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__g(mark(x0)))
A__SEL(s(s(y0)), cons(y1, f(x0))) → A__SEL(s(mark(y0)), a__f(mark(x0)))
A__G(s(g(x0))) → A__G(a__g(mark(x0)))
A__SEL(s(X), cons(Y, Z)) → MARK(X)
A__G(s(s(x0))) → A__G(s(mark(x0)))
A__SEL(s(f(y0)), cons(y1, g(x0))) → A__SEL(a__f(mark(y0)), a__g(mark(x0)))
A__SEL(s(g(x0)), cons(y1, y2)) → A__SEL(a__g(mark(x0)), mark(y2))
A__SEL(s(f(y0)), cons(y1, f(x0))) → A__SEL(a__f(mark(y0)), a__f(mark(x0)))
MARK(sel(f(y0), sel(x0, x1))) → A__SEL(a__f(mark(y0)), a__sel(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), f(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__f(mark(x0)))
MARK(sel(f(y0), f(x0))) → A__SEL(a__f(mark(y0)), a__f(mark(x0)))
MARK(sel(0, sel(x0, x1))) → A__SEL(0, a__sel(mark(x0), mark(x1)))
MARK(g(s(x0))) → A__G(s(mark(x0)))
MARK(sel(f(y0), g(x0))) → A__SEL(a__f(mark(y0)), a__g(mark(x0)))
A__SEL(s(f(y0)), cons(y1, sel(x0, x1))) → A__SEL(a__f(mark(y0)), a__sel(mark(x0), mark(x1)))
MARK(sel(g(y0), f(x0))) → A__SEL(a__g(mark(y0)), a__f(mark(x0)))
A__G(s(X)) → MARK(X)
MARK(f(X)) → A__F(mark(X))
MARK(sel(X1, X2)) → MARK(X2)

The TRS R consists of the following rules:

a__f(X) → cons(mark(X), f(g(X)))
a__g(0) → s(0)
a__g(s(X)) → s(s(a__g(mark(X))))
a__sel(0, cons(X, Y)) → mark(X)
a__sel(s(X), cons(Y, Z)) → a__sel(mark(X), mark(Z))
mark(f(X)) → a__f(mark(X))
mark(g(X)) → a__g(mark(X))
mark(sel(X1, X2)) → a__sel(mark(X1), mark(X2))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0) → 0
mark(s(X)) → s(mark(X))
a__f(X) → f(X)
a__g(X) → g(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(g(x0)), cons(y1, y2)) → A__SEL(a__g(mark(x0)), mark(y2)) at position [1] we obtained the following new rules:

A__SEL(s(g(y0)), cons(y1, sel(x0, x1))) → A__SEL(a__g(mark(y0)), a__sel(mark(x0), mark(x1)))
A__SEL(s(g(y0)), cons(y1, f(x0))) → A__SEL(a__g(mark(y0)), a__f(mark(x0)))
A__SEL(s(g(y0)), cons(y1, cons(x0, x1))) → A__SEL(a__g(mark(y0)), cons(mark(x0), x1))
A__SEL(s(g(y0)), cons(y1, g(x0))) → A__SEL(a__g(mark(y0)), a__g(mark(x0)))
A__SEL(s(g(y0)), cons(y1, s(x0))) → A__SEL(a__g(mark(y0)), s(mark(x0)))
A__SEL(s(g(y0)), cons(y1, 0)) → A__SEL(a__g(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

                                            ↳ QDP

                                              ↳ Narrowing

                                                ↳ QDP

                                                  ↳ DependencyGraphProof

                                                    ↳ QDP

                                                      ↳ Narrowing

                                                        ↳ QDP

                                                          ↳ DependencyGraphProof

                                                            ↳ QDP

                                                              ↳ Narrowing

                                                                ↳ QDP

                                                                  ↳ DependencyGraphProof

                                                                    ↳ QDP

                                                                      ↳ Narrowing

                                                                        ↳ QDP

                                                                          ↳ DependencyGraphProof

                                                                            ↳ QDP

                                                                              ↳ Narrowing

                                                                                ↳ QDP

                                                                                  ↳ DependencyGraphProof

                                                                                    ↳ QDP

                                                                                      ↳ Narrowing

                                                                                        ↳ QDP

                                                                                          ↳ DependencyGraphProof

                                                                                            ↳ QDP

                                                                                              ↳ Narrowing

                                                                                                ↳ QDP

                                                                                                  ↳ DependencyGraphProof

                                                                                                    ↳ QDP

                                                                                                      ↳ Narrowing

                                                                                                        ↳ QDP

                                                                                                          ↳ DependencyGraphProof

                                                                                                            ↳ QDP

                                                                                                              ↳ Narrowing

                                                                                                                ↳ QDP

                                                                                                                  ↳ DependencyGraphProof

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

MARK(sel(sel(y0, y1), sel(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), a__sel(mark(x0), mark(x1)))
MARK(sel(s(y0), cons(x0, x1))) → A__SEL(s(mark(y0)), cons(mark(x0), x1))
A__SEL(s(g(y0)), cons(y1, g(x0))) → A__SEL(a__g(mark(y0)), a__g(mark(x0)))
MARK(sel(g(y0), g(x0))) → A__SEL(a__g(mark(y0)), a__g(mark(x0)))
MARK(sel(f(y0), cons(x0, x1))) → A__SEL(a__f(mark(y0)), cons(mark(x0), x1))
MARK(cons(X1, X2)) → MARK(X1)
A__F(X) → MARK(X)
MARK(g(X)) → MARK(X)
A__SEL(s(X), cons(Y, Z)) → MARK(Z)
MARK(g(f(x0))) → A__G(a__f(mark(x0)))
A__SEL(s(0), cons(y0, f(x0))) → A__SEL(0, a__f(mark(x0)))
MARK(sel(0, g(x0))) → A__SEL(0, a__g(mark(x0)))
MARK(g(g(x0))) → A__G(a__g(mark(x0)))
MARK(sel(s(y0), g(x0))) → A__SEL(s(mark(y0)), a__g(mark(x0)))
A__SEL(s(0), cons(y0, g(x0))) → A__SEL(0, a__g(mark(x0)))
MARK(f(X)) → 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)))
A__G(s(f(x0))) → A__G(a__f(mark(x0)))
MARK(sel(X1, X2)) → MARK(X1)
MARK(g(sel(x0, x1))) → A__G(a__sel(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), cons(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), cons(mark(x0), x1))
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__SEL(s(sel(y0, y1)), cons(y2, f(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__f(mark(x0)))
MARK(sel(s(y0), f(x0))) → A__SEL(s(mark(y0)), a__f(mark(x0)))
A__SEL(s(g(y0)), cons(y1, sel(x0, x1))) → A__SEL(a__g(mark(y0)), a__sel(mark(x0), mark(x1)))
MARK(sel(0, cons(x0, x1))) → A__SEL(0, cons(mark(x0), x1))
A__SEL(s(f(y0)), cons(y1, cons(x0, x1))) → A__SEL(a__f(mark(y0)), cons(mark(x0), x1))
A__SEL(s(0), cons(y0, sel(x0, x1))) → A__SEL(0, a__sel(mark(x0), mark(x1)))
MARK(sel(g(y0), cons(x0, x1))) → A__SEL(a__g(mark(y0)), cons(mark(x0), x1))
MARK(s(X)) → MARK(X)
A__SEL(s(g(y0)), cons(y1, s(x0))) → A__SEL(a__g(mark(y0)), s(mark(x0)))
A__SEL(s(s(y0)), cons(y1, g(x0))) → A__SEL(s(mark(y0)), a__g(mark(x0)))
MARK(sel(g(y0), sel(x0, x1))) → A__SEL(a__g(mark(y0)), a__sel(mark(x0), mark(x1)))
MARK(sel(s(y0), sel(x0, x1))) → A__SEL(s(mark(y0)), a__sel(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), g(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__g(mark(x0)))
A__SEL(0, cons(X, Y)) → MARK(X)
MARK(sel(0, f(x0))) → A__SEL(0, a__f(mark(x0)))
A__SEL(s(g(y0)), cons(y1, cons(x0, x1))) → A__SEL(a__g(mark(y0)), cons(mark(x0), x1))
A__G(s(sel(x0, x1))) → A__G(a__sel(mark(x0), mark(x1)))
A__SEL(s(0), cons(y0, cons(x0, x1))) → A__SEL(0, cons(mark(x0), x1))
A__SEL(s(sel(y0, y1)), cons(y2, g(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__g(mark(x0)))
A__SEL(s(s(y0)), cons(y1, f(x0))) → A__SEL(s(mark(y0)), a__f(mark(x0)))
A__G(s(g(x0))) → A__G(a__g(mark(x0)))
A__SEL(s(X), cons(Y, Z)) → MARK(X)
A__SEL(s(g(y0)), cons(y1, 0)) → A__SEL(a__g(mark(y0)), 0)
A__G(s(s(x0))) → A__G(s(mark(x0)))
A__SEL(s(g(y0)), cons(y1, f(x0))) → A__SEL(a__g(mark(y0)), a__f(mark(x0)))
A__SEL(s(f(y0)), cons(y1, g(x0))) → A__SEL(a__f(mark(y0)), a__g(mark(x0)))
A__SEL(s(f(y0)), cons(y1, f(x0))) → A__SEL(a__f(mark(y0)), a__f(mark(x0)))
MARK(sel(sel(y0, y1), f(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__f(mark(x0)))
MARK(sel(f(y0), sel(x0, x1))) → A__SEL(a__f(mark(y0)), a__sel(mark(x0), mark(x1)))
MARK(sel(f(y0), f(x0))) → A__SEL(a__f(mark(y0)), a__f(mark(x0)))
MARK(g(s(x0))) → A__G(s(mark(x0)))
MARK(sel(0, sel(x0, x1))) → A__SEL(0, a__sel(mark(x0), mark(x1)))
MARK(sel(g(y0), f(x0))) → A__SEL(a__g(mark(y0)), a__f(mark(x0)))
A__SEL(s(f(y0)), cons(y1, sel(x0, x1))) → A__SEL(a__f(mark(y0)), a__sel(mark(x0), mark(x1)))
MARK(sel(f(y0), g(x0))) → A__SEL(a__f(mark(y0)), a__g(mark(x0)))
A__G(s(X)) → MARK(X)
MARK(f(X)) → A__F(mark(X))
MARK(sel(X1, X2)) → MARK(X2)

The TRS R consists of the following rules:

a__f(X) → cons(mark(X), f(g(X)))
a__g(0) → s(0)
a__g(s(X)) → s(s(a__g(mark(X))))
a__sel(0, cons(X, Y)) → mark(X)
a__sel(s(X), cons(Y, Z)) → a__sel(mark(X), mark(Z))
mark(f(X)) → a__f(mark(X))
mark(g(X)) → a__g(mark(X))
mark(sel(X1, X2)) → a__sel(mark(X1), mark(X2))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0) → 0
mark(s(X)) → s(mark(X))
a__f(X) → f(X)
a__g(X) → g(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

                                                                                      ↳ Narrowing

                                                                                        ↳ QDP

                                                                                          ↳ DependencyGraphProof

                                                                                            ↳ QDP

                                                                                              ↳ Narrowing

                                                                                                ↳ QDP

                                                                                                  ↳ DependencyGraphProof

                                                                                                    ↳ QDP

                                                                                                      ↳ Narrowing

                                                                                                        ↳ QDP

                                                                                                          ↳ DependencyGraphProof

                                                                                                            ↳ QDP

                                                                                                              ↳ Narrowing

                                                                                                                ↳ QDP

                                                                                                                  ↳ DependencyGraphProof

                                                                                                                    ↳ QDP

                                                                                                                      ↳ QDPOrderProof

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

MARK(sel(sel(y0, y1), sel(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), a__sel(mark(x0), mark(x1)))
MARK(sel(s(y0), cons(x0, x1))) → A__SEL(s(mark(y0)), cons(mark(x0), x1))
MARK(sel(g(y0), g(x0))) → A__SEL(a__g(mark(y0)), a__g(mark(x0)))
A__SEL(s(g(y0)), cons(y1, g(x0))) → A__SEL(a__g(mark(y0)), a__g(mark(x0)))
MARK(sel(f(y0), cons(x0, x1))) → A__SEL(a__f(mark(y0)), cons(mark(x0), x1))
MARK(cons(X1, X2)) → MARK(X1)
A__F(X) → MARK(X)
MARK(g(X)) → MARK(X)
A__SEL(s(X), cons(Y, Z)) → MARK(Z)
MARK(g(f(x0))) → A__G(a__f(mark(x0)))
MARK(sel(0, g(x0))) → A__SEL(0, a__g(mark(x0)))
A__SEL(s(0), cons(y0, f(x0))) → A__SEL(0, a__f(mark(x0)))
MARK(g(g(x0))) → A__G(a__g(mark(x0)))
MARK(sel(s(y0), g(x0))) → A__SEL(s(mark(y0)), a__g(mark(x0)))
A__SEL(s(0), cons(y0, g(x0))) → A__SEL(0, a__g(mark(x0)))
MARK(f(X)) → 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)
A__G(s(f(x0))) → A__G(a__f(mark(x0)))
MARK(g(sel(x0, x1))) → A__G(a__sel(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), cons(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), cons(mark(x0), x1))
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, f(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__f(mark(x0)))
A__SEL(s(s(y0)), cons(y1, cons(x0, x1))) → A__SEL(s(mark(y0)), cons(mark(x0), x1))
MARK(sel(s(y0), f(x0))) → A__SEL(s(mark(y0)), a__f(mark(x0)))
A__SEL(s(g(y0)), cons(y1, sel(x0, x1))) → A__SEL(a__g(mark(y0)), a__sel(mark(x0), mark(x1)))
MARK(sel(0, cons(x0, x1))) → A__SEL(0, cons(mark(x0), x1))
A__SEL(s(f(y0)), cons(y1, cons(x0, x1))) → A__SEL(a__f(mark(y0)), cons(mark(x0), x1))
A__SEL(s(0), cons(y0, sel(x0, x1))) → A__SEL(0, a__sel(mark(x0), mark(x1)))
MARK(sel(g(y0), cons(x0, x1))) → A__SEL(a__g(mark(y0)), cons(mark(x0), x1))
MARK(s(X)) → MARK(X)
A__SEL(s(s(y0)), cons(y1, g(x0))) → A__SEL(s(mark(y0)), a__g(mark(x0)))
MARK(sel(g(y0), sel(x0, x1))) → A__SEL(a__g(mark(y0)), a__sel(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), g(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__g(mark(x0)))
MARK(sel(s(y0), sel(x0, x1))) → A__SEL(s(mark(y0)), a__sel(mark(x0), mark(x1)))
A__SEL(0, cons(X, Y)) → MARK(X)
MARK(sel(0, f(x0))) → A__SEL(0, a__f(mark(x0)))
A__SEL(s(g(y0)), cons(y1, cons(x0, x1))) → A__SEL(a__g(mark(y0)), cons(mark(x0), x1))
A__G(s(sel(x0, x1))) → A__G(a__sel(mark(x0), mark(x1)))
A__SEL(s(0), cons(y0, cons(x0, x1))) → A__SEL(0, cons(mark(x0), x1))
A__SEL(s(sel(y0, y1)), cons(y2, g(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__g(mark(x0)))
A__SEL(s(s(y0)), cons(y1, f(x0))) → A__SEL(s(mark(y0)), a__f(mark(x0)))
A__G(s(g(x0))) → A__G(a__g(mark(x0)))
A__SEL(s(X), cons(Y, Z)) → MARK(X)
A__G(s(s(x0))) → A__G(s(mark(x0)))
A__SEL(s(f(y0)), cons(y1, g(x0))) → A__SEL(a__f(mark(y0)), a__g(mark(x0)))
A__SEL(s(g(y0)), cons(y1, f(x0))) → A__SEL(a__g(mark(y0)), a__f(mark(x0)))
A__SEL(s(f(y0)), cons(y1, f(x0))) → A__SEL(a__f(mark(y0)), a__f(mark(x0)))
MARK(sel(f(y0), sel(x0, x1))) → A__SEL(a__f(mark(y0)), a__sel(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), f(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__f(mark(x0)))
MARK(sel(f(y0), f(x0))) → A__SEL(a__f(mark(y0)), a__f(mark(x0)))
MARK(sel(0, sel(x0, x1))) → A__SEL(0, a__sel(mark(x0), mark(x1)))
MARK(g(s(x0))) → A__G(s(mark(x0)))
MARK(sel(f(y0), g(x0))) → A__SEL(a__f(mark(y0)), a__g(mark(x0)))
MARK(sel(g(y0), f(x0))) → A__SEL(a__g(mark(y0)), a__f(mark(x0)))
A__SEL(s(f(y0)), cons(y1, sel(x0, x1))) → A__SEL(a__f(mark(y0)), a__sel(mark(x0), mark(x1)))
A__G(s(X)) → MARK(X)
MARK(f(X)) → A__F(mark(X))
MARK(sel(X1, X2)) → MARK(X2)

The TRS R consists of the following rules:

a__f(X) → cons(mark(X), f(g(X)))
a__g(0) → s(0)
a__g(s(X)) → s(s(a__g(mark(X))))
a__sel(0, cons(X, Y)) → mark(X)
a__sel(s(X), cons(Y, Z)) → a__sel(mark(X), mark(Z))
mark(f(X)) → a__f(mark(X))
mark(g(X)) → a__g(mark(X))
mark(sel(X1, X2)) → a__sel(mark(X1), mark(X2))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0) → 0
mark(s(X)) → s(mark(X))
a__f(X) → f(X)
a__g(X) → g(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(g(f(x0))) → A__G(a__f(mark(x0)))
A__G(s(f(x0))) → A__G(a__f(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(sel(g(y0), g(x0))) → A__SEL(a__g(mark(y0)), a__g(mark(x0)))
A__SEL(s(g(y0)), cons(y1, g(x0))) → A__SEL(a__g(mark(y0)), a__g(mark(x0)))
MARK(sel(f(y0), cons(x0, x1))) → A__SEL(a__f(mark(y0)), cons(mark(x0), x1))
MARK(cons(X1, X2)) → MARK(X1)
A__F(X) → MARK(X)
MARK(g(X)) → MARK(X)
A__SEL(s(X), cons(Y, Z)) → MARK(Z)
MARK(sel(0, g(x0))) → A__SEL(0, a__g(mark(x0)))
A__SEL(s(0), cons(y0, f(x0))) → A__SEL(0, a__f(mark(x0)))
MARK(g(g(x0))) → A__G(a__g(mark(x0)))
MARK(sel(s(y0), g(x0))) → A__SEL(s(mark(y0)), a__g(mark(x0)))
A__SEL(s(0), cons(y0, g(x0))) → A__SEL(0, a__g(mark(x0)))
MARK(f(X)) → 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(g(sel(x0, x1))) → A__G(a__sel(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), cons(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), cons(mark(x0), x1))
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, f(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__f(mark(x0)))
A__SEL(s(s(y0)), cons(y1, cons(x0, x1))) → A__SEL(s(mark(y0)), cons(mark(x0), x1))
MARK(sel(s(y0), f(x0))) → A__SEL(s(mark(y0)), a__f(mark(x0)))
A__SEL(s(g(y0)), cons(y1, sel(x0, x1))) → A__SEL(a__g(mark(y0)), a__sel(mark(x0), mark(x1)))
MARK(sel(0, cons(x0, x1))) → A__SEL(0, cons(mark(x0), x1))
A__SEL(s(f(y0)), cons(y1, cons(x0, x1))) → A__SEL(a__f(mark(y0)), cons(mark(x0), x1))
A__SEL(s(0), cons(y0, sel(x0, x1))) → A__SEL(0, a__sel(mark(x0), mark(x1)))
MARK(sel(g(y0), cons(x0, x1))) → A__SEL(a__g(mark(y0)), cons(mark(x0), x1))
MARK(s(X)) → MARK(X)
A__SEL(s(s(y0)), cons(y1, g(x0))) → A__SEL(s(mark(y0)), a__g(mark(x0)))
MARK(sel(g(y0), sel(x0, x1))) → A__SEL(a__g(mark(y0)), a__sel(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), g(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__g(mark(x0)))
MARK(sel(s(y0), sel(x0, x1))) → A__SEL(s(mark(y0)), a__sel(mark(x0), mark(x1)))
A__SEL(0, cons(X, Y)) → MARK(X)
MARK(sel(0, f(x0))) → A__SEL(0, a__f(mark(x0)))
A__SEL(s(g(y0)), cons(y1, cons(x0, x1))) → A__SEL(a__g(mark(y0)), cons(mark(x0), x1))
A__G(s(sel(x0, x1))) → A__G(a__sel(mark(x0), mark(x1)))
A__SEL(s(0), cons(y0, cons(x0, x1))) → A__SEL(0, cons(mark(x0), x1))
A__SEL(s(sel(y0, y1)), cons(y2, g(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__g(mark(x0)))
A__SEL(s(s(y0)), cons(y1, f(x0))) → A__SEL(s(mark(y0)), a__f(mark(x0)))
A__G(s(g(x0))) → A__G(a__g(mark(x0)))
A__SEL(s(X), cons(Y, Z)) → MARK(X)
A__G(s(s(x0))) → A__G(s(mark(x0)))
A__SEL(s(f(y0)), cons(y1, g(x0))) → A__SEL(a__f(mark(y0)), a__g(mark(x0)))
A__SEL(s(g(y0)), cons(y1, f(x0))) → A__SEL(a__g(mark(y0)), a__f(mark(x0)))
A__SEL(s(f(y0)), cons(y1, f(x0))) → A__SEL(a__f(mark(y0)), a__f(mark(x0)))
MARK(sel(f(y0), sel(x0, x1))) → A__SEL(a__f(mark(y0)), a__sel(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), f(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__f(mark(x0)))
MARK(sel(f(y0), f(x0))) → A__SEL(a__f(mark(y0)), a__f(mark(x0)))
MARK(sel(0, sel(x0, x1))) → A__SEL(0, a__sel(mark(x0), mark(x1)))
MARK(g(s(x0))) → A__G(s(mark(x0)))
MARK(sel(f(y0), g(x0))) → A__SEL(a__f(mark(y0)), a__g(mark(x0)))
MARK(sel(g(y0), f(x0))) → A__SEL(a__g(mark(y0)), a__f(mark(x0)))
A__SEL(s(f(y0)), cons(y1, sel(x0, x1))) → A__SEL(a__f(mark(y0)), a__sel(mark(x0), mark(x1)))
A__G(s(X)) → MARK(X)
MARK(f(X)) → A__F(mark(X))
MARK(sel(X1, X2)) → MARK(X2)

Used ordering: Polynomial interpretation [25]:

POL(0) = 0
POL(A__F(x₁)) = 1
POL(A__G(x₁)) = x₁
POL(A__SEL(x₁, x₂)) = 1
POL(MARK(x₁)) = 1
POL(a__f(x₁)) = 0
POL(a__g(x₁)) = 1
POL(a__sel(x₁, x₂)) = 1
POL(cons(x₁, x₂)) = 0
POL(f(x₁)) = 0
POL(g(x₁)) = 0
POL(mark(x₁)) = 1
POL(s(x₁)) = 1
POL(sel(x₁, x₂)) = 0

The following usable rules [17] were oriented:

a__f(X) → cons(mark(X), f(g(X)))
a__g(0) → s(0)
mark(f(X)) → a__f(mark(X))
a__g(s(X)) → s(s(a__g(mark(X))))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0) → 0
mark(g(X)) → a__g(mark(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))
a__sel(X1, X2) → sel(X1, X2)
a__g(X) → g(X)
a__f(X) → f(X)
mark(s(X)) → s(mark(X))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ Narrowing

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ QDP

              ↳ Narrowing

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ QDP

                      ↳ Narrowing

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ QDP

                              ↳ Narrowing

                                ↳ QDP

                                  ↳ DependencyGraphProof

                                    ↳ QDP

                                      ↳ Narrowing

                                        ↳ QDP

                                          ↳ DependencyGraphProof

                                            ↳ QDP

                                              ↳ Narrowing

                                                ↳ QDP

                                                  ↳ DependencyGraphProof

                                                    ↳ QDP

                                                      ↳ Narrowing

                                                        ↳ QDP

                                                          ↳ DependencyGraphProof

                                                            ↳ QDP

                                                              ↳ Narrowing

                                                                ↳ QDP

                                                                  ↳ DependencyGraphProof

                                                                    ↳ QDP

                                                                      ↳ Narrowing

                                                                        ↳ QDP

                                                                          ↳ DependencyGraphProof

                                                                            ↳ QDP

                                                                              ↳ Narrowing

                                                                                ↳ QDP

                                                                                  ↳ DependencyGraphProof

                                                                                    ↳ QDP

                                                                                      ↳ Narrowing

                                                                                        ↳ QDP

                                                                                          ↳ DependencyGraphProof

                                                                                            ↳ QDP

                                                                                              ↳ Narrowing

                                                                                                ↳ QDP

                                                                                                  ↳ DependencyGraphProof

                                                                                                    ↳ QDP

                                                                                                      ↳ Narrowing

                                                                                                        ↳ QDP

                                                                                                          ↳ DependencyGraphProof

                                                                                                            ↳ QDP

                                                                                                              ↳ Narrowing

                                                                                                                ↳ QDP

                                                                                                                  ↳ DependencyGraphProof

                                                                                                                    ↳ QDP

                                                                                                                      ↳ QDPOrderProof

                                                                                                                        ↳ QDP

                                                                                                                          ↳ QDPOrderProof

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

MARK(sel(sel(y0, y1), sel(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), a__sel(mark(x0), mark(x1)))
MARK(sel(s(y0), cons(x0, x1))) → A__SEL(s(mark(y0)), cons(mark(x0), x1))
MARK(sel(g(y0), g(x0))) → A__SEL(a__g(mark(y0)), a__g(mark(x0)))
A__SEL(s(g(y0)), cons(y1, g(x0))) → A__SEL(a__g(mark(y0)), a__g(mark(x0)))
MARK(sel(f(y0), cons(x0, x1))) → A__SEL(a__f(mark(y0)), cons(mark(x0), x1))
MARK(cons(X1, X2)) → MARK(X1)
A__F(X) → MARK(X)
MARK(g(X)) → MARK(X)
A__SEL(s(X), cons(Y, Z)) → MARK(Z)
MARK(sel(0, g(x0))) → A__SEL(0, a__g(mark(x0)))
A__SEL(s(0), cons(y0, f(x0))) → A__SEL(0, a__f(mark(x0)))
MARK(g(g(x0))) → A__G(a__g(mark(x0)))
MARK(sel(s(y0), g(x0))) → A__SEL(s(mark(y0)), a__g(mark(x0)))
A__SEL(s(0), cons(y0, g(x0))) → A__SEL(0, a__g(mark(x0)))
MARK(f(X)) → 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(g(sel(x0, x1))) → A__G(a__sel(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), cons(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), cons(mark(x0), x1))
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__SEL(s(sel(y0, y1)), cons(y2, f(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__f(mark(x0)))
MARK(sel(s(y0), f(x0))) → A__SEL(s(mark(y0)), a__f(mark(x0)))
A__SEL(s(g(y0)), cons(y1, sel(x0, x1))) → A__SEL(a__g(mark(y0)), a__sel(mark(x0), mark(x1)))
MARK(sel(0, cons(x0, x1))) → A__SEL(0, cons(mark(x0), x1))
A__SEL(s(f(y0)), cons(y1, cons(x0, x1))) → A__SEL(a__f(mark(y0)), cons(mark(x0), x1))
A__SEL(s(0), cons(y0, sel(x0, x1))) → A__SEL(0, a__sel(mark(x0), mark(x1)))
MARK(sel(g(y0), cons(x0, x1))) → A__SEL(a__g(mark(y0)), cons(mark(x0), x1))
MARK(s(X)) → MARK(X)
A__SEL(s(s(y0)), cons(y1, g(x0))) → A__SEL(s(mark(y0)), a__g(mark(x0)))
MARK(sel(g(y0), sel(x0, x1))) → A__SEL(a__g(mark(y0)), a__sel(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), g(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__g(mark(x0)))
MARK(sel(s(y0), sel(x0, x1))) → A__SEL(s(mark(y0)), a__sel(mark(x0), mark(x1)))
A__SEL(0, cons(X, Y)) → MARK(X)
MARK(sel(0, f(x0))) → A__SEL(0, a__f(mark(x0)))
A__SEL(s(g(y0)), cons(y1, cons(x0, x1))) → A__SEL(a__g(mark(y0)), cons(mark(x0), x1))
A__G(s(sel(x0, x1))) → A__G(a__sel(mark(x0), mark(x1)))
A__SEL(s(0), cons(y0, cons(x0, x1))) → A__SEL(0, cons(mark(x0), x1))
A__SEL(s(sel(y0, y1)), cons(y2, g(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__g(mark(x0)))
A__SEL(s(s(y0)), cons(y1, f(x0))) → A__SEL(s(mark(y0)), a__f(mark(x0)))
A__G(s(g(x0))) → A__G(a__g(mark(x0)))
A__SEL(s(X), cons(Y, Z)) → MARK(X)
A__G(s(s(x0))) → A__G(s(mark(x0)))
A__SEL(s(f(y0)), cons(y1, g(x0))) → A__SEL(a__f(mark(y0)), a__g(mark(x0)))
A__SEL(s(g(y0)), cons(y1, f(x0))) → A__SEL(a__g(mark(y0)), a__f(mark(x0)))
A__SEL(s(f(y0)), cons(y1, f(x0))) → A__SEL(a__f(mark(y0)), a__f(mark(x0)))
MARK(sel(f(y0), sel(x0, x1))) → A__SEL(a__f(mark(y0)), a__sel(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), f(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__f(mark(x0)))
MARK(sel(f(y0), f(x0))) → A__SEL(a__f(mark(y0)), a__f(mark(x0)))
MARK(g(s(x0))) → A__G(s(mark(x0)))
MARK(sel(0, sel(x0, x1))) → A__SEL(0, a__sel(mark(x0), mark(x1)))
A__SEL(s(f(y0)), cons(y1, sel(x0, x1))) → A__SEL(a__f(mark(y0)), a__sel(mark(x0), mark(x1)))
MARK(sel(g(y0), f(x0))) → A__SEL(a__g(mark(y0)), a__f(mark(x0)))
MARK(sel(f(y0), g(x0))) → A__SEL(a__f(mark(y0)), a__g(mark(x0)))
A__G(s(X)) → MARK(X)
MARK(f(X)) → A__F(mark(X))
MARK(sel(X1, X2)) → MARK(X2)

The TRS R consists of the following rules:

a__f(X) → cons(mark(X), f(g(X)))
a__g(0) → s(0)
a__g(s(X)) → s(s(a__g(mark(X))))
a__sel(0, cons(X, Y)) → mark(X)
a__sel(s(X), cons(Y, Z)) → a__sel(mark(X), mark(Z))
mark(f(X)) → a__f(mark(X))
mark(g(X)) → a__g(mark(X))
mark(sel(X1, X2)) → a__sel(mark(X1), mark(X2))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0) → 0
mark(s(X)) → s(mark(X))
a__f(X) → f(X)
a__g(X) → g(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(g(y0), g(x0))) → A__SEL(a__g(mark(y0)), a__g(mark(x0)))
A__SEL(s(g(y0)), cons(y1, g(x0))) → A__SEL(a__g(mark(y0)), a__g(mark(x0)))
MARK(sel(0, g(x0))) → A__SEL(0, a__g(mark(x0)))
MARK(sel(s(y0), g(x0))) → A__SEL(s(mark(y0)), a__g(mark(x0)))
A__SEL(s(0), cons(y0, g(x0))) → A__SEL(0, a__g(mark(x0)))
A__SEL(s(s(y0)), cons(y1, g(x0))) → A__SEL(s(mark(y0)), a__g(mark(x0)))
MARK(sel(sel(y0, y1), g(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__g(mark(x0)))
A__SEL(s(sel(y0, y1)), cons(y2, g(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__g(mark(x0)))
A__SEL(s(f(y0)), cons(y1, g(x0))) → A__SEL(a__f(mark(y0)), a__g(mark(x0)))
MARK(sel(f(y0), g(x0))) → A__SEL(a__f(mark(y0)), a__g(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(sel(f(y0), cons(x0, x1))) → A__SEL(a__f(mark(y0)), cons(mark(x0), x1))
MARK(cons(X1, X2)) → MARK(X1)
A__F(X) → MARK(X)
MARK(g(X)) → MARK(X)
A__SEL(s(X), cons(Y, Z)) → MARK(Z)
A__SEL(s(0), cons(y0, f(x0))) → A__SEL(0, a__f(mark(x0)))
MARK(g(g(x0))) → A__G(a__g(mark(x0)))
MARK(f(X)) → 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(g(sel(x0, x1))) → A__G(a__sel(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), cons(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), cons(mark(x0), x1))
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__SEL(s(sel(y0, y1)), cons(y2, f(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__f(mark(x0)))
MARK(sel(s(y0), f(x0))) → A__SEL(s(mark(y0)), a__f(mark(x0)))
A__SEL(s(g(y0)), cons(y1, sel(x0, x1))) → A__SEL(a__g(mark(y0)), a__sel(mark(x0), mark(x1)))
MARK(sel(0, cons(x0, x1))) → A__SEL(0, cons(mark(x0), x1))
A__SEL(s(f(y0)), cons(y1, cons(x0, x1))) → A__SEL(a__f(mark(y0)), cons(mark(x0), x1))
A__SEL(s(0), cons(y0, sel(x0, x1))) → A__SEL(0, a__sel(mark(x0), mark(x1)))
MARK(sel(g(y0), cons(x0, x1))) → A__SEL(a__g(mark(y0)), cons(mark(x0), x1))
MARK(s(X)) → MARK(X)
MARK(sel(g(y0), sel(x0, x1))) → A__SEL(a__g(mark(y0)), a__sel(mark(x0), mark(x1)))
MARK(sel(s(y0), sel(x0, x1))) → A__SEL(s(mark(y0)), a__sel(mark(x0), mark(x1)))
A__SEL(0, cons(X, Y)) → MARK(X)
MARK(sel(0, f(x0))) → A__SEL(0, a__f(mark(x0)))
A__SEL(s(g(y0)), cons(y1, cons(x0, x1))) → A__SEL(a__g(mark(y0)), cons(mark(x0), x1))
A__G(s(sel(x0, x1))) → A__G(a__sel(mark(x0), mark(x1)))
A__SEL(s(0), cons(y0, cons(x0, x1))) → A__SEL(0, cons(mark(x0), x1))
A__SEL(s(s(y0)), cons(y1, f(x0))) → A__SEL(s(mark(y0)), a__f(mark(x0)))
A__G(s(g(x0))) → A__G(a__g(mark(x0)))
A__SEL(s(X), cons(Y, Z)) → MARK(X)
A__G(s(s(x0))) → A__G(s(mark(x0)))
A__SEL(s(g(y0)), cons(y1, f(x0))) → A__SEL(a__g(mark(y0)), a__f(mark(x0)))
A__SEL(s(f(y0)), cons(y1, f(x0))) → A__SEL(a__f(mark(y0)), a__f(mark(x0)))
MARK(sel(f(y0), sel(x0, x1))) → A__SEL(a__f(mark(y0)), a__sel(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), f(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__f(mark(x0)))
MARK(sel(f(y0), f(x0))) → A__SEL(a__f(mark(y0)), a__f(mark(x0)))
MARK(g(s(x0))) → A__G(s(mark(x0)))
MARK(sel(0, sel(x0, x1))) → A__SEL(0, a__sel(mark(x0), mark(x1)))
A__SEL(s(f(y0)), cons(y1, sel(x0, x1))) → A__SEL(a__f(mark(y0)), a__sel(mark(x0), mark(x1)))
MARK(sel(g(y0), f(x0))) → A__SEL(a__g(mark(y0)), a__f(mark(x0)))
A__G(s(X)) → MARK(X)
MARK(f(X)) → A__F(mark(X))
MARK(sel(X1, X2)) → MARK(X2)

Used ordering: Polynomial interpretation [25]:

POL(0) = 0
POL(A__F(x₁)) = 1
POL(A__G(x₁)) = 1
POL(A__SEL(x₁, x₂)) = x₂
POL(MARK(x₁)) = 1
POL(a__f(x₁)) = 1
POL(a__g(x₁)) = 0
POL(a__sel(x₁, x₂)) = 1
POL(cons(x₁, x₂)) = 1
POL(f(x₁)) = 0
POL(g(x₁)) = 0
POL(mark(x₁)) = 1
POL(s(x₁)) = 0
POL(sel(x₁, x₂)) = 0

The following usable rules [17] were oriented:

a__f(X) → cons(mark(X), f(g(X)))
a__g(0) → s(0)
mark(f(X)) → a__f(mark(X))
a__g(s(X)) → s(s(a__g(mark(X))))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0) → 0
mark(g(X)) → a__g(mark(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))
a__sel(X1, X2) → sel(X1, X2)
a__g(X) → g(X)
a__f(X) → f(X)
mark(s(X)) → s(mark(X))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ Narrowing

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ QDP

              ↳ Narrowing

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ QDP

                      ↳ Narrowing

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ QDP

                              ↳ Narrowing

                                ↳ QDP

                                  ↳ DependencyGraphProof

                                    ↳ QDP

                                      ↳ Narrowing

                                        ↳ QDP

                                          ↳ DependencyGraphProof

                                            ↳ QDP

                                              ↳ Narrowing

                                                ↳ QDP

                                                  ↳ DependencyGraphProof

                                                    ↳ QDP

                                                      ↳ Narrowing

                                                        ↳ QDP

                                                          ↳ DependencyGraphProof

                                                            ↳ QDP

                                                              ↳ Narrowing

                                                                ↳ QDP

                                                                  ↳ DependencyGraphProof

                                                                    ↳ QDP

                                                                      ↳ Narrowing

                                                                        ↳ QDP

                                                                          ↳ DependencyGraphProof

                                                                            ↳ QDP

                                                                              ↳ Narrowing

                                                                                ↳ QDP

                                                                                  ↳ DependencyGraphProof

                                                                                    ↳ QDP

                                                                                      ↳ Narrowing

                                                                                        ↳ QDP

                                                                                          ↳ DependencyGraphProof

                                                                                            ↳ QDP

                                                                                              ↳ Narrowing

                                                                                                ↳ QDP

                                                                                                  ↳ DependencyGraphProof

                                                                                                    ↳ QDP

                                                                                                      ↳ Narrowing

                                                                                                        ↳ QDP

                                                                                                          ↳ DependencyGraphProof

                                                                                                            ↳ QDP

                                                                                                              ↳ Narrowing

                                                                                                                ↳ QDP

                                                                                                                  ↳ DependencyGraphProof

                                                                                                                    ↳ QDP

                                                                                                                      ↳ QDPOrderProof

                                                                                                                        ↳ QDP

                                                                                                                          ↳ QDPOrderProof

                                                                                                                            ↳ QDP

                                                                                                                              ↳ QDPOrderProof

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

MARK(sel(sel(y0, y1), sel(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), a__sel(mark(x0), mark(x1)))
MARK(sel(g(y0), cons(x0, x1))) → A__SEL(a__g(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(f(y0), cons(x0, x1))) → A__SEL(a__f(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(sel(g(y0), sel(x0, x1))) → A__SEL(a__g(mark(y0)), a__sel(mark(x0), mark(x1)))
MARK(cons(X1, X2)) → MARK(X1)
A__F(X) → MARK(X)
A__SEL(0, cons(X, Y)) → MARK(X)
MARK(g(X)) → MARK(X)
MARK(sel(0, f(x0))) → A__SEL(0, a__f(mark(x0)))
A__SEL(s(X), cons(Y, Z)) → MARK(Z)
A__SEL(s(g(y0)), cons(y1, cons(x0, x1))) → A__SEL(a__g(mark(y0)), cons(mark(x0), x1))
A__SEL(s(0), cons(y0, f(x0))) → A__SEL(0, a__f(mark(x0)))
A__G(s(sel(x0, x1))) → A__G(a__sel(mark(x0), mark(x1)))
A__SEL(s(0), cons(y0, cons(x0, x1))) → A__SEL(0, cons(mark(x0), x1))
A__SEL(s(s(y0)), cons(y1, f(x0))) → A__SEL(s(mark(y0)), a__f(mark(x0)))
MARK(g(g(x0))) → A__G(a__g(mark(x0)))
A__G(s(g(x0))) → A__G(a__g(mark(x0)))
A__SEL(s(X), cons(Y, Z)) → MARK(X)
A__G(s(s(x0))) → A__G(s(mark(x0)))
MARK(f(X)) → 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(g(sel(x0, x1))) → A__G(a__sel(mark(x0), mark(x1)))
A__SEL(s(g(y0)), cons(y1, f(x0))) → A__SEL(a__g(mark(y0)), a__f(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(f(y0)), cons(y1, f(x0))) → A__SEL(a__f(mark(y0)), a__f(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(s(y0)), cons(y1, cons(x0, x1))) → A__SEL(s(mark(y0)), cons(mark(x0), x1))
A__SEL(s(sel(y0, y1)), cons(y2, f(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__f(mark(x0)))
MARK(sel(sel(y0, y1), f(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__f(mark(x0)))
MARK(sel(f(y0), sel(x0, x1))) → A__SEL(a__f(mark(y0)), a__sel(mark(x0), mark(x1)))
MARK(sel(f(y0), f(x0))) → A__SEL(a__f(mark(y0)), a__f(mark(x0)))
MARK(sel(s(y0), f(x0))) → A__SEL(s(mark(y0)), a__f(mark(x0)))
A__SEL(s(g(y0)), cons(y1, sel(x0, x1))) → A__SEL(a__g(mark(y0)), a__sel(mark(x0), mark(x1)))
MARK(sel(0, sel(x0, x1))) → A__SEL(0, a__sel(mark(x0), mark(x1)))
MARK(g(s(x0))) → A__G(s(mark(x0)))
MARK(sel(0, cons(x0, x1))) → A__SEL(0, cons(mark(x0), x1))
MARK(sel(g(y0), f(x0))) → A__SEL(a__g(mark(y0)), a__f(mark(x0)))
A__SEL(s(f(y0)), cons(y1, sel(x0, x1))) → A__SEL(a__f(mark(y0)), a__sel(mark(x0), mark(x1)))
A__SEL(s(f(y0)), cons(y1, cons(x0, x1))) → A__SEL(a__f(mark(y0)), cons(mark(x0), x1))
A__G(s(X)) → MARK(X)
MARK(f(X)) → A__F(mark(X))
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__f(X) → cons(mark(X), f(g(X)))
a__g(0) → s(0)
a__g(s(X)) → s(s(a__g(mark(X))))
a__sel(0, cons(X, Y)) → mark(X)
a__sel(s(X), cons(Y, Z)) → a__sel(mark(X), mark(Z))
mark(f(X)) → a__f(mark(X))
mark(g(X)) → a__g(mark(X))
mark(sel(X1, X2)) → a__sel(mark(X1), mark(X2))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0) → 0
mark(s(X)) → s(mark(X))
a__f(X) → f(X)
a__g(X) → g(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(f(y0), cons(x0, x1))) → A__SEL(a__f(mark(y0)), cons(mark(x0), x1))
A__SEL(s(f(y0)), cons(y1, f(x0))) → A__SEL(a__f(mark(y0)), a__f(mark(x0)))
MARK(sel(f(y0), sel(x0, x1))) → A__SEL(a__f(mark(y0)), a__sel(mark(x0), mark(x1)))
MARK(sel(f(y0), f(x0))) → A__SEL(a__f(mark(y0)), a__f(mark(x0)))
A__SEL(s(f(y0)), cons(y1, sel(x0, x1))) → A__SEL(a__f(mark(y0)), a__sel(mark(x0), mark(x1)))
A__SEL(s(f(y0)), cons(y1, cons(x0, x1))) → A__SEL(a__f(mark(y0)), cons(mark(x0), x1))

The remaining pairs can at least be oriented weakly.

MARK(sel(sel(y0, y1), sel(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), a__sel(mark(x0), mark(x1)))
MARK(sel(g(y0), cons(x0, x1))) → A__SEL(a__g(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(sel(g(y0), sel(x0, x1))) → A__SEL(a__g(mark(y0)), a__sel(mark(x0), mark(x1)))
MARK(cons(X1, X2)) → MARK(X1)
A__F(X) → MARK(X)
A__SEL(0, cons(X, Y)) → MARK(X)
MARK(g(X)) → MARK(X)
MARK(sel(0, f(x0))) → A__SEL(0, a__f(mark(x0)))
A__SEL(s(X), cons(Y, Z)) → MARK(Z)
A__SEL(s(g(y0)), cons(y1, cons(x0, x1))) → A__SEL(a__g(mark(y0)), cons(mark(x0), x1))
A__SEL(s(0), cons(y0, f(x0))) → A__SEL(0, a__f(mark(x0)))
A__G(s(sel(x0, x1))) → A__G(a__sel(mark(x0), mark(x1)))
A__SEL(s(0), cons(y0, cons(x0, x1))) → A__SEL(0, cons(mark(x0), x1))
A__SEL(s(s(y0)), cons(y1, f(x0))) → A__SEL(s(mark(y0)), a__f(mark(x0)))
MARK(g(g(x0))) → A__G(a__g(mark(x0)))
A__G(s(g(x0))) → A__G(a__g(mark(x0)))
A__SEL(s(X), cons(Y, Z)) → MARK(X)
A__G(s(s(x0))) → A__G(s(mark(x0)))
MARK(f(X)) → 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(g(sel(x0, x1))) → A__G(a__sel(mark(x0), mark(x1)))
A__SEL(s(g(y0)), cons(y1, f(x0))) → A__SEL(a__g(mark(y0)), a__f(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__SEL(s(sel(y0, y1)), cons(y2, f(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__f(mark(x0)))
MARK(sel(sel(y0, y1), f(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__f(mark(x0)))
MARK(sel(s(y0), f(x0))) → A__SEL(s(mark(y0)), a__f(mark(x0)))
A__SEL(s(g(y0)), cons(y1, sel(x0, x1))) → A__SEL(a__g(mark(y0)), a__sel(mark(x0), mark(x1)))
MARK(sel(0, sel(x0, x1))) → A__SEL(0, a__sel(mark(x0), mark(x1)))
MARK(g(s(x0))) → A__G(s(mark(x0)))
MARK(sel(0, cons(x0, x1))) → A__SEL(0, cons(mark(x0), x1))
MARK(sel(g(y0), f(x0))) → A__SEL(a__g(mark(y0)), a__f(mark(x0)))
A__G(s(X)) → MARK(X)
MARK(f(X)) → A__F(mark(X))
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__F(x₁)) = 1
POL(A__G(x₁)) = 1
POL(A__SEL(x₁, x₂)) = x₁
POL(MARK(x₁)) = 1
POL(a__f(x₁)) = 0
POL(a__g(x₁)) = x₁
POL(a__sel(x₁, x₂)) = 1
POL(cons(x₁, x₂)) = 0
POL(f(x₁)) = 0
POL(g(x₁)) = 0
POL(mark(x₁)) = 1
POL(s(x₁)) = 1
POL(sel(x₁, x₂)) = 0

The following usable rules [17] were oriented:

a__f(X) → cons(mark(X), f(g(X)))
a__g(0) → s(0)
mark(f(X)) → a__f(mark(X))
a__g(s(X)) → s(s(a__g(mark(X))))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0) → 0
mark(g(X)) → a__g(mark(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))
a__sel(X1, X2) → sel(X1, X2)
a__g(X) → g(X)
a__f(X) → f(X)
mark(s(X)) → s(mark(X))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ Narrowing

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ QDP

              ↳ Narrowing

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ QDP

                      ↳ Narrowing

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ QDP

                              ↳ Narrowing

                                ↳ QDP

                                  ↳ DependencyGraphProof

                                    ↳ QDP

                                      ↳ Narrowing

                                        ↳ QDP

                                          ↳ DependencyGraphProof

                                            ↳ QDP

                                              ↳ Narrowing

                                                ↳ QDP

                                                  ↳ DependencyGraphProof

                                                    ↳ QDP

                                                      ↳ Narrowing

                                                        ↳ QDP

                                                          ↳ DependencyGraphProof

                                                            ↳ QDP

                                                              ↳ Narrowing

                                                                ↳ QDP

                                                                  ↳ DependencyGraphProof

                                                                    ↳ QDP

                                                                      ↳ Narrowing

                                                                        ↳ QDP

                                                                          ↳ DependencyGraphProof

                                                                            ↳ QDP

                                                                              ↳ Narrowing

                                                                                ↳ QDP

                                                                                  ↳ DependencyGraphProof

                                                                                    ↳ QDP

                                                                                      ↳ Narrowing

                                                                                        ↳ QDP

                                                                                          ↳ DependencyGraphProof

                                                                                            ↳ QDP

                                                                                              ↳ Narrowing

                                                                                                ↳ QDP

                                                                                                  ↳ DependencyGraphProof

                                                                                                    ↳ QDP

                                                                                                      ↳ Narrowing

                                                                                                        ↳ QDP

                                                                                                          ↳ DependencyGraphProof

                                                                                                            ↳ QDP

                                                                                                              ↳ Narrowing

                                                                                                                ↳ QDP

                                                                                                                  ↳ DependencyGraphProof

                                                                                                                    ↳ QDP

                                                                                                                      ↳ QDPOrderProof

                                                                                                                        ↳ QDP

                                                                                                                          ↳ QDPOrderProof

                                                                                                                            ↳ QDP

                                                                                                                              ↳ QDPOrderProof

                                                                                                                                ↳ QDP

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

MARK(sel(g(y0), cons(x0, x1))) → A__SEL(a__g(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(s(X)) → MARK(X)
MARK(sel(s(y0), cons(x0, x1))) → A__SEL(s(mark(y0)), cons(mark(x0), x1))
MARK(sel(g(y0), sel(x0, x1))) → A__SEL(a__g(mark(y0)), a__sel(mark(x0), mark(x1)))
MARK(sel(s(y0), sel(x0, x1))) → A__SEL(s(mark(y0)), a__sel(mark(x0), mark(x1)))
MARK(cons(X1, X2)) → MARK(X1)
A__F(X) → MARK(X)
A__SEL(0, cons(X, Y)) → MARK(X)
MARK(g(X)) → MARK(X)
A__SEL(s(X), cons(Y, Z)) → MARK(Z)
MARK(sel(0, f(x0))) → A__SEL(0, a__f(mark(x0)))
A__SEL(s(g(y0)), cons(y1, cons(x0, x1))) → A__SEL(a__g(mark(y0)), cons(mark(x0), x1))
A__SEL(s(0), cons(y0, f(x0))) → A__SEL(0, a__f(mark(x0)))
A__G(s(sel(x0, x1))) → A__G(a__sel(mark(x0), mark(x1)))
A__SEL(s(0), cons(y0, cons(x0, x1))) → A__SEL(0, cons(mark(x0), x1))
A__SEL(s(s(y0)), cons(y1, f(x0))) → A__SEL(s(mark(y0)), a__f(mark(x0)))
MARK(g(g(x0))) → A__G(a__g(mark(x0)))
A__G(s(g(x0))) → A__G(a__g(mark(x0)))
A__SEL(s(X), cons(Y, Z)) → MARK(X)
A__G(s(s(x0))) → A__G(s(mark(x0)))
MARK(f(X)) → 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(g(sel(x0, x1))) → A__G(a__sel(mark(x0), mark(x1)))
A__SEL(s(g(y0)), cons(y1, f(x0))) → A__SEL(a__g(mark(y0)), a__f(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, f(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__f(mark(x0)))
A__SEL(s(s(y0)), cons(y1, cons(x0, x1))) → A__SEL(s(mark(y0)), cons(mark(x0), x1))
MARK(sel(sel(y0, y1), f(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__f(mark(x0)))
MARK(sel(s(y0), f(x0))) → A__SEL(s(mark(y0)), a__f(mark(x0)))
A__SEL(s(g(y0)), cons(y1, sel(x0, x1))) → A__SEL(a__g(mark(y0)), a__sel(mark(x0), mark(x1)))
MARK(g(s(x0))) → A__G(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(g(y0), f(x0))) → A__SEL(a__g(mark(y0)), a__f(mark(x0)))
A__G(s(X)) → MARK(X)
MARK(f(X)) → A__F(mark(X))
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__f(X) → cons(mark(X), f(g(X)))
a__g(0) → s(0)
a__g(s(X)) → s(s(a__g(mark(X))))
a__sel(0, cons(X, Y)) → mark(X)
a__sel(s(X), cons(Y, Z)) → a__sel(mark(X), mark(Z))
mark(f(X)) → a__f(mark(X))
mark(g(X)) → a__g(mark(X))
mark(sel(X1, X2)) → a__sel(mark(X1), mark(X2))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(0) → 0
mark(s(X)) → s(mark(X))
a__f(X) → f(X)
a__g(X) → g(X)
a__sel(X1, X2) → sel(X1, X2)

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