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
      ↳ QDPOrderProof

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.
We use the reduction pair processor [15].


The following pairs can be oriented strictly and are deleted.


A__SEL(0, cons(X, Y)) → MARK(X)
The remaining pairs can at least be oriented weakly.

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

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

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
      ↳ QDPOrderProof
        ↳ QDP
          ↳ QDPOrderProof

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)
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))
A__SEL(s(X), cons(Y, Z)) → MARK(X)
MARK(sel(X1, X2)) → MARK(X2)
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.
We use the reduction pair processor [15].


The following pairs can be oriented strictly and are deleted.


MARK(sel(X1, X2)) → MARK(X1)
A__SEL(s(X), cons(Y, Z)) → MARK(Z)
A__SEL(s(X), cons(Y, Z)) → MARK(X)
MARK(sel(X1, X2)) → MARK(X2)
The remaining pairs can at least be oriented weakly.

MARK(g(X)) → A__G(mark(X))
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)
MARK(g(X)) → MARK(X)
A__G(s(X)) → MARK(X)
MARK(f(X)) → A__F(mark(X))
MARK(f(X)) → MARK(X)
Used ordering: Polynomial interpretation with max and min functions [25]:

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

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
      ↳ QDPOrderProof
        ↳ QDP
          ↳ QDPOrderProof
            ↳ QDP
              ↳ DependencyGraphProof

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

MARK(g(X)) → A__G(mark(X))
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))
A__G(s(X)) → MARK(X)
MARK(cons(X1, X2)) → MARK(X1)
MARK(f(X)) → A__F(mark(X))
A__F(X) → MARK(X)
MARK(f(X)) → MARK(X)
MARK(g(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 2 SCCs with 1 less node.

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ QDPOrderProof
        ↳ QDP
          ↳ QDPOrderProof
            ↳ QDP
              ↳ DependencyGraphProof
                ↳ AND
                  ↳ QDP
                    ↳ Narrowing
                  ↳ QDP

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

A__SEL(s(X), cons(Y, Z)) → A__SEL(mark(X), mark(Z))

The TRS R consists of the following rules:

a__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
      ↳ QDPOrderProof
        ↳ QDP
          ↳ QDPOrderProof
            ↳ QDP
              ↳ DependencyGraphProof
                ↳ AND
                  ↳ QDP
                    ↳ Narrowing
                      ↳ QDP
                        ↳ DependencyGraphProof
                  ↳ QDP

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

A__SEL(s(sel(x0, x1)), cons(y1, y2)) → A__SEL(a__sel(mark(x0), mark(x1)), mark(y2))
A__SEL(s(0), cons(y1, y2)) → A__SEL(0, mark(y2))
A__SEL(s(s(x0)), cons(y1, y2)) → A__SEL(s(mark(x0)), mark(y2))
A__SEL(s(g(x0)), cons(y1, y2)) → A__SEL(a__g(mark(x0)), mark(y2))
A__SEL(s(f(x0)), cons(y1, y2)) → A__SEL(a__f(mark(x0)), mark(y2))
A__SEL(s(cons(x0, x1)), cons(y1, y2)) → A__SEL(cons(mark(x0), x1), mark(y2))

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
      ↳ QDPOrderProof
        ↳ QDP
          ↳ QDPOrderProof
            ↳ QDP
              ↳ DependencyGraphProof
                ↳ AND
                  ↳ QDP
                    ↳ Narrowing
                      ↳ QDP
                        ↳ DependencyGraphProof
                          ↳ QDP
                            ↳ Narrowing
                  ↳ QDP

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

A__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(f(x0)), cons(y1, y2)) → A__SEL(a__f(mark(x0)), mark(y2))

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
      ↳ QDPOrderProof
        ↳ QDP
          ↳ QDPOrderProof
            ↳ QDP
              ↳ DependencyGraphProof
                ↳ AND
                  ↳ QDP
                    ↳ Narrowing
                      ↳ QDP
                        ↳ DependencyGraphProof
                          ↳ QDP
                            ↳ Narrowing
                              ↳ QDP
                                ↳ DependencyGraphProof
                  ↳ QDP

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

A__SEL(s(sel(x0, x1)), cons(y1, y2)) → A__SEL(a__sel(mark(x0), mark(x1)), mark(y2))
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, 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__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, 0)) → A__SEL(a__f(mark(y0)), 0)
A__SEL(s(f(y0)), cons(y1, s(x0))) → A__SEL(a__f(mark(y0)), s(mark(x0)))

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
      ↳ QDPOrderProof
        ↳ QDP
          ↳ QDPOrderProof
            ↳ QDP
              ↳ DependencyGraphProof
                ↳ AND
                  ↳ QDP
                    ↳ Narrowing
                      ↳ QDP
                        ↳ DependencyGraphProof
                          ↳ QDP
                            ↳ Narrowing
                              ↳ QDP
                                ↳ DependencyGraphProof
                                  ↳ QDP
                                    ↳ Narrowing
                  ↳ QDP

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

A__SEL(s(sel(x0, x1)), cons(y1, y2)) → A__SEL(a__sel(mark(x0), mark(x1)), mark(y2))
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, 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__SEL(s(f(y0)), cons(y1, f(x0))) → A__SEL(a__f(mark(y0)), a__f(mark(x0)))

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
      ↳ QDPOrderProof
        ↳ QDP
          ↳ QDPOrderProof
            ↳ QDP
              ↳ DependencyGraphProof
                ↳ AND
                  ↳ QDP
                    ↳ Narrowing
                      ↳ QDP
                        ↳ DependencyGraphProof
                          ↳ QDP
                            ↳ Narrowing
                              ↳ QDP
                                ↳ DependencyGraphProof
                                  ↳ QDP
                                    ↳ Narrowing
                                      ↳ QDP
                                        ↳ DependencyGraphProof
                  ↳ QDP

Q DP problem:
The TRS P consists of the following 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(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, 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(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__SEL(s(sel(y0, y1)), cons(y2, g(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__g(mark(x0)))

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
      ↳ QDPOrderProof
        ↳ QDP
          ↳ QDPOrderProof
            ↳ QDP
              ↳ DependencyGraphProof
                ↳ AND
                  ↳ QDP
                    ↳ Narrowing
                      ↳ QDP
                        ↳ DependencyGraphProof
                          ↳ QDP
                            ↳ Narrowing
                              ↳ QDP
                                ↳ DependencyGraphProof
                                  ↳ QDP
                                    ↳ Narrowing
                                      ↳ QDP
                                        ↳ DependencyGraphProof
                                          ↳ QDP
                                            ↳ Narrowing
                  ↳ QDP

Q DP problem:
The TRS P consists of the following 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(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, 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__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, g(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__g(mark(x0)))
A__SEL(s(sel(y0, y1)), cons(y2, cons(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), cons(mark(x0), x1))
A__SEL(s(sel(y0, y1)), cons(y2, f(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__f(mark(x0)))

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
      ↳ QDPOrderProof
        ↳ QDP
          ↳ QDPOrderProof
            ↳ QDP
              ↳ DependencyGraphProof
                ↳ AND
                  ↳ QDP
                    ↳ Narrowing
                      ↳ QDP
                        ↳ DependencyGraphProof
                          ↳ QDP
                            ↳ Narrowing
                              ↳ QDP
                                ↳ DependencyGraphProof
                                  ↳ QDP
                                    ↳ Narrowing
                                      ↳ QDP
                                        ↳ DependencyGraphProof
                                          ↳ QDP
                                            ↳ Narrowing
                                              ↳ QDP
                                                ↳ DependencyGraphProof
                  ↳ QDP

Q DP problem:
The TRS P consists of the following 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(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(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(s(y0)), cons(y1, g(x0))) → A__SEL(s(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)))
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)))
A__SEL(s(s(y0)), cons(y1, 0)) → A__SEL(s(mark(y0)), 0)
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__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)))

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
      ↳ QDPOrderProof
        ↳ QDP
          ↳ QDPOrderProof
            ↳ QDP
              ↳ DependencyGraphProof
                ↳ AND
                  ↳ QDP
                    ↳ Narrowing
                      ↳ QDP
                        ↳ DependencyGraphProof
                          ↳ QDP
                            ↳ Narrowing
                              ↳ QDP
                                ↳ DependencyGraphProof
                                  ↳ QDP
                                    ↳ Narrowing
                                      ↳ QDP
                                        ↳ DependencyGraphProof
                                          ↳ QDP
                                            ↳ Narrowing
                                              ↳ QDP
                                                ↳ DependencyGraphProof
                                                  ↳ QDP
                                                    ↳ Narrowing
                  ↳ QDP

Q DP problem:
The TRS P consists of the following 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(s(y0)), cons(y1, sel(x0, x1))) → A__SEL(s(mark(y0)), 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(g(x0)), cons(y1, y2)) → A__SEL(a__g(mark(x0)), mark(y2))
A__SEL(s(s(y0)), cons(y1, g(x0))) → A__SEL(s(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)))
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))
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(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)))

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
      ↳ QDPOrderProof
        ↳ QDP
          ↳ QDPOrderProof
            ↳ QDP
              ↳ DependencyGraphProof
                ↳ AND
                  ↳ QDP
                    ↳ Narrowing
                      ↳ QDP
                        ↳ DependencyGraphProof
                          ↳ QDP
                            ↳ Narrowing
                              ↳ QDP
                                ↳ DependencyGraphProof
                                  ↳ QDP
                                    ↳ Narrowing
                                      ↳ QDP
                                        ↳ DependencyGraphProof
                                          ↳ QDP
                                            ↳ Narrowing
                                              ↳ QDP
                                                ↳ DependencyGraphProof
                                                  ↳ QDP
                                                    ↳ Narrowing
                                                      ↳ QDP
                                                        ↳ DependencyGraphProof
                  ↳ QDP

Q DP problem:
The TRS P consists of the following 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(s(y0)), cons(y1, sel(x0, x1))) → A__SEL(s(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(f(y0)), cons(y1, g(x0))) → A__SEL(a__f(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)))
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)))
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)))
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, cons(x0, x1))) → A__SEL(a__g(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, cons(x0, x1))) → A__SEL(a__f(mark(y0)), 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__SEL(s(g(y0)), cons(y1, 0)) → A__SEL(a__g(mark(y0)), 0)

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
      ↳ QDPOrderProof
        ↳ QDP
          ↳ QDPOrderProof
            ↳ QDP
              ↳ DependencyGraphProof
                ↳ AND
                  ↳ QDP
                    ↳ Narrowing
                      ↳ QDP
                        ↳ DependencyGraphProof
                          ↳ QDP
                            ↳ Narrowing
                              ↳ QDP
                                ↳ DependencyGraphProof
                                  ↳ QDP
                                    ↳ Narrowing
                                      ↳ QDP
                                        ↳ DependencyGraphProof
                                          ↳ QDP
                                            ↳ Narrowing
                                              ↳ QDP
                                                ↳ DependencyGraphProof
                                                  ↳ QDP
                                                    ↳ Narrowing
                                                      ↳ QDP
                                                        ↳ DependencyGraphProof
                                                          ↳ QDP
                                                            ↳ QDPOrderProof
                  ↳ QDP

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

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

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.


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)))
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)))
The remaining pairs can at least be oriented weakly.

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__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, g(x0))) → A__SEL(a__g(mark(y0)), a__g(mark(x0)))
A__SEL(s(s(y0)), cons(y1, g(x0))) → A__SEL(s(mark(y0)), a__g(mark(x0)))
A__SEL(s(sel(y0, y1)), cons(y2, cons(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), cons(mark(x0), x1))
A__SEL(s(sel(y0, y1)), cons(y2, 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))
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, cons(x0, x1))) → A__SEL(a__g(mark(y0)), 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)))
Used ordering: Polynomial interpretation [25]:

POL(0) = 1   
POL(A__SEL(x1, x2)) = x1   
POL(a__f(x1)) = 0   
POL(a__g(x1)) = 1   
POL(a__sel(x1, x2)) = x1   
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
      ↳ QDPOrderProof
        ↳ QDP
          ↳ QDPOrderProof
            ↳ QDP
              ↳ DependencyGraphProof
                ↳ AND
                  ↳ QDP
                    ↳ Narrowing
                      ↳ QDP
                        ↳ DependencyGraphProof
                          ↳ QDP
                            ↳ Narrowing
                              ↳ QDP
                                ↳ DependencyGraphProof
                                  ↳ QDP
                                    ↳ Narrowing
                                      ↳ QDP
                                        ↳ DependencyGraphProof
                                          ↳ QDP
                                            ↳ Narrowing
                                              ↳ QDP
                                                ↳ DependencyGraphProof
                                                  ↳ QDP
                                                    ↳ Narrowing
                                                      ↳ QDP
                                                        ↳ DependencyGraphProof
                                                          ↳ QDP
                                                            ↳ QDPOrderProof
                                                              ↳ QDP
                                                                ↳ QDPOrderProof
                  ↳ QDP

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

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__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(s(y0)), cons(y1, g(x0))) → A__SEL(s(mark(y0)), 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(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, 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(sel(y0, y1)), cons(y2, f(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__f(mark(x0)))

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.


A__SEL(s(g(y0)), cons(y1, g(x0))) → A__SEL(a__g(mark(y0)), a__g(mark(x0)))
A__SEL(s(s(y0)), cons(y1, g(x0))) → A__SEL(s(mark(y0)), 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)))
The remaining pairs can at least be oriented weakly.

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__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(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, 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(sel(y0, y1)), cons(y2, f(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__f(mark(x0)))
Used ordering: Polynomial interpretation [25]:

POL(0) = 0   
POL(A__SEL(x1, x2)) = x2   
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
      ↳ QDPOrderProof
        ↳ QDP
          ↳ QDPOrderProof
            ↳ QDP
              ↳ DependencyGraphProof
                ↳ AND
                  ↳ 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:

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__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(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, f(x0))) → A__SEL(s(mark(y0)), a__f(mark(x0)))
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))

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.


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__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, cons(x0, x1))) → A__SEL(a__g(mark(y0)), 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))
The remaining pairs can at least be oriented weakly.

A__SEL(s(g(y0)), cons(y1, f(x0))) → A__SEL(a__g(mark(y0)), a__f(mark(x0)))
A__SEL(s(s(y0)), cons(y1, f(x0))) → A__SEL(s(mark(y0)), a__f(mark(x0)))
A__SEL(s(sel(y0, y1)), cons(y2, f(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__f(mark(x0)))
Used ordering: Matrix interpretation [3]:
Non-tuple symbols:
M( sel(x1, x2) ) =
/1\
\1/
+
/00\
\00/
·x1+
/01\
\01/
·x2

M( cons(x1, x2) ) =
/1\
\0/
+
/00\
\01/
·x1+
/10\
\11/
·x2

M( f(x1) ) =
/0\
\0/
+
/00\
\01/
·x1

M( a__f(x1) ) =
/1\
\0/
+
/00\
\01/
·x1

M( g(x1) ) =
/0\
\0/
+
/00\
\00/
·x1

M( a__sel(x1, x2) ) =
/1\
\1/
+
/00\
\00/
·x1+
/01\
\01/
·x2

M( a__g(x1) ) =
/0\
\0/
+
/00\
\00/
·x1

M( mark(x1) ) =
/1\
\0/
+
/01\
\01/
·x1

M( s(x1) ) =
/0\
\0/
+
/00\
\00/
·x1

M( 0 ) =
/0\
\0/

Tuple symbols:
M( A__SEL(x1, x2) ) = 0+
[0,0]
·x1+
[1,0]
·x2


Matrix type:
We used a basic matrix type which is not further parametrizeable.


As matrix orders are CE-compatible, we used usable rules w.r.t. argument filtering in the order.
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
      ↳ QDPOrderProof
        ↳ QDP
          ↳ QDPOrderProof
            ↳ QDP
              ↳ DependencyGraphProof
                ↳ AND
                  ↳ 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
                                                                        ↳ QDPOrderProof
                  ↳ QDP

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

A__SEL(s(g(y0)), cons(y1, f(x0))) → A__SEL(a__g(mark(y0)), a__f(mark(x0)))
A__SEL(s(s(y0)), cons(y1, f(x0))) → A__SEL(s(mark(y0)), a__f(mark(x0)))
A__SEL(s(sel(y0, y1)), cons(y2, f(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__f(mark(x0)))

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.


A__SEL(s(sel(y0, y1)), cons(y2, f(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__f(mark(x0)))
The remaining pairs can at least be oriented weakly.

A__SEL(s(g(y0)), cons(y1, f(x0))) → A__SEL(a__g(mark(y0)), a__f(mark(x0)))
A__SEL(s(s(y0)), cons(y1, f(x0))) → A__SEL(s(mark(y0)), a__f(mark(x0)))
Used ordering: Matrix interpretation [3]:
Non-tuple symbols:
M( sel(x1, x2) ) =
/1\
\0/
+
/00\
\00/
·x1+
/01\
\01/
·x2

M( cons(x1, x2) ) =
/0\
\0/
+
/00\
\11/
·x1+
/00\
\01/
·x2

M( f(x1) ) =
/0\
\0/
+
/00\
\11/
·x1

M( g(x1) ) =
/0\
\0/
+
/00\
\00/
·x1

M( a__f(x1) ) =
/0\
\0/
+
/00\
\11/
·x1

M( a__sel(x1, x2) ) =
/1\
\0/
+
/00\
\00/
·x1+
/01\
\01/
·x2

M( a__g(x1) ) =
/0\
\0/
+
/00\
\00/
·x1

M( mark(x1) ) =
/0\
\0/
+
/10\
\01/
·x1

M( s(x1) ) =
/0\
\0/
+
/10\
\10/
·x1

M( 0 ) =
/0\
\0/

Tuple symbols:
M( A__SEL(x1, x2) ) = 0+
[0,1]
·x1+
[0,0]
·x2


Matrix type:
We used a basic matrix type which is not further parametrizeable.


As matrix orders are CE-compatible, we used usable rules w.r.t. argument filtering in the order.
The following usable rules [17] were oriented:

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



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ QDPOrderProof
        ↳ QDP
          ↳ QDPOrderProof
            ↳ QDP
              ↳ DependencyGraphProof
                ↳ AND
                  ↳ 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
                                                                        ↳ QDPOrderProof
                                                                          ↳ QDP
                  ↳ QDP

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

A__SEL(s(g(y0)), cons(y1, f(x0))) → A__SEL(a__g(mark(y0)), a__f(mark(x0)))
A__SEL(s(s(y0)), cons(y1, f(x0))) → A__SEL(s(mark(y0)), a__f(mark(x0)))

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
      ↳ QDPOrderProof
        ↳ QDP
          ↳ QDPOrderProof
            ↳ QDP
              ↳ DependencyGraphProof
                ↳ AND
                  ↳ QDP
                  ↳ QDP
                    ↳ QDPOrderProof

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

MARK(g(X)) → A__G(mark(X))
A__G(s(X)) → A__G(mark(X))
MARK(s(X)) → MARK(X)
A__G(s(X)) → MARK(X)
MARK(cons(X1, X2)) → MARK(X1)
MARK(f(X)) → A__F(mark(X))
A__F(X) → MARK(X)
MARK(g(X)) → 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.
We use the reduction pair processor [15].


The following pairs can be oriented strictly and are deleted.


MARK(f(X)) → A__F(mark(X))
MARK(f(X)) → MARK(X)
The remaining pairs can at least be oriented weakly.

MARK(g(X)) → A__G(mark(X))
A__G(s(X)) → A__G(mark(X))
MARK(s(X)) → MARK(X)
A__G(s(X)) → MARK(X)
MARK(cons(X1, X2)) → MARK(X1)
A__F(X) → MARK(X)
MARK(g(X)) → MARK(X)
Used ordering: Polynomial interpretation with max and min functions [25]:

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

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
      ↳ QDPOrderProof
        ↳ QDP
          ↳ QDPOrderProof
            ↳ QDP
              ↳ DependencyGraphProof
                ↳ AND
                  ↳ QDP
                  ↳ QDP
                    ↳ QDPOrderProof
                      ↳ QDP
                        ↳ DependencyGraphProof

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

MARK(g(X)) → A__G(mark(X))
A__G(s(X)) → A__G(mark(X))
MARK(s(X)) → MARK(X)
A__G(s(X)) → MARK(X)
MARK(cons(X1, X2)) → MARK(X1)
A__F(X) → MARK(X)
MARK(g(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 1 less node.

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ QDPOrderProof
        ↳ QDP
          ↳ QDPOrderProof
            ↳ QDP
              ↳ DependencyGraphProof
                ↳ AND
                  ↳ QDP
                  ↳ QDP
                    ↳ QDPOrderProof
                      ↳ 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(X)) → A__G(mark(X))
MARK(s(X)) → MARK(X)
A__G(s(X)) → MARK(X)
MARK(cons(X1, X2)) → MARK(X1)
MARK(g(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
      ↳ QDPOrderProof
        ↳ QDP
          ↳ QDPOrderProof
            ↳ QDP
              ↳ DependencyGraphProof
                ↳ AND
                  ↳ QDP
                  ↳ QDP
                    ↳ QDPOrderProof
                      ↳ QDP
                        ↳ DependencyGraphProof
                          ↳ 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)))
A__G(s(0)) → A__G(0)
MARK(s(X)) → MARK(X)
A__G(s(sel(x0, x1))) → A__G(a__sel(mark(x0), mark(x1)))
A__G(s(X)) → MARK(X)
A__G(s(cons(x0, x1))) → A__G(cons(mark(x0), x1))
MARK(cons(X1, X2)) → MARK(X1)
A__G(s(g(x0))) → A__G(a__g(mark(x0)))
A__G(s(s(x0))) → A__G(s(mark(x0)))
MARK(g(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
      ↳ QDPOrderProof
        ↳ QDP
          ↳ QDPOrderProof
            ↳ QDP
              ↳ DependencyGraphProof
                ↳ AND
                  ↳ QDP
                  ↳ QDP
                    ↳ QDPOrderProof
                      ↳ QDP
                        ↳ DependencyGraphProof
                          ↳ 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(s(X)) → MARK(X)
A__G(s(sel(x0, x1))) → A__G(a__sel(mark(x0), mark(x1)))
A__G(s(X)) → MARK(X)
MARK(cons(X1, X2)) → MARK(X1)
A__G(s(g(x0))) → A__G(a__g(mark(x0)))
A__G(s(s(x0))) → A__G(s(mark(x0)))
MARK(g(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
      ↳ QDPOrderProof
        ↳ QDP
          ↳ QDPOrderProof
            ↳ QDP
              ↳ DependencyGraphProof
                ↳ AND
                  ↳ QDP
                  ↳ QDP
                    ↳ QDPOrderProof
                      ↳ QDP
                        ↳ DependencyGraphProof
                          ↳ QDP
                            ↳ Narrowing
                              ↳ QDP
                                ↳ DependencyGraphProof
                                  ↳ QDP
                                    ↳ Narrowing
                                      ↳ QDP
                                        ↳ DependencyGraphProof

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

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(s(X)) → MARK(X)
MARK(cons(X1, X2)) → MARK(X1)
MARK(g(X)) → MARK(X)
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(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)))
MARK(g(0)) → A__G(0)
A__G(s(s(x0))) → A__G(s(mark(x0)))

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
      ↳ QDPOrderProof
        ↳ QDP
          ↳ QDPOrderProof
            ↳ QDP
              ↳ DependencyGraphProof
                ↳ AND
                  ↳ QDP
                  ↳ QDP
                    ↳ QDPOrderProof
                      ↳ QDP
                        ↳ DependencyGraphProof
                          ↳ QDP
                            ↳ Narrowing
                              ↳ QDP
                                ↳ DependencyGraphProof
                                  ↳ QDP
                                    ↳ Narrowing
                                      ↳ QDP
                                        ↳ DependencyGraphProof
                                          ↳ QDP
                                            ↳ QDPOrderProof

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

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(g(f(x0))) → A__G(a__f(mark(x0)))
MARK(g(s(x0))) → A__G(s(mark(x0)))
MARK(s(X)) → MARK(X)
A__G(s(sel(x0, x1))) → A__G(a__sel(mark(x0), mark(x1)))
A__G(s(X)) → MARK(X)
MARK(cons(X1, X2)) → MARK(X1)
MARK(g(g(x0))) → A__G(a__g(mark(x0)))
A__G(s(g(x0))) → A__G(a__g(mark(x0)))
A__G(s(s(x0))) → A__G(s(mark(x0)))
MARK(g(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.
We use the reduction pair processor [15].


The following pairs can be oriented strictly and are deleted.


A__G(s(f(x0))) → A__G(a__f(mark(x0)))
MARK(g(f(x0))) → A__G(a__f(mark(x0)))
The remaining pairs can at least be oriented weakly.

MARK(g(sel(x0, x1))) → A__G(a__sel(mark(x0), mark(x1)))
MARK(g(s(x0))) → A__G(s(mark(x0)))
MARK(s(X)) → MARK(X)
A__G(s(sel(x0, x1))) → A__G(a__sel(mark(x0), mark(x1)))
A__G(s(X)) → MARK(X)
MARK(cons(X1, X2)) → MARK(X1)
MARK(g(g(x0))) → A__G(a__g(mark(x0)))
A__G(s(g(x0))) → A__G(a__g(mark(x0)))
A__G(s(s(x0))) → A__G(s(mark(x0)))
MARK(g(X)) → MARK(X)
Used ordering: Polynomial interpretation with max and min functions [25]:

POL(0) = 1   
POL(A__G(x1)) = 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
      ↳ QDPOrderProof
        ↳ QDP
          ↳ QDPOrderProof
            ↳ QDP
              ↳ DependencyGraphProof
                ↳ AND
                  ↳ QDP
                  ↳ QDP
                    ↳ QDPOrderProof
                      ↳ QDP
                        ↳ DependencyGraphProof
                          ↳ QDP
                            ↳ Narrowing
                              ↳ QDP
                                ↳ DependencyGraphProof
                                  ↳ QDP
                                    ↳ Narrowing
                                      ↳ QDP
                                        ↳ DependencyGraphProof
                                          ↳ QDP
                                            ↳ QDPOrderProof
                                              ↳ QDP

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

MARK(g(sel(x0, x1))) → A__G(a__sel(mark(x0), mark(x1)))
MARK(s(X)) → MARK(X)
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(cons(X1, X2)) → MARK(X1)
MARK(g(g(x0))) → A__G(a__g(mark(x0)))
A__G(s(g(x0))) → A__G(a__g(mark(x0)))
MARK(g(X)) → MARK(X)
A__G(s(s(x0))) → A__G(s(mark(x0)))

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.