Termination w.r.t. Q proof of /tmp/tmpXcb2qW/Ex7_BLR02

Termination w.r.t. Q of the following Term Rewriting System could not be shown:

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

a__from(X) → cons(mark(X), from(s(X)))
a__head(cons(X, XS)) → mark(X)
a__2nd(cons(X, XS)) → a__head(mark(XS))
a__take(0, XS) → nil
a__take(s(N), cons(X, XS)) → cons(mark(X), take(N, XS))
a__sel(0, cons(X, XS)) → mark(X)
a__sel(s(N), cons(X, XS)) → a__sel(mark(N), mark(XS))
mark(from(X)) → a__from(mark(X))
mark(head(X)) → a__head(mark(X))
mark(2nd(X)) → a__2nd(mark(X))
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(sel(X1, X2)) → a__sel(mark(X1), mark(X2))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(s(X)) → s(mark(X))
mark(0) → 0
mark(nil) → nil
a__from(X) → from(X)
a__head(X) → head(X)
a__2nd(X) → 2nd(X)
a__take(X1, X2) → take(X1, X2)
a__sel(X1, X2) → sel(X1, X2)

Q is empty.

↳ QTRS

  ↳ DependencyPairsProof

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

a__from(X) → cons(mark(X), from(s(X)))
a__head(cons(X, XS)) → mark(X)
a__2nd(cons(X, XS)) → a__head(mark(XS))
a__take(0, XS) → nil
a__take(s(N), cons(X, XS)) → cons(mark(X), take(N, XS))
a__sel(0, cons(X, XS)) → mark(X)
a__sel(s(N), cons(X, XS)) → a__sel(mark(N), mark(XS))
mark(from(X)) → a__from(mark(X))
mark(head(X)) → a__head(mark(X))
mark(2nd(X)) → a__2nd(mark(X))
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(sel(X1, X2)) → a__sel(mark(X1), mark(X2))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(s(X)) → s(mark(X))
mark(0) → 0
mark(nil) → nil
a__from(X) → from(X)
a__head(X) → head(X)
a__2nd(X) → 2nd(X)
a__take(X1, X2) → take(X1, X2)
a__sel(X1, X2) → sel(X1, X2)

Q is empty.

Using Dependency Pairs [1,15] we result in the following initial DP problem:
Q DP problem:
The TRS P consists of the following rules:

MARK(take(X1, X2)) → MARK(X2)
MARK(sel(X1, X2)) → MARK(X1)
MARK(from(X)) → MARK(X)
MARK(sel(X1, X2)) → A__SEL(mark(X1), mark(X2))
MARK(take(X1, X2)) → A__TAKE(mark(X1), mark(X2))
A__2ND(cons(X, XS)) → A__HEAD(mark(XS))
MARK(from(X)) → A__FROM(mark(X))
MARK(s(X)) → MARK(X)
A__SEL(s(N), cons(X, XS)) → A__SEL(mark(N), mark(XS))
MARK(head(X)) → MARK(X)
MARK(cons(X1, X2)) → MARK(X1)
A__FROM(X) → MARK(X)
A__SEL(0, cons(X, XS)) → MARK(X)
A__HEAD(cons(X, XS)) → MARK(X)
A__2ND(cons(X, XS)) → MARK(XS)
A__SEL(s(N), cons(X, XS)) → MARK(XS)
MARK(2nd(X)) → MARK(X)
MARK(take(X1, X2)) → MARK(X1)
MARK(head(X)) → A__HEAD(mark(X))
MARK(2nd(X)) → A__2ND(mark(X))
A__TAKE(s(N), cons(X, XS)) → MARK(X)
MARK(sel(X1, X2)) → MARK(X2)
A__SEL(s(N), cons(X, XS)) → MARK(N)

The TRS R consists of the following rules:

a__from(X) → cons(mark(X), from(s(X)))
a__head(cons(X, XS)) → mark(X)
a__2nd(cons(X, XS)) → a__head(mark(XS))
a__take(0, XS) → nil
a__take(s(N), cons(X, XS)) → cons(mark(X), take(N, XS))
a__sel(0, cons(X, XS)) → mark(X)
a__sel(s(N), cons(X, XS)) → a__sel(mark(N), mark(XS))
mark(from(X)) → a__from(mark(X))
mark(head(X)) → a__head(mark(X))
mark(2nd(X)) → a__2nd(mark(X))
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(sel(X1, X2)) → a__sel(mark(X1), mark(X2))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(s(X)) → s(mark(X))
mark(0) → 0
mark(nil) → nil
a__from(X) → from(X)
a__head(X) → head(X)
a__2nd(X) → 2nd(X)
a__take(X1, X2) → take(X1, X2)
a__sel(X1, X2) → sel(X1, X2)

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ QDPOrderProof

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

MARK(take(X1, X2)) → MARK(X2)
MARK(sel(X1, X2)) → MARK(X1)
MARK(from(X)) → MARK(X)
MARK(sel(X1, X2)) → A__SEL(mark(X1), mark(X2))
MARK(take(X1, X2)) → A__TAKE(mark(X1), mark(X2))
A__2ND(cons(X, XS)) → A__HEAD(mark(XS))
MARK(from(X)) → A__FROM(mark(X))
MARK(s(X)) → MARK(X)
A__SEL(s(N), cons(X, XS)) → A__SEL(mark(N), mark(XS))
MARK(head(X)) → MARK(X)
MARK(cons(X1, X2)) → MARK(X1)
A__FROM(X) → MARK(X)
A__SEL(0, cons(X, XS)) → MARK(X)
A__HEAD(cons(X, XS)) → MARK(X)
A__2ND(cons(X, XS)) → MARK(XS)
A__SEL(s(N), cons(X, XS)) → MARK(XS)
MARK(2nd(X)) → MARK(X)
MARK(take(X1, X2)) → MARK(X1)
MARK(head(X)) → A__HEAD(mark(X))
MARK(2nd(X)) → A__2ND(mark(X))
A__TAKE(s(N), cons(X, XS)) → MARK(X)
MARK(sel(X1, X2)) → MARK(X2)
A__SEL(s(N), cons(X, XS)) → MARK(N)

The TRS R consists of the following rules:

a__from(X) → cons(mark(X), from(s(X)))
a__head(cons(X, XS)) → mark(X)
a__2nd(cons(X, XS)) → a__head(mark(XS))
a__take(0, XS) → nil
a__take(s(N), cons(X, XS)) → cons(mark(X), take(N, XS))
a__sel(0, cons(X, XS)) → mark(X)
a__sel(s(N), cons(X, XS)) → a__sel(mark(N), mark(XS))
mark(from(X)) → a__from(mark(X))
mark(head(X)) → a__head(mark(X))
mark(2nd(X)) → a__2nd(mark(X))
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(sel(X1, X2)) → a__sel(mark(X1), mark(X2))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(s(X)) → s(mark(X))
mark(0) → 0
mark(nil) → nil
a__from(X) → from(X)
a__head(X) → head(X)
a__2nd(X) → 2nd(X)
a__take(X1, X2) → take(X1, X2)
a__sel(X1, X2) → sel(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].

The following pairs can be oriented strictly and are deleted.

MARK(sel(X1, X2)) → MARK(X1)
MARK(sel(X1, X2)) → A__SEL(mark(X1), mark(X2))
MARK(2nd(X)) → MARK(X)
MARK(2nd(X)) → A__2ND(mark(X))
MARK(sel(X1, X2)) → MARK(X2)

The remaining pairs can at least be oriented weakly.

MARK(take(X1, X2)) → MARK(X2)
MARK(from(X)) → MARK(X)
MARK(take(X1, X2)) → A__TAKE(mark(X1), mark(X2))
A__2ND(cons(X, XS)) → A__HEAD(mark(XS))
MARK(from(X)) → A__FROM(mark(X))
MARK(s(X)) → MARK(X)
A__SEL(s(N), cons(X, XS)) → A__SEL(mark(N), mark(XS))
MARK(head(X)) → MARK(X)
MARK(cons(X1, X2)) → MARK(X1)
A__FROM(X) → MARK(X)
A__SEL(0, cons(X, XS)) → MARK(X)
A__HEAD(cons(X, XS)) → MARK(X)
A__2ND(cons(X, XS)) → MARK(XS)
A__SEL(s(N), cons(X, XS)) → MARK(XS)
MARK(take(X1, X2)) → MARK(X1)
MARK(head(X)) → A__HEAD(mark(X))
A__TAKE(s(N), cons(X, XS)) → MARK(X)
A__SEL(s(N), cons(X, XS)) → MARK(N)

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

POL(0) = 0
POL(2nd(x₁)) = 1 + x₁
POL(A__2ND(x₁)) = x₁
POL(A__FROM(x₁)) = x₁
POL(A__HEAD(x₁)) = x₁
POL(A__SEL(x₁, x₂)) = x₁ + x₂
POL(A__TAKE(x₁, x₂)) = x₂
POL(MARK(x₁)) = x₁
POL(a__2nd(x₁)) = 1 + x₁
POL(a__from(x₁)) = x₁
POL(a__head(x₁)) = x₁
POL(a__sel(x₁, x₂)) = 1 + x₁ + x₂
POL(a__take(x₁, x₂)) = x₁ + x₂
POL(cons(x₁, x₂)) = max(x₁, x₂)
POL(from(x₁)) = x₁
POL(head(x₁)) = x₁
POL(mark(x₁)) = x₁
POL(nil) = 0
POL(s(x₁)) = x₁
POL(sel(x₁, x₂)) = 1 + x₁ + x₂
POL(take(x₁, x₂)) = x₁ + x₂

The following usable rules [17] were oriented:

mark(s(X)) → s(mark(X))
mark(0) → 0
mark(nil) → nil
a__from(X) → from(X)
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(cons(X1, X2)) → cons(mark(X1), X2)
a__head(X) → head(X)
a__2nd(X) → 2nd(X)
a__take(X1, X2) → take(X1, X2)
a__sel(X1, X2) → sel(X1, X2)
a__from(X) → cons(mark(X), from(s(X)))
a__take(s(N), cons(X, XS)) → cons(mark(X), take(N, XS))
a__take(0, XS) → nil
a__sel(s(N), cons(X, XS)) → a__sel(mark(N), mark(XS))
a__2nd(cons(X, XS)) → a__head(mark(XS))
a__sel(0, cons(X, XS)) → mark(X)
mark(2nd(X)) → a__2nd(mark(X))
mark(head(X)) → a__head(mark(X))
mark(sel(X1, X2)) → a__sel(mark(X1), mark(X2))
a__head(cons(X, XS)) → mark(X)
mark(from(X)) → a__from(mark(X))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ QDPOrderProof

        ↳ QDP

          ↳ DependencyGraphProof

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

MARK(take(X1, X2)) → MARK(X2)
MARK(from(X)) → MARK(X)
MARK(take(X1, X2)) → A__TAKE(mark(X1), mark(X2))
A__2ND(cons(X, XS)) → A__HEAD(mark(XS))
MARK(from(X)) → A__FROM(mark(X))
MARK(s(X)) → MARK(X)
A__SEL(s(N), cons(X, XS)) → A__SEL(mark(N), mark(XS))
MARK(head(X)) → MARK(X)
MARK(cons(X1, X2)) → MARK(X1)
A__FROM(X) → MARK(X)
A__SEL(0, cons(X, XS)) → MARK(X)
A__HEAD(cons(X, XS)) → MARK(X)
A__SEL(s(N), cons(X, XS)) → MARK(XS)
A__2ND(cons(X, XS)) → MARK(XS)
MARK(head(X)) → A__HEAD(mark(X))
MARK(take(X1, X2)) → MARK(X1)
A__TAKE(s(N), cons(X, XS)) → MARK(X)
A__SEL(s(N), cons(X, XS)) → MARK(N)

The TRS R consists of the following rules:

a__from(X) → cons(mark(X), from(s(X)))
a__head(cons(X, XS)) → mark(X)
a__2nd(cons(X, XS)) → a__head(mark(XS))
a__take(0, XS) → nil
a__take(s(N), cons(X, XS)) → cons(mark(X), take(N, XS))
a__sel(0, cons(X, XS)) → mark(X)
a__sel(s(N), cons(X, XS)) → a__sel(mark(N), mark(XS))
mark(from(X)) → a__from(mark(X))
mark(head(X)) → a__head(mark(X))
mark(2nd(X)) → a__2nd(mark(X))
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(sel(X1, X2)) → a__sel(mark(X1), mark(X2))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(s(X)) → s(mark(X))
mark(0) → 0
mark(nil) → nil
a__from(X) → from(X)
a__head(X) → head(X)
a__2nd(X) → 2nd(X)
a__take(X1, X2) → take(X1, X2)
a__sel(X1, X2) → sel(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 2 SCCs with 5 less nodes.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ QDPOrderProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

                ↳ QDPOrderProof

              ↳ QDP

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

MARK(take(X1, X2)) → MARK(X2)
A__HEAD(cons(X, XS)) → MARK(X)
MARK(from(X)) → MARK(X)
MARK(take(X1, X2)) → A__TAKE(mark(X1), mark(X2))
MARK(from(X)) → A__FROM(mark(X))
MARK(s(X)) → MARK(X)
MARK(head(X)) → MARK(X)
MARK(take(X1, X2)) → MARK(X1)
MARK(head(X)) → A__HEAD(mark(X))
MARK(cons(X1, X2)) → MARK(X1)
A__TAKE(s(N), cons(X, XS)) → MARK(X)
A__FROM(X) → MARK(X)

The TRS R consists of the following rules:

a__from(X) → cons(mark(X), from(s(X)))
a__head(cons(X, XS)) → mark(X)
a__2nd(cons(X, XS)) → a__head(mark(XS))
a__take(0, XS) → nil
a__take(s(N), cons(X, XS)) → cons(mark(X), take(N, XS))
a__sel(0, cons(X, XS)) → mark(X)
a__sel(s(N), cons(X, XS)) → a__sel(mark(N), mark(XS))
mark(from(X)) → a__from(mark(X))
mark(head(X)) → a__head(mark(X))
mark(2nd(X)) → a__2nd(mark(X))
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(sel(X1, X2)) → a__sel(mark(X1), mark(X2))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(s(X)) → s(mark(X))
mark(0) → 0
mark(nil) → nil
a__from(X) → from(X)
a__head(X) → head(X)
a__2nd(X) → 2nd(X)
a__take(X1, X2) → take(X1, X2)
a__sel(X1, X2) → sel(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].

The following pairs can be oriented strictly and are deleted.

MARK(take(X1, X2)) → MARK(X2)
MARK(take(X1, X2)) → A__TAKE(mark(X1), mark(X2))
MARK(take(X1, X2)) → MARK(X1)

The remaining pairs can at least be oriented weakly.

A__HEAD(cons(X, XS)) → MARK(X)
MARK(from(X)) → MARK(X)
MARK(from(X)) → A__FROM(mark(X))
MARK(s(X)) → MARK(X)
MARK(head(X)) → MARK(X)
MARK(head(X)) → A__HEAD(mark(X))
MARK(cons(X1, X2)) → MARK(X1)
A__TAKE(s(N), cons(X, XS)) → MARK(X)
A__FROM(X) → MARK(X)

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

POL(0) = 0
POL(2nd(x₁)) = x₁
POL(A__FROM(x₁)) = x₁
POL(A__HEAD(x₁)) = x₁
POL(A__TAKE(x₁, x₂)) = x₂
POL(MARK(x₁)) = x₁
POL(a__2nd(x₁)) = x₁
POL(a__from(x₁)) = x₁
POL(a__head(x₁)) = x₁
POL(a__sel(x₁, x₂)) = x₂
POL(a__take(x₁, x₂)) = 1 + x₁ + x₂
POL(cons(x₁, x₂)) = max(x₁, x₂)
POL(from(x₁)) = x₁
POL(head(x₁)) = x₁
POL(mark(x₁)) = x₁
POL(nil) = 0
POL(s(x₁)) = x₁
POL(sel(x₁, x₂)) = x₂
POL(take(x₁, x₂)) = 1 + x₁ + x₂

The following usable rules [17] were oriented:

mark(s(X)) → s(mark(X))
mark(0) → 0
mark(nil) → nil
a__from(X) → from(X)
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(cons(X1, X2)) → cons(mark(X1), X2)
a__head(X) → head(X)
a__2nd(X) → 2nd(X)
a__take(X1, X2) → take(X1, X2)
a__sel(X1, X2) → sel(X1, X2)
a__from(X) → cons(mark(X), from(s(X)))
a__take(s(N), cons(X, XS)) → cons(mark(X), take(N, XS))
a__take(0, XS) → nil
a__sel(s(N), cons(X, XS)) → a__sel(mark(N), mark(XS))
a__2nd(cons(X, XS)) → a__head(mark(XS))
a__sel(0, cons(X, XS)) → mark(X)
mark(2nd(X)) → a__2nd(mark(X))
mark(head(X)) → a__head(mark(X))
mark(sel(X1, X2)) → a__sel(mark(X1), mark(X2))
a__head(cons(X, XS)) → mark(X)
mark(from(X)) → a__from(mark(X))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ QDPOrderProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

                ↳ QDPOrderProof

                  ↳ QDP

                    ↳ DependencyGraphProof

              ↳ QDP

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

A__HEAD(cons(X, XS)) → MARK(X)
MARK(from(X)) → MARK(X)
MARK(from(X)) → A__FROM(mark(X))
MARK(s(X)) → MARK(X)
MARK(head(X)) → MARK(X)
MARK(head(X)) → A__HEAD(mark(X))
MARK(cons(X1, X2)) → MARK(X1)
A__TAKE(s(N), cons(X, XS)) → MARK(X)
A__FROM(X) → MARK(X)

The TRS R consists of the following rules:

a__from(X) → cons(mark(X), from(s(X)))
a__head(cons(X, XS)) → mark(X)
a__2nd(cons(X, XS)) → a__head(mark(XS))
a__take(0, XS) → nil
a__take(s(N), cons(X, XS)) → cons(mark(X), take(N, XS))
a__sel(0, cons(X, XS)) → mark(X)
a__sel(s(N), cons(X, XS)) → a__sel(mark(N), mark(XS))
mark(from(X)) → a__from(mark(X))
mark(head(X)) → a__head(mark(X))
mark(2nd(X)) → a__2nd(mark(X))
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(sel(X1, X2)) → a__sel(mark(X1), mark(X2))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(s(X)) → s(mark(X))
mark(0) → 0
mark(nil) → nil
a__from(X) → from(X)
a__head(X) → head(X)
a__2nd(X) → 2nd(X)
a__take(X1, X2) → take(X1, X2)
a__sel(X1, X2) → sel(X1, X2)

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ QDPOrderProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

                ↳ QDPOrderProof

                  ↳ QDP

                    ↳ DependencyGraphProof

                      ↳ QDP

                        ↳ QDPOrderProof

              ↳ QDP

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

A__HEAD(cons(X, XS)) → MARK(X)
MARK(from(X)) → MARK(X)
MARK(from(X)) → A__FROM(mark(X))
MARK(s(X)) → MARK(X)
MARK(head(X)) → MARK(X)
MARK(head(X)) → A__HEAD(mark(X))
MARK(cons(X1, X2)) → MARK(X1)
A__FROM(X) → MARK(X)

The TRS R consists of the following rules:

a__from(X) → cons(mark(X), from(s(X)))
a__head(cons(X, XS)) → mark(X)
a__2nd(cons(X, XS)) → a__head(mark(XS))
a__take(0, XS) → nil
a__take(s(N), cons(X, XS)) → cons(mark(X), take(N, XS))
a__sel(0, cons(X, XS)) → mark(X)
a__sel(s(N), cons(X, XS)) → a__sel(mark(N), mark(XS))
mark(from(X)) → a__from(mark(X))
mark(head(X)) → a__head(mark(X))
mark(2nd(X)) → a__2nd(mark(X))
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(sel(X1, X2)) → a__sel(mark(X1), mark(X2))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(s(X)) → s(mark(X))
mark(0) → 0
mark(nil) → nil
a__from(X) → from(X)
a__head(X) → head(X)
a__2nd(X) → 2nd(X)
a__take(X1, X2) → take(X1, X2)
a__sel(X1, X2) → sel(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].

The following pairs can be oriented strictly and are deleted.

MARK(head(X)) → MARK(X)
MARK(head(X)) → A__HEAD(mark(X))

The remaining pairs can at least be oriented weakly.

A__HEAD(cons(X, XS)) → MARK(X)
MARK(from(X)) → MARK(X)
MARK(from(X)) → A__FROM(mark(X))
MARK(s(X)) → MARK(X)
MARK(cons(X1, X2)) → MARK(X1)
A__FROM(X) → MARK(X)

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

POL(0) = 0
POL(2nd(x₁)) = 1 + x₁
POL(A__FROM(x₁)) = x₁
POL(A__HEAD(x₁)) = x₁
POL(MARK(x₁)) = x₁
POL(a__2nd(x₁)) = 1 + x₁
POL(a__from(x₁)) = x₁
POL(a__head(x₁)) = 1 + x₁
POL(a__sel(x₁, x₂)) = x₂
POL(a__take(x₁, x₂)) = x₂
POL(cons(x₁, x₂)) = max(x₁, x₂)
POL(from(x₁)) = x₁
POL(head(x₁)) = 1 + x₁
POL(mark(x₁)) = x₁
POL(nil) = 0
POL(s(x₁)) = x₁
POL(sel(x₁, x₂)) = x₂
POL(take(x₁, x₂)) = x₂

The following usable rules [17] were oriented:

mark(s(X)) → s(mark(X))
mark(0) → 0
mark(nil) → nil
a__from(X) → from(X)
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(cons(X1, X2)) → cons(mark(X1), X2)
a__head(X) → head(X)
a__2nd(X) → 2nd(X)
a__take(X1, X2) → take(X1, X2)
a__sel(X1, X2) → sel(X1, X2)
a__from(X) → cons(mark(X), from(s(X)))
a__take(s(N), cons(X, XS)) → cons(mark(X), take(N, XS))
a__take(0, XS) → nil
a__sel(s(N), cons(X, XS)) → a__sel(mark(N), mark(XS))
a__2nd(cons(X, XS)) → a__head(mark(XS))
a__sel(0, cons(X, XS)) → mark(X)
mark(2nd(X)) → a__2nd(mark(X))
mark(head(X)) → a__head(mark(X))
mark(sel(X1, X2)) → a__sel(mark(X1), mark(X2))
a__head(cons(X, XS)) → mark(X)
mark(from(X)) → a__from(mark(X))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ QDPOrderProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

                ↳ QDPOrderProof

                  ↳ QDP

                    ↳ DependencyGraphProof

                      ↳ QDP

                        ↳ QDPOrderProof

                          ↳ QDP

                            ↳ DependencyGraphProof

              ↳ QDP

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

A__HEAD(cons(X, XS)) → MARK(X)
MARK(from(X)) → MARK(X)
MARK(from(X)) → A__FROM(mark(X))
MARK(s(X)) → MARK(X)
MARK(cons(X1, X2)) → MARK(X1)
A__FROM(X) → MARK(X)

The TRS R consists of the following rules:

a__from(X) → cons(mark(X), from(s(X)))
a__head(cons(X, XS)) → mark(X)
a__2nd(cons(X, XS)) → a__head(mark(XS))
a__take(0, XS) → nil
a__take(s(N), cons(X, XS)) → cons(mark(X), take(N, XS))
a__sel(0, cons(X, XS)) → mark(X)
a__sel(s(N), cons(X, XS)) → a__sel(mark(N), mark(XS))
mark(from(X)) → a__from(mark(X))
mark(head(X)) → a__head(mark(X))
mark(2nd(X)) → a__2nd(mark(X))
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(sel(X1, X2)) → a__sel(mark(X1), mark(X2))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(s(X)) → s(mark(X))
mark(0) → 0
mark(nil) → nil
a__from(X) → from(X)
a__head(X) → head(X)
a__2nd(X) → 2nd(X)
a__take(X1, X2) → take(X1, X2)
a__sel(X1, X2) → sel(X1, X2)

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ QDPOrderProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

                ↳ QDPOrderProof

                  ↳ QDP

                    ↳ DependencyGraphProof

                      ↳ QDP

                        ↳ QDPOrderProof

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ QDPOrderProof

              ↳ QDP

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

MARK(from(X)) → MARK(X)
MARK(from(X)) → A__FROM(mark(X))
MARK(s(X)) → MARK(X)
MARK(cons(X1, X2)) → MARK(X1)
A__FROM(X) → MARK(X)

The TRS R consists of the following rules:

a__from(X) → cons(mark(X), from(s(X)))
a__head(cons(X, XS)) → mark(X)
a__2nd(cons(X, XS)) → a__head(mark(XS))
a__take(0, XS) → nil
a__take(s(N), cons(X, XS)) → cons(mark(X), take(N, XS))
a__sel(0, cons(X, XS)) → mark(X)
a__sel(s(N), cons(X, XS)) → a__sel(mark(N), mark(XS))
mark(from(X)) → a__from(mark(X))
mark(head(X)) → a__head(mark(X))
mark(2nd(X)) → a__2nd(mark(X))
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(sel(X1, X2)) → a__sel(mark(X1), mark(X2))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(s(X)) → s(mark(X))
mark(0) → 0
mark(nil) → nil
a__from(X) → from(X)
a__head(X) → head(X)
a__2nd(X) → 2nd(X)
a__take(X1, X2) → take(X1, X2)
a__sel(X1, X2) → sel(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].

The following pairs can be oriented strictly and are deleted.

MARK(from(X)) → MARK(X)
MARK(from(X)) → A__FROM(mark(X))

The remaining pairs can at least be oriented weakly.

MARK(s(X)) → MARK(X)
MARK(cons(X1, X2)) → MARK(X1)
A__FROM(X) → MARK(X)

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

POL(0) = 0
POL(2nd(x₁)) = x₁
POL(A__FROM(x₁)) = x₁
POL(MARK(x₁)) = x₁
POL(a__2nd(x₁)) = x₁
POL(a__from(x₁)) = 1 + x₁
POL(a__head(x₁)) = x₁
POL(a__sel(x₁, x₂)) = x₁ + x₂
POL(a__take(x₁, x₂)) = x₂
POL(cons(x₁, x₂)) = max(x₁, x₂)
POL(from(x₁)) = 1 + x₁
POL(head(x₁)) = x₁
POL(mark(x₁)) = x₁
POL(nil) = 0
POL(s(x₁)) = x₁
POL(sel(x₁, x₂)) = x₁ + x₂
POL(take(x₁, x₂)) = x₂

The following usable rules [17] were oriented:

mark(s(X)) → s(mark(X))
mark(0) → 0
mark(nil) → nil
a__from(X) → from(X)
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(cons(X1, X2)) → cons(mark(X1), X2)
a__head(X) → head(X)
a__2nd(X) → 2nd(X)
a__take(X1, X2) → take(X1, X2)
a__sel(X1, X2) → sel(X1, X2)
a__from(X) → cons(mark(X), from(s(X)))
a__take(s(N), cons(X, XS)) → cons(mark(X), take(N, XS))
a__take(0, XS) → nil
a__sel(s(N), cons(X, XS)) → a__sel(mark(N), mark(XS))
a__2nd(cons(X, XS)) → a__head(mark(XS))
a__sel(0, cons(X, XS)) → mark(X)
mark(2nd(X)) → a__2nd(mark(X))
mark(head(X)) → a__head(mark(X))
mark(sel(X1, X2)) → a__sel(mark(X1), mark(X2))
a__head(cons(X, XS)) → mark(X)
mark(from(X)) → a__from(mark(X))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ QDPOrderProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

                ↳ QDPOrderProof

                  ↳ QDP

                    ↳ DependencyGraphProof

                      ↳ QDP

                        ↳ QDPOrderProof

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ QDPOrderProof

                                  ↳ QDP

                                    ↳ DependencyGraphProof

              ↳ QDP

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

MARK(s(X)) → MARK(X)
MARK(cons(X1, X2)) → MARK(X1)
A__FROM(X) → MARK(X)

The TRS R consists of the following rules:

a__from(X) → cons(mark(X), from(s(X)))
a__head(cons(X, XS)) → mark(X)
a__2nd(cons(X, XS)) → a__head(mark(XS))
a__take(0, XS) → nil
a__take(s(N), cons(X, XS)) → cons(mark(X), take(N, XS))
a__sel(0, cons(X, XS)) → mark(X)
a__sel(s(N), cons(X, XS)) → a__sel(mark(N), mark(XS))
mark(from(X)) → a__from(mark(X))
mark(head(X)) → a__head(mark(X))
mark(2nd(X)) → a__2nd(mark(X))
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(sel(X1, X2)) → a__sel(mark(X1), mark(X2))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(s(X)) → s(mark(X))
mark(0) → 0
mark(nil) → nil
a__from(X) → from(X)
a__head(X) → head(X)
a__2nd(X) → 2nd(X)
a__take(X1, X2) → take(X1, X2)
a__sel(X1, X2) → sel(X1, X2)

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ QDPOrderProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

                ↳ QDPOrderProof

                  ↳ QDP

                    ↳ DependencyGraphProof

                      ↳ QDP

                        ↳ QDPOrderProof

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ QDPOrderProof

                                  ↳ QDP

                                    ↳ DependencyGraphProof

                                      ↳ QDP

                                        ↳ UsableRulesProof

              ↳ QDP

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

MARK(s(X)) → MARK(X)
MARK(cons(X1, X2)) → MARK(X1)

The TRS R consists of the following rules:

a__from(X) → cons(mark(X), from(s(X)))
a__head(cons(X, XS)) → mark(X)
a__2nd(cons(X, XS)) → a__head(mark(XS))
a__take(0, XS) → nil
a__take(s(N), cons(X, XS)) → cons(mark(X), take(N, XS))
a__sel(0, cons(X, XS)) → mark(X)
a__sel(s(N), cons(X, XS)) → a__sel(mark(N), mark(XS))
mark(from(X)) → a__from(mark(X))
mark(head(X)) → a__head(mark(X))
mark(2nd(X)) → a__2nd(mark(X))
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(sel(X1, X2)) → a__sel(mark(X1), mark(X2))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(s(X)) → s(mark(X))
mark(0) → 0
mark(nil) → nil
a__from(X) → from(X)
a__head(X) → head(X)
a__2nd(X) → 2nd(X)
a__take(X1, X2) → take(X1, X2)
a__sel(X1, X2) → sel(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We can use the usable rules and reduction pair processor [15] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its argument. Then, we can delete all non-usable rules [17] from R.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ QDPOrderProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

                ↳ QDPOrderProof

                  ↳ QDP

                    ↳ DependencyGraphProof

                      ↳ QDP

                        ↳ QDPOrderProof

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ QDPOrderProof

                                  ↳ QDP

                                    ↳ DependencyGraphProof

                                      ↳ QDP

                                        ↳ UsableRulesProof

                                          ↳ QDP

                                            ↳ QDPSizeChangeProof

              ↳ QDP

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

MARK(s(X)) → MARK(X)
MARK(cons(X1, X2)) → MARK(X1)

R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By using the subterm criterion [20] together with the size-change analysis [32] we have proven that there are no infinite chains for this DP problem.

From the DPs we obtained the following set of size-change graphs:

MARK(s(X)) → MARK(X)
The graph contains the following edges 1 > 1
MARK(cons(X1, X2)) → MARK(X1)
The graph contains the following edges 1 > 1

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ QDPOrderProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

                ↳ Narrowing

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

A__SEL(s(N), cons(X, XS)) → A__SEL(mark(N), mark(XS))

The TRS R consists of the following rules:

a__from(X) → cons(mark(X), from(s(X)))
a__head(cons(X, XS)) → mark(X)
a__2nd(cons(X, XS)) → a__head(mark(XS))
a__take(0, XS) → nil
a__take(s(N), cons(X, XS)) → cons(mark(X), take(N, XS))
a__sel(0, cons(X, XS)) → mark(X)
a__sel(s(N), cons(X, XS)) → a__sel(mark(N), mark(XS))
mark(from(X)) → a__from(mark(X))
mark(head(X)) → a__head(mark(X))
mark(2nd(X)) → a__2nd(mark(X))
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(sel(X1, X2)) → a__sel(mark(X1), mark(X2))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(s(X)) → s(mark(X))
mark(0) → 0
mark(nil) → nil
a__from(X) → from(X)
a__head(X) → head(X)
a__2nd(X) → 2nd(X)
a__take(X1, X2) → take(X1, X2)
a__sel(X1, X2) → sel(X1, X2)

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

A__SEL(s(from(x0)), cons(y1, y2)) → A__SEL(a__from(mark(x0)), mark(y2))
A__SEL(s(0), cons(y1, y2)) → A__SEL(0, mark(y2))
A__SEL(s(2nd(x0)), cons(y1, y2)) → A__SEL(a__2nd(mark(x0)), 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(nil), cons(y1, y2)) → A__SEL(nil, mark(y2))
A__SEL(s(take(x0, x1)), cons(y1, y2)) → A__SEL(a__take(mark(x0), mark(x1)), mark(y2))
A__SEL(s(head(x0)), cons(y1, y2)) → A__SEL(a__head(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

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

                ↳ Narrowing

                  ↳ QDP

                    ↳ DependencyGraphProof

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

A__SEL(s(from(x0)), cons(y1, y2)) → A__SEL(a__from(mark(x0)), mark(y2))
A__SEL(s(sel(x0, x1)), cons(y1, y2)) → A__SEL(a__sel(mark(x0), mark(x1)), mark(y2))
A__SEL(s(2nd(x0)), cons(y1, y2)) → A__SEL(a__2nd(mark(x0)), mark(y2))
A__SEL(s(0), cons(y1, y2)) → A__SEL(0, mark(y2))
A__SEL(s(s(x0)), cons(y1, y2)) → A__SEL(s(mark(x0)), mark(y2))
A__SEL(s(nil), cons(y1, y2)) → A__SEL(nil, mark(y2))
A__SEL(s(take(x0, x1)), cons(y1, y2)) → A__SEL(a__take(mark(x0), mark(x1)), mark(y2))
A__SEL(s(head(x0)), cons(y1, y2)) → A__SEL(a__head(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__from(X) → cons(mark(X), from(s(X)))
a__head(cons(X, XS)) → mark(X)
a__2nd(cons(X, XS)) → a__head(mark(XS))
a__take(0, XS) → nil
a__take(s(N), cons(X, XS)) → cons(mark(X), take(N, XS))
a__sel(0, cons(X, XS)) → mark(X)
a__sel(s(N), cons(X, XS)) → a__sel(mark(N), mark(XS))
mark(from(X)) → a__from(mark(X))
mark(head(X)) → a__head(mark(X))
mark(2nd(X)) → a__2nd(mark(X))
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(sel(X1, X2)) → a__sel(mark(X1), mark(X2))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(s(X)) → s(mark(X))
mark(0) → 0
mark(nil) → nil
a__from(X) → from(X)
a__head(X) → head(X)
a__2nd(X) → 2nd(X)
a__take(X1, X2) → take(X1, X2)
a__sel(X1, X2) → sel(X1, X2)

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ QDPOrderProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

                ↳ Narrowing

                  ↳ QDP

                    ↳ DependencyGraphProof

                      ↳ QDP

                        ↳ Narrowing

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

A__SEL(s(from(x0)), cons(y1, y2)) → A__SEL(a__from(mark(x0)), mark(y2))
A__SEL(s(sel(x0, x1)), cons(y1, y2)) → A__SEL(a__sel(mark(x0), mark(x1)), mark(y2))
A__SEL(s(2nd(x0)), cons(y1, y2)) → A__SEL(a__2nd(mark(x0)), mark(y2))
A__SEL(s(s(x0)), cons(y1, y2)) → A__SEL(s(mark(x0)), mark(y2))
A__SEL(s(take(x0, x1)), cons(y1, y2)) → A__SEL(a__take(mark(x0), mark(x1)), mark(y2))
A__SEL(s(head(x0)), cons(y1, y2)) → A__SEL(a__head(mark(x0)), mark(y2))

The TRS R consists of the following rules:

a__from(X) → cons(mark(X), from(s(X)))
a__head(cons(X, XS)) → mark(X)
a__2nd(cons(X, XS)) → a__head(mark(XS))
a__take(0, XS) → nil
a__take(s(N), cons(X, XS)) → cons(mark(X), take(N, XS))
a__sel(0, cons(X, XS)) → mark(X)
a__sel(s(N), cons(X, XS)) → a__sel(mark(N), mark(XS))
mark(from(X)) → a__from(mark(X))
mark(head(X)) → a__head(mark(X))
mark(2nd(X)) → a__2nd(mark(X))
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(sel(X1, X2)) → a__sel(mark(X1), mark(X2))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(s(X)) → s(mark(X))
mark(0) → 0
mark(nil) → nil
a__from(X) → from(X)
a__head(X) → head(X)
a__2nd(X) → 2nd(X)
a__take(X1, X2) → take(X1, X2)
a__sel(X1, X2) → sel(X1, X2)

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

A__SEL(s(from(y0)), cons(y1, cons(x0, x1))) → A__SEL(a__from(mark(y0)), cons(mark(x0), x1))
A__SEL(s(from(y0)), cons(y1, head(x0))) → A__SEL(a__from(mark(y0)), a__head(mark(x0)))
A__SEL(s(from(y0)), cons(y1, nil)) → A__SEL(a__from(mark(y0)), nil)
A__SEL(s(from(y0)), cons(y1, sel(x0, x1))) → A__SEL(a__from(mark(y0)), a__sel(mark(x0), mark(x1)))
A__SEL(s(from(y0)), cons(y1, from(x0))) → A__SEL(a__from(mark(y0)), a__from(mark(x0)))
A__SEL(s(from(y0)), cons(y1, 0)) → A__SEL(a__from(mark(y0)), 0)
A__SEL(s(from(y0)), cons(y1, take(x0, x1))) → A__SEL(a__from(mark(y0)), a__take(mark(x0), mark(x1)))
A__SEL(s(from(y0)), cons(y1, 2nd(x0))) → A__SEL(a__from(mark(y0)), a__2nd(mark(x0)))
A__SEL(s(from(y0)), cons(y1, s(x0))) → A__SEL(a__from(mark(y0)), s(mark(x0)))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ QDPOrderProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

                ↳ Narrowing

                  ↳ QDP

                    ↳ DependencyGraphProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

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

A__SEL(s(2nd(x0)), cons(y1, y2)) → A__SEL(a__2nd(mark(x0)), mark(y2))
A__SEL(s(from(y0)), cons(y1, nil)) → A__SEL(a__from(mark(y0)), nil)
A__SEL(s(s(x0)), cons(y1, y2)) → A__SEL(s(mark(x0)), mark(y2))
A__SEL(s(from(y0)), cons(y1, take(x0, x1))) → A__SEL(a__from(mark(y0)), a__take(mark(x0), mark(x1)))
A__SEL(s(take(x0, x1)), cons(y1, y2)) → A__SEL(a__take(mark(x0), mark(x1)), mark(y2))
A__SEL(s(from(y0)), cons(y1, cons(x0, x1))) → A__SEL(a__from(mark(y0)), cons(mark(x0), x1))
A__SEL(s(from(y0)), cons(y1, head(x0))) → A__SEL(a__from(mark(y0)), a__head(mark(x0)))
A__SEL(s(sel(x0, x1)), cons(y1, y2)) → A__SEL(a__sel(mark(x0), mark(x1)), mark(y2))
A__SEL(s(from(y0)), cons(y1, sel(x0, x1))) → A__SEL(a__from(mark(y0)), a__sel(mark(x0), mark(x1)))
A__SEL(s(from(y0)), cons(y1, from(x0))) → A__SEL(a__from(mark(y0)), a__from(mark(x0)))
A__SEL(s(from(y0)), cons(y1, 0)) → A__SEL(a__from(mark(y0)), 0)
A__SEL(s(from(y0)), cons(y1, 2nd(x0))) → A__SEL(a__from(mark(y0)), a__2nd(mark(x0)))
A__SEL(s(from(y0)), cons(y1, s(x0))) → A__SEL(a__from(mark(y0)), s(mark(x0)))
A__SEL(s(head(x0)), cons(y1, y2)) → A__SEL(a__head(mark(x0)), mark(y2))

The TRS R consists of the following rules:

a__from(X) → cons(mark(X), from(s(X)))
a__head(cons(X, XS)) → mark(X)
a__2nd(cons(X, XS)) → a__head(mark(XS))
a__take(0, XS) → nil
a__take(s(N), cons(X, XS)) → cons(mark(X), take(N, XS))
a__sel(0, cons(X, XS)) → mark(X)
a__sel(s(N), cons(X, XS)) → a__sel(mark(N), mark(XS))
mark(from(X)) → a__from(mark(X))
mark(head(X)) → a__head(mark(X))
mark(2nd(X)) → a__2nd(mark(X))
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(sel(X1, X2)) → a__sel(mark(X1), mark(X2))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(s(X)) → s(mark(X))
mark(0) → 0
mark(nil) → nil
a__from(X) → from(X)
a__head(X) → head(X)
a__2nd(X) → 2nd(X)
a__take(X1, X2) → take(X1, X2)
a__sel(X1, X2) → sel(X1, X2)

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ QDPOrderProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

                ↳ Narrowing

                  ↳ QDP

                    ↳ DependencyGraphProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ Narrowing

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

A__SEL(s(from(y0)), cons(y1, head(x0))) → A__SEL(a__from(mark(y0)), a__head(mark(x0)))
A__SEL(s(from(y0)), cons(y1, cons(x0, x1))) → A__SEL(a__from(mark(y0)), cons(mark(x0), x1))
A__SEL(s(sel(x0, x1)), cons(y1, y2)) → A__SEL(a__sel(mark(x0), mark(x1)), mark(y2))
A__SEL(s(2nd(x0)), cons(y1, y2)) → A__SEL(a__2nd(mark(x0)), mark(y2))
A__SEL(s(s(x0)), cons(y1, y2)) → A__SEL(s(mark(x0)), mark(y2))
A__SEL(s(from(y0)), cons(y1, sel(x0, x1))) → A__SEL(a__from(mark(y0)), a__sel(mark(x0), mark(x1)))
A__SEL(s(from(y0)), cons(y1, from(x0))) → A__SEL(a__from(mark(y0)), a__from(mark(x0)))
A__SEL(s(from(y0)), cons(y1, take(x0, x1))) → A__SEL(a__from(mark(y0)), a__take(mark(x0), mark(x1)))
A__SEL(s(from(y0)), cons(y1, 2nd(x0))) → A__SEL(a__from(mark(y0)), a__2nd(mark(x0)))
A__SEL(s(take(x0, x1)), cons(y1, y2)) → A__SEL(a__take(mark(x0), mark(x1)), mark(y2))
A__SEL(s(head(x0)), cons(y1, y2)) → A__SEL(a__head(mark(x0)), mark(y2))

The TRS R consists of the following rules:

a__from(X) → cons(mark(X), from(s(X)))
a__head(cons(X, XS)) → mark(X)
a__2nd(cons(X, XS)) → a__head(mark(XS))
a__take(0, XS) → nil
a__take(s(N), cons(X, XS)) → cons(mark(X), take(N, XS))
a__sel(0, cons(X, XS)) → mark(X)
a__sel(s(N), cons(X, XS)) → a__sel(mark(N), mark(XS))
mark(from(X)) → a__from(mark(X))
mark(head(X)) → a__head(mark(X))
mark(2nd(X)) → a__2nd(mark(X))
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(sel(X1, X2)) → a__sel(mark(X1), mark(X2))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(s(X)) → s(mark(X))
mark(0) → 0
mark(nil) → nil
a__from(X) → from(X)
a__head(X) → head(X)
a__2nd(X) → 2nd(X)
a__take(X1, X2) → take(X1, X2)
a__sel(X1, X2) → sel(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule A__SEL(s(2nd(x0)), cons(y1, y2)) → A__SEL(a__2nd(mark(x0)), mark(y2)) at position [1] we obtained the following new rules:

A__SEL(s(2nd(y0)), cons(y1, head(x0))) → A__SEL(a__2nd(mark(y0)), a__head(mark(x0)))
A__SEL(s(2nd(y0)), cons(y1, cons(x0, x1))) → A__SEL(a__2nd(mark(y0)), cons(mark(x0), x1))
A__SEL(s(2nd(y0)), cons(y1, 2nd(x0))) → A__SEL(a__2nd(mark(y0)), a__2nd(mark(x0)))
A__SEL(s(2nd(y0)), cons(y1, take(x0, x1))) → A__SEL(a__2nd(mark(y0)), a__take(mark(x0), mark(x1)))
A__SEL(s(2nd(y0)), cons(y1, 0)) → A__SEL(a__2nd(mark(y0)), 0)
A__SEL(s(2nd(y0)), cons(y1, sel(x0, x1))) → A__SEL(a__2nd(mark(y0)), a__sel(mark(x0), mark(x1)))
A__SEL(s(2nd(y0)), cons(y1, from(x0))) → A__SEL(a__2nd(mark(y0)), a__from(mark(x0)))
A__SEL(s(2nd(y0)), cons(y1, nil)) → A__SEL(a__2nd(mark(y0)), nil)
A__SEL(s(2nd(y0)), cons(y1, s(x0))) → A__SEL(a__2nd(mark(y0)), s(mark(x0)))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ QDPOrderProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ 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(2nd(y0)), cons(y1, cons(x0, x1))) → A__SEL(a__2nd(mark(y0)), cons(mark(x0), x1))
A__SEL(s(s(x0)), cons(y1, y2)) → A__SEL(s(mark(x0)), mark(y2))
A__SEL(s(2nd(y0)), cons(y1, 0)) → A__SEL(a__2nd(mark(y0)), 0)
A__SEL(s(from(y0)), cons(y1, take(x0, x1))) → A__SEL(a__from(mark(y0)), a__take(mark(x0), mark(x1)))
A__SEL(s(2nd(y0)), cons(y1, sel(x0, x1))) → A__SEL(a__2nd(mark(y0)), a__sel(mark(x0), mark(x1)))
A__SEL(s(take(x0, x1)), cons(y1, y2)) → A__SEL(a__take(mark(x0), mark(x1)), mark(y2))
A__SEL(s(2nd(y0)), cons(y1, nil)) → A__SEL(a__2nd(mark(y0)), nil)
A__SEL(s(2nd(y0)), cons(y1, head(x0))) → A__SEL(a__2nd(mark(y0)), a__head(mark(x0)))
A__SEL(s(from(y0)), cons(y1, head(x0))) → A__SEL(a__from(mark(y0)), a__head(mark(x0)))
A__SEL(s(from(y0)), cons(y1, cons(x0, x1))) → A__SEL(a__from(mark(y0)), cons(mark(x0), x1))
A__SEL(s(2nd(y0)), cons(y1, 2nd(x0))) → A__SEL(a__2nd(mark(y0)), a__2nd(mark(x0)))
A__SEL(s(2nd(y0)), cons(y1, take(x0, x1))) → A__SEL(a__2nd(mark(y0)), a__take(mark(x0), mark(x1)))
A__SEL(s(sel(x0, x1)), cons(y1, y2)) → A__SEL(a__sel(mark(x0), mark(x1)), mark(y2))
A__SEL(s(from(y0)), cons(y1, sel(x0, x1))) → A__SEL(a__from(mark(y0)), a__sel(mark(x0), mark(x1)))
A__SEL(s(from(y0)), cons(y1, from(x0))) → A__SEL(a__from(mark(y0)), a__from(mark(x0)))
A__SEL(s(2nd(y0)), cons(y1, from(x0))) → A__SEL(a__2nd(mark(y0)), a__from(mark(x0)))
A__SEL(s(from(y0)), cons(y1, 2nd(x0))) → A__SEL(a__from(mark(y0)), a__2nd(mark(x0)))
A__SEL(s(head(x0)), cons(y1, y2)) → A__SEL(a__head(mark(x0)), mark(y2))
A__SEL(s(2nd(y0)), cons(y1, s(x0))) → A__SEL(a__2nd(mark(y0)), s(mark(x0)))

The TRS R consists of the following rules:

a__from(X) → cons(mark(X), from(s(X)))
a__head(cons(X, XS)) → mark(X)
a__2nd(cons(X, XS)) → a__head(mark(XS))
a__take(0, XS) → nil
a__take(s(N), cons(X, XS)) → cons(mark(X), take(N, XS))
a__sel(0, cons(X, XS)) → mark(X)
a__sel(s(N), cons(X, XS)) → a__sel(mark(N), mark(XS))
mark(from(X)) → a__from(mark(X))
mark(head(X)) → a__head(mark(X))
mark(2nd(X)) → a__2nd(mark(X))
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(sel(X1, X2)) → a__sel(mark(X1), mark(X2))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(s(X)) → s(mark(X))
mark(0) → 0
mark(nil) → nil
a__from(X) → from(X)
a__head(X) → head(X)
a__2nd(X) → 2nd(X)
a__take(X1, X2) → take(X1, X2)
a__sel(X1, X2) → sel(X1, X2)

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ QDPOrderProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

                ↳ Narrowing

                  ↳ QDP

                    ↳ DependencyGraphProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ Narrowing

                                  ↳ QDP

                                    ↳ DependencyGraphProof

                                      ↳ QDP

                                        ↳ Narrowing

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

A__SEL(s(2nd(y0)), cons(y1, cons(x0, x1))) → A__SEL(a__2nd(mark(y0)), cons(mark(x0), x1))
A__SEL(s(s(x0)), cons(y1, y2)) → A__SEL(s(mark(x0)), mark(y2))
A__SEL(s(from(y0)), cons(y1, take(x0, x1))) → A__SEL(a__from(mark(y0)), a__take(mark(x0), mark(x1)))
A__SEL(s(2nd(y0)), cons(y1, sel(x0, x1))) → A__SEL(a__2nd(mark(y0)), a__sel(mark(x0), mark(x1)))
A__SEL(s(take(x0, x1)), cons(y1, y2)) → A__SEL(a__take(mark(x0), mark(x1)), mark(y2))
A__SEL(s(2nd(y0)), cons(y1, head(x0))) → A__SEL(a__2nd(mark(y0)), a__head(mark(x0)))
A__SEL(s(from(y0)), cons(y1, cons(x0, x1))) → A__SEL(a__from(mark(y0)), cons(mark(x0), x1))
A__SEL(s(from(y0)), cons(y1, head(x0))) → A__SEL(a__from(mark(y0)), a__head(mark(x0)))
A__SEL(s(2nd(y0)), cons(y1, 2nd(x0))) → A__SEL(a__2nd(mark(y0)), a__2nd(mark(x0)))
A__SEL(s(2nd(y0)), cons(y1, take(x0, x1))) → A__SEL(a__2nd(mark(y0)), a__take(mark(x0), mark(x1)))
A__SEL(s(sel(x0, x1)), cons(y1, y2)) → A__SEL(a__sel(mark(x0), mark(x1)), mark(y2))
A__SEL(s(from(y0)), cons(y1, sel(x0, x1))) → A__SEL(a__from(mark(y0)), a__sel(mark(x0), mark(x1)))
A__SEL(s(from(y0)), cons(y1, from(x0))) → A__SEL(a__from(mark(y0)), a__from(mark(x0)))
A__SEL(s(2nd(y0)), cons(y1, from(x0))) → A__SEL(a__2nd(mark(y0)), a__from(mark(x0)))
A__SEL(s(from(y0)), cons(y1, 2nd(x0))) → A__SEL(a__from(mark(y0)), a__2nd(mark(x0)))
A__SEL(s(head(x0)), cons(y1, y2)) → A__SEL(a__head(mark(x0)), mark(y2))

The TRS R consists of the following rules:

a__from(X) → cons(mark(X), from(s(X)))
a__head(cons(X, XS)) → mark(X)
a__2nd(cons(X, XS)) → a__head(mark(XS))
a__take(0, XS) → nil
a__take(s(N), cons(X, XS)) → cons(mark(X), take(N, XS))
a__sel(0, cons(X, XS)) → mark(X)
a__sel(s(N), cons(X, XS)) → a__sel(mark(N), mark(XS))
mark(from(X)) → a__from(mark(X))
mark(head(X)) → a__head(mark(X))
mark(2nd(X)) → a__2nd(mark(X))
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(sel(X1, X2)) → a__sel(mark(X1), mark(X2))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(s(X)) → s(mark(X))
mark(0) → 0
mark(nil) → nil
a__from(X) → from(X)
a__head(X) → head(X)
a__2nd(X) → 2nd(X)
a__take(X1, X2) → take(X1, X2)
a__sel(X1, X2) → sel(X1, X2)

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

A__SEL(s(sel(y0, y1)), cons(y2, sel(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), a__sel(mark(x0), mark(x1)))
A__SEL(s(sel(y0, y1)), cons(y2, head(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__head(mark(x0)))
A__SEL(s(sel(y0, y1)), cons(y2, from(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__from(mark(x0)))
A__SEL(s(sel(y0, y1)), cons(y2, cons(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), cons(mark(x0), x1))
A__SEL(s(sel(y0, y1)), cons(y2, s(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), s(mark(x0)))
A__SEL(s(sel(y0, y1)), cons(y2, 0)) → A__SEL(a__sel(mark(y0), mark(y1)), 0)
A__SEL(s(sel(y0, y1)), cons(y2, nil)) → A__SEL(a__sel(mark(y0), mark(y1)), nil)
A__SEL(s(sel(y0, y1)), cons(y2, take(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), a__take(mark(x0), mark(x1)))
A__SEL(s(sel(y0, y1)), cons(y2, 2nd(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__2nd(mark(x0)))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ QDPOrderProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ 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(sel(y0, y1)), cons(y2, head(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__head(mark(x0)))
A__SEL(s(sel(y0, y1)), cons(y2, from(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__from(mark(x0)))
A__SEL(s(2nd(y0)), cons(y1, sel(x0, x1))) → A__SEL(a__2nd(mark(y0)), a__sel(mark(x0), mark(x1)))
A__SEL(s(take(x0, x1)), cons(y1, y2)) → A__SEL(a__take(mark(x0), mark(x1)), mark(y2))
A__SEL(s(2nd(y0)), cons(y1, head(x0))) → A__SEL(a__2nd(mark(y0)), a__head(mark(x0)))
A__SEL(s(sel(y0, y1)), cons(y2, 0)) → A__SEL(a__sel(mark(y0), mark(y1)), 0)
A__SEL(s(from(y0)), cons(y1, cons(x0, x1))) → A__SEL(a__from(mark(y0)), cons(mark(x0), x1))
A__SEL(s(from(y0)), cons(y1, head(x0))) → A__SEL(a__from(mark(y0)), a__head(mark(x0)))
A__SEL(s(2nd(y0)), cons(y1, 2nd(x0))) → A__SEL(a__2nd(mark(y0)), a__2nd(mark(x0)))
A__SEL(s(2nd(y0)), cons(y1, take(x0, x1))) → A__SEL(a__2nd(mark(y0)), a__take(mark(x0), mark(x1)))
A__SEL(s(sel(y0, y1)), cons(y2, take(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), a__take(mark(x0), mark(x1)))
A__SEL(s(sel(y0, y1)), cons(y2, 2nd(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__2nd(mark(x0)))
A__SEL(s(2nd(y0)), cons(y1, from(x0))) → A__SEL(a__2nd(mark(y0)), a__from(mark(x0)))
A__SEL(s(from(y0)), cons(y1, 2nd(x0))) → A__SEL(a__from(mark(y0)), a__2nd(mark(x0)))
A__SEL(s(head(x0)), cons(y1, y2)) → A__SEL(a__head(mark(x0)), mark(y2))
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(2nd(y0)), cons(y1, cons(x0, x1))) → A__SEL(a__2nd(mark(y0)), cons(mark(x0), x1))
A__SEL(s(s(x0)), cons(y1, y2)) → A__SEL(s(mark(x0)), mark(y2))
A__SEL(s(from(y0)), cons(y1, take(x0, x1))) → A__SEL(a__from(mark(y0)), a__take(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, s(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), s(mark(x0)))
A__SEL(s(sel(y0, y1)), cons(y2, nil)) → A__SEL(a__sel(mark(y0), mark(y1)), nil)
A__SEL(s(from(y0)), cons(y1, sel(x0, x1))) → A__SEL(a__from(mark(y0)), a__sel(mark(x0), mark(x1)))
A__SEL(s(from(y0)), cons(y1, from(x0))) → A__SEL(a__from(mark(y0)), a__from(mark(x0)))

The TRS R consists of the following rules:

a__from(X) → cons(mark(X), from(s(X)))
a__head(cons(X, XS)) → mark(X)
a__2nd(cons(X, XS)) → a__head(mark(XS))
a__take(0, XS) → nil
a__take(s(N), cons(X, XS)) → cons(mark(X), take(N, XS))
a__sel(0, cons(X, XS)) → mark(X)
a__sel(s(N), cons(X, XS)) → a__sel(mark(N), mark(XS))
mark(from(X)) → a__from(mark(X))
mark(head(X)) → a__head(mark(X))
mark(2nd(X)) → a__2nd(mark(X))
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(sel(X1, X2)) → a__sel(mark(X1), mark(X2))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(s(X)) → s(mark(X))
mark(0) → 0
mark(nil) → nil
a__from(X) → from(X)
a__head(X) → head(X)
a__2nd(X) → 2nd(X)
a__take(X1, X2) → take(X1, X2)
a__sel(X1, X2) → sel(X1, X2)

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ QDPOrderProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

                ↳ Narrowing

                  ↳ QDP

                    ↳ DependencyGraphProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ Narrowing

                                  ↳ QDP

                                    ↳ DependencyGraphProof

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Narrowing

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

A__SEL(s(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(2nd(y0)), cons(y1, cons(x0, x1))) → A__SEL(a__2nd(mark(y0)), cons(mark(x0), x1))
A__SEL(s(sel(y0, y1)), cons(y2, head(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__head(mark(x0)))
A__SEL(s(s(x0)), cons(y1, y2)) → A__SEL(s(mark(x0)), mark(y2))
A__SEL(s(sel(y0, y1)), cons(y2, from(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__from(mark(x0)))
A__SEL(s(from(y0)), cons(y1, take(x0, x1))) → A__SEL(a__from(mark(y0)), a__take(mark(x0), mark(x1)))
A__SEL(s(2nd(y0)), cons(y1, sel(x0, x1))) → A__SEL(a__2nd(mark(y0)), 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(take(x0, x1)), cons(y1, y2)) → A__SEL(a__take(mark(x0), mark(x1)), mark(y2))
A__SEL(s(2nd(y0)), cons(y1, head(x0))) → A__SEL(a__2nd(mark(y0)), a__head(mark(x0)))
A__SEL(s(from(y0)), cons(y1, cons(x0, x1))) → A__SEL(a__from(mark(y0)), cons(mark(x0), x1))
A__SEL(s(from(y0)), cons(y1, head(x0))) → A__SEL(a__from(mark(y0)), a__head(mark(x0)))
A__SEL(s(2nd(y0)), cons(y1, 2nd(x0))) → A__SEL(a__2nd(mark(y0)), a__2nd(mark(x0)))
A__SEL(s(sel(y0, y1)), cons(y2, take(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), a__take(mark(x0), mark(x1)))
A__SEL(s(2nd(y0)), cons(y1, take(x0, x1))) → A__SEL(a__2nd(mark(y0)), a__take(mark(x0), mark(x1)))
A__SEL(s(sel(y0, y1)), cons(y2, 2nd(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__2nd(mark(x0)))
A__SEL(s(from(y0)), cons(y1, sel(x0, x1))) → A__SEL(a__from(mark(y0)), a__sel(mark(x0), mark(x1)))
A__SEL(s(from(y0)), cons(y1, from(x0))) → A__SEL(a__from(mark(y0)), a__from(mark(x0)))
A__SEL(s(2nd(y0)), cons(y1, from(x0))) → A__SEL(a__2nd(mark(y0)), a__from(mark(x0)))
A__SEL(s(from(y0)), cons(y1, 2nd(x0))) → A__SEL(a__from(mark(y0)), a__2nd(mark(x0)))
A__SEL(s(head(x0)), cons(y1, y2)) → A__SEL(a__head(mark(x0)), mark(y2))

The TRS R consists of the following rules:

a__from(X) → cons(mark(X), from(s(X)))
a__head(cons(X, XS)) → mark(X)
a__2nd(cons(X, XS)) → a__head(mark(XS))
a__take(0, XS) → nil
a__take(s(N), cons(X, XS)) → cons(mark(X), take(N, XS))
a__sel(0, cons(X, XS)) → mark(X)
a__sel(s(N), cons(X, XS)) → a__sel(mark(N), mark(XS))
mark(from(X)) → a__from(mark(X))
mark(head(X)) → a__head(mark(X))
mark(2nd(X)) → a__2nd(mark(X))
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(sel(X1, X2)) → a__sel(mark(X1), mark(X2))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(s(X)) → s(mark(X))
mark(0) → 0
mark(nil) → nil
a__from(X) → from(X)
a__head(X) → head(X)
a__2nd(X) → 2nd(X)
a__take(X1, X2) → take(X1, X2)
a__sel(X1, X2) → sel(X1, X2)

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

A__SEL(s(s(y0)), cons(y1, s(x0))) → A__SEL(s(mark(y0)), s(mark(x0)))
A__SEL(s(s(y0)), cons(y1, sel(x0, x1))) → A__SEL(s(mark(y0)), a__sel(mark(x0), mark(x1)))
A__SEL(s(s(y0)), cons(y1, 0)) → A__SEL(s(mark(y0)), 0)
A__SEL(s(s(y0)), cons(y1, nil)) → A__SEL(s(mark(y0)), nil)
A__SEL(s(s(y0)), cons(y1, head(x0))) → A__SEL(s(mark(y0)), a__head(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, from(x0))) → A__SEL(s(mark(y0)), a__from(mark(x0)))
A__SEL(s(s(y0)), cons(y1, take(x0, x1))) → A__SEL(s(mark(y0)), a__take(mark(x0), mark(x1)))
A__SEL(s(s(y0)), cons(y1, 2nd(x0))) → A__SEL(s(mark(y0)), a__2nd(mark(x0)))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ QDPOrderProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ 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)))
A__SEL(s(sel(y0, y1)), cons(y2, head(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__head(mark(x0)))
A__SEL(s(sel(y0, y1)), cons(y2, from(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__from(mark(x0)))
A__SEL(s(s(y0)), cons(y1, head(x0))) → A__SEL(s(mark(y0)), a__head(mark(x0)))
A__SEL(s(2nd(y0)), cons(y1, sel(x0, x1))) → A__SEL(a__2nd(mark(y0)), a__sel(mark(x0), mark(x1)))
A__SEL(s(take(x0, x1)), cons(y1, y2)) → A__SEL(a__take(mark(x0), mark(x1)), mark(y2))
A__SEL(s(2nd(y0)), cons(y1, head(x0))) → A__SEL(a__2nd(mark(y0)), a__head(mark(x0)))
A__SEL(s(from(y0)), cons(y1, cons(x0, x1))) → A__SEL(a__from(mark(y0)), cons(mark(x0), x1))
A__SEL(s(from(y0)), cons(y1, head(x0))) → A__SEL(a__from(mark(y0)), a__head(mark(x0)))
A__SEL(s(2nd(y0)), cons(y1, 2nd(x0))) → A__SEL(a__2nd(mark(y0)), a__2nd(mark(x0)))
A__SEL(s(sel(y0, y1)), cons(y2, take(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), a__take(mark(x0), mark(x1)))
A__SEL(s(2nd(y0)), cons(y1, take(x0, x1))) → A__SEL(a__2nd(mark(y0)), a__take(mark(x0), mark(x1)))
A__SEL(s(sel(y0, y1)), cons(y2, 2nd(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__2nd(mark(x0)))
A__SEL(s(s(y0)), cons(y1, nil)) → A__SEL(s(mark(y0)), nil)
A__SEL(s(2nd(y0)), cons(y1, from(x0))) → A__SEL(a__2nd(mark(y0)), a__from(mark(x0)))
A__SEL(s(from(y0)), cons(y1, 2nd(x0))) → A__SEL(a__from(mark(y0)), a__2nd(mark(x0)))
A__SEL(s(s(y0)), cons(y1, from(x0))) → A__SEL(s(mark(y0)), a__from(mark(x0)))
A__SEL(s(head(x0)), cons(y1, y2)) → A__SEL(a__head(mark(x0)), mark(y2))
A__SEL(s(s(y0)), cons(y1, 2nd(x0))) → A__SEL(s(mark(y0)), a__2nd(mark(x0)))
A__SEL(s(s(y0)), cons(y1, take(x0, x1))) → A__SEL(s(mark(y0)), a__take(mark(x0), mark(x1)))
A__SEL(s(sel(y0, y1)), cons(y2, sel(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), a__sel(mark(x0), mark(x1)))
A__SEL(s(s(y0)), cons(y1, sel(x0, x1))) → A__SEL(s(mark(y0)), a__sel(mark(x0), mark(x1)))
A__SEL(s(2nd(y0)), cons(y1, cons(x0, x1))) → A__SEL(a__2nd(mark(y0)), cons(mark(x0), x1))
A__SEL(s(from(y0)), cons(y1, take(x0, x1))) → A__SEL(a__from(mark(y0)), a__take(mark(x0), mark(x1)))
A__SEL(s(sel(y0, y1)), cons(y2, cons(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), cons(mark(x0), x1))
A__SEL(s(s(y0)), cons(y1, cons(x0, x1))) → A__SEL(s(mark(y0)), cons(mark(x0), x1))
A__SEL(s(s(y0)), cons(y1, 0)) → A__SEL(s(mark(y0)), 0)
A__SEL(s(from(y0)), cons(y1, sel(x0, x1))) → A__SEL(a__from(mark(y0)), a__sel(mark(x0), mark(x1)))
A__SEL(s(from(y0)), cons(y1, from(x0))) → A__SEL(a__from(mark(y0)), a__from(mark(x0)))

The TRS R consists of the following rules:

a__from(X) → cons(mark(X), from(s(X)))
a__head(cons(X, XS)) → mark(X)
a__2nd(cons(X, XS)) → a__head(mark(XS))
a__take(0, XS) → nil
a__take(s(N), cons(X, XS)) → cons(mark(X), take(N, XS))
a__sel(0, cons(X, XS)) → mark(X)
a__sel(s(N), cons(X, XS)) → a__sel(mark(N), mark(XS))
mark(from(X)) → a__from(mark(X))
mark(head(X)) → a__head(mark(X))
mark(2nd(X)) → a__2nd(mark(X))
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(sel(X1, X2)) → a__sel(mark(X1), mark(X2))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(s(X)) → s(mark(X))
mark(0) → 0
mark(nil) → nil
a__from(X) → from(X)
a__head(X) → head(X)
a__2nd(X) → 2nd(X)
a__take(X1, X2) → take(X1, X2)
a__sel(X1, X2) → sel(X1, X2)

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ QDPOrderProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

                ↳ Narrowing

                  ↳ QDP

                    ↳ DependencyGraphProof

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ Narrowing

                                  ↳ QDP

                                    ↳ DependencyGraphProof

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ DependencyGraphProof

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ Narrowing

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

A__SEL(s(sel(y0, y1)), cons(y2, head(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__head(mark(x0)))
A__SEL(s(sel(y0, y1)), cons(y2, from(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__from(mark(x0)))
A__SEL(s(s(y0)), cons(y1, head(x0))) → A__SEL(s(mark(y0)), a__head(mark(x0)))
A__SEL(s(2nd(y0)), cons(y1, sel(x0, x1))) → A__SEL(a__2nd(mark(y0)), a__sel(mark(x0), mark(x1)))
A__SEL(s(take(x0, x1)), cons(y1, y2)) → A__SEL(a__take(mark(x0), mark(x1)), mark(y2))
A__SEL(s(2nd(y0)), cons(y1, head(x0))) → A__SEL(a__2nd(mark(y0)), a__head(mark(x0)))
A__SEL(s(from(y0)), cons(y1, head(x0))) → A__SEL(a__from(mark(y0)), a__head(mark(x0)))
A__SEL(s(from(y0)), cons(y1, cons(x0, x1))) → A__SEL(a__from(mark(y0)), cons(mark(x0), x1))
A__SEL(s(2nd(y0)), cons(y1, 2nd(x0))) → A__SEL(a__2nd(mark(y0)), a__2nd(mark(x0)))
A__SEL(s(2nd(y0)), cons(y1, take(x0, x1))) → A__SEL(a__2nd(mark(y0)), a__take(mark(x0), mark(x1)))
A__SEL(s(sel(y0, y1)), cons(y2, take(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), a__take(mark(x0), mark(x1)))
A__SEL(s(sel(y0, y1)), cons(y2, 2nd(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__2nd(mark(x0)))
A__SEL(s(from(y0)), cons(y1, 2nd(x0))) → A__SEL(a__from(mark(y0)), a__2nd(mark(x0)))
A__SEL(s(2nd(y0)), cons(y1, from(x0))) → A__SEL(a__2nd(mark(y0)), a__from(mark(x0)))
A__SEL(s(head(x0)), cons(y1, y2)) → A__SEL(a__head(mark(x0)), mark(y2))
A__SEL(s(s(y0)), cons(y1, from(x0))) → A__SEL(s(mark(y0)), a__from(mark(x0)))
A__SEL(s(s(y0)), cons(y1, 2nd(x0))) → A__SEL(s(mark(y0)), a__2nd(mark(x0)))
A__SEL(s(s(y0)), cons(y1, take(x0, x1))) → A__SEL(s(mark(y0)), a__take(mark(x0), mark(x1)))
A__SEL(s(sel(y0, y1)), cons(y2, sel(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), a__sel(mark(x0), mark(x1)))
A__SEL(s(s(y0)), cons(y1, sel(x0, x1))) → A__SEL(s(mark(y0)), a__sel(mark(x0), mark(x1)))
A__SEL(s(2nd(y0)), cons(y1, cons(x0, x1))) → A__SEL(a__2nd(mark(y0)), cons(mark(x0), x1))
A__SEL(s(from(y0)), cons(y1, take(x0, x1))) → A__SEL(a__from(mark(y0)), a__take(mark(x0), mark(x1)))
A__SEL(s(sel(y0, y1)), cons(y2, cons(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), cons(mark(x0), x1))
A__SEL(s(s(y0)), cons(y1, cons(x0, x1))) → A__SEL(s(mark(y0)), cons(mark(x0), x1))
A__SEL(s(from(y0)), cons(y1, sel(x0, x1))) → A__SEL(a__from(mark(y0)), a__sel(mark(x0), mark(x1)))
A__SEL(s(from(y0)), cons(y1, from(x0))) → A__SEL(a__from(mark(y0)), a__from(mark(x0)))

The TRS R consists of the following rules:

a__from(X) → cons(mark(X), from(s(X)))
a__head(cons(X, XS)) → mark(X)
a__2nd(cons(X, XS)) → a__head(mark(XS))
a__take(0, XS) → nil
a__take(s(N), cons(X, XS)) → cons(mark(X), take(N, XS))
a__sel(0, cons(X, XS)) → mark(X)
a__sel(s(N), cons(X, XS)) → a__sel(mark(N), mark(XS))
mark(from(X)) → a__from(mark(X))
mark(head(X)) → a__head(mark(X))
mark(2nd(X)) → a__2nd(mark(X))
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(sel(X1, X2)) → a__sel(mark(X1), mark(X2))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(s(X)) → s(mark(X))
mark(0) → 0
mark(nil) → nil
a__from(X) → from(X)
a__head(X) → head(X)
a__2nd(X) → 2nd(X)
a__take(X1, X2) → take(X1, X2)
a__sel(X1, X2) → sel(X1, X2)

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

A__SEL(s(take(y0, y1)), cons(y2, cons(x0, x1))) → A__SEL(a__take(mark(y0), mark(y1)), cons(mark(x0), x1))
A__SEL(s(take(y0, y1)), cons(y2, head(x0))) → A__SEL(a__take(mark(y0), mark(y1)), a__head(mark(x0)))
A__SEL(s(take(y0, y1)), cons(y2, s(x0))) → A__SEL(a__take(mark(y0), mark(y1)), s(mark(x0)))
A__SEL(s(take(y0, y1)), cons(y2, take(x0, x1))) → A__SEL(a__take(mark(y0), mark(y1)), a__take(mark(x0), mark(x1)))
A__SEL(s(take(y0, y1)), cons(y2, 2nd(x0))) → A__SEL(a__take(mark(y0), mark(y1)), a__2nd(mark(x0)))
A__SEL(s(take(y0, y1)), cons(y2, from(x0))) → A__SEL(a__take(mark(y0), mark(y1)), a__from(mark(x0)))
A__SEL(s(take(y0, y1)), cons(y2, sel(x0, x1))) → A__SEL(a__take(mark(y0), mark(y1)), a__sel(mark(x0), mark(x1)))
A__SEL(s(take(y0, y1)), cons(y2, nil)) → A__SEL(a__take(mark(y0), mark(y1)), nil)
A__SEL(s(take(y0, y1)), cons(y2, 0)) → A__SEL(a__take(mark(y0), mark(y1)), 0)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ QDPOrderProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ 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(sel(y0, y1)), cons(y2, head(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__head(mark(x0)))
A__SEL(s(sel(y0, y1)), cons(y2, from(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__from(mark(x0)))
A__SEL(s(take(y0, y1)), cons(y2, s(x0))) → A__SEL(a__take(mark(y0), mark(y1)), s(mark(x0)))
A__SEL(s(s(y0)), cons(y1, head(x0))) → A__SEL(s(mark(y0)), a__head(mark(x0)))
A__SEL(s(2nd(y0)), cons(y1, sel(x0, x1))) → A__SEL(a__2nd(mark(y0)), a__sel(mark(x0), mark(x1)))
A__SEL(s(2nd(y0)), cons(y1, head(x0))) → A__SEL(a__2nd(mark(y0)), a__head(mark(x0)))
A__SEL(s(from(y0)), cons(y1, cons(x0, x1))) → A__SEL(a__from(mark(y0)), cons(mark(x0), x1))
A__SEL(s(from(y0)), cons(y1, head(x0))) → A__SEL(a__from(mark(y0)), a__head(mark(x0)))
A__SEL(s(2nd(y0)), cons(y1, 2nd(x0))) → A__SEL(a__2nd(mark(y0)), a__2nd(mark(x0)))
A__SEL(s(take(y0, y1)), cons(y2, sel(x0, x1))) → A__SEL(a__take(mark(y0), mark(y1)), a__sel(mark(x0), mark(x1)))
A__SEL(s(sel(y0, y1)), cons(y2, take(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), a__take(mark(x0), mark(x1)))
A__SEL(s(2nd(y0)), cons(y1, take(x0, x1))) → A__SEL(a__2nd(mark(y0)), a__take(mark(x0), mark(x1)))
A__SEL(s(sel(y0, y1)), cons(y2, 2nd(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__2nd(mark(x0)))
A__SEL(s(2nd(y0)), cons(y1, from(x0))) → A__SEL(a__2nd(mark(y0)), a__from(mark(x0)))
A__SEL(s(from(y0)), cons(y1, 2nd(x0))) → A__SEL(a__from(mark(y0)), a__2nd(mark(x0)))
A__SEL(s(s(y0)), cons(y1, from(x0))) → A__SEL(s(mark(y0)), a__from(mark(x0)))
A__SEL(s(head(x0)), cons(y1, y2)) → A__SEL(a__head(mark(x0)), mark(y2))
A__SEL(s(s(y0)), cons(y1, take(x0, x1))) → A__SEL(s(mark(y0)), a__take(mark(x0), mark(x1)))
A__SEL(s(s(y0)), cons(y1, 2nd(x0))) → A__SEL(s(mark(y0)), a__2nd(mark(x0)))
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(2nd(y0)), cons(y1, cons(x0, x1))) → A__SEL(a__2nd(mark(y0)), cons(mark(x0), x1))
A__SEL(s(take(y0, y1)), cons(y2, head(x0))) → A__SEL(a__take(mark(y0), mark(y1)), a__head(mark(x0)))
A__SEL(s(take(y0, y1)), cons(y2, cons(x0, x1))) → A__SEL(a__take(mark(y0), mark(y1)), cons(mark(x0), x1))
A__SEL(s(from(y0)), cons(y1, take(x0, x1))) → A__SEL(a__from(mark(y0)), a__take(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(take(y0, y1)), cons(y2, take(x0, x1))) → A__SEL(a__take(mark(y0), mark(y1)), a__take(mark(x0), mark(x1)))
A__SEL(s(take(y0, y1)), cons(y2, from(x0))) → A__SEL(a__take(mark(y0), mark(y1)), a__from(mark(x0)))
A__SEL(s(take(y0, y1)), cons(y2, 2nd(x0))) → A__SEL(a__take(mark(y0), mark(y1)), a__2nd(mark(x0)))
A__SEL(s(s(y0)), cons(y1, cons(x0, x1))) → A__SEL(s(mark(y0)), cons(mark(x0), x1))
A__SEL(s(from(y0)), cons(y1, sel(x0, x1))) → A__SEL(a__from(mark(y0)), a__sel(mark(x0), mark(x1)))
A__SEL(s(from(y0)), cons(y1, from(x0))) → A__SEL(a__from(mark(y0)), a__from(mark(x0)))
A__SEL(s(take(y0, y1)), cons(y2, nil)) → A__SEL(a__take(mark(y0), mark(y1)), nil)
A__SEL(s(take(y0, y1)), cons(y2, 0)) → A__SEL(a__take(mark(y0), mark(y1)), 0)

The TRS R consists of the following rules:

a__from(X) → cons(mark(X), from(s(X)))
a__head(cons(X, XS)) → mark(X)
a__2nd(cons(X, XS)) → a__head(mark(XS))
a__take(0, XS) → nil
a__take(s(N), cons(X, XS)) → cons(mark(X), take(N, XS))
a__sel(0, cons(X, XS)) → mark(X)
a__sel(s(N), cons(X, XS)) → a__sel(mark(N), mark(XS))
mark(from(X)) → a__from(mark(X))
mark(head(X)) → a__head(mark(X))
mark(2nd(X)) → a__2nd(mark(X))
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(sel(X1, X2)) → a__sel(mark(X1), mark(X2))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(s(X)) → s(mark(X))
mark(0) → 0
mark(nil) → nil
a__from(X) → from(X)
a__head(X) → head(X)
a__2nd(X) → 2nd(X)
a__take(X1, X2) → take(X1, X2)
a__sel(X1, X2) → sel(X1, X2)

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ QDPOrderProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

                ↳ Narrowing

                  ↳ 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:

A__SEL(s(sel(y0, y1)), cons(y2, head(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__head(mark(x0)))
A__SEL(s(sel(y0, y1)), cons(y2, from(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__from(mark(x0)))
A__SEL(s(s(y0)), cons(y1, head(x0))) → A__SEL(s(mark(y0)), a__head(mark(x0)))
A__SEL(s(2nd(y0)), cons(y1, sel(x0, x1))) → A__SEL(a__2nd(mark(y0)), a__sel(mark(x0), mark(x1)))
A__SEL(s(2nd(y0)), cons(y1, head(x0))) → A__SEL(a__2nd(mark(y0)), a__head(mark(x0)))
A__SEL(s(from(y0)), cons(y1, head(x0))) → A__SEL(a__from(mark(y0)), a__head(mark(x0)))
A__SEL(s(from(y0)), cons(y1, cons(x0, x1))) → A__SEL(a__from(mark(y0)), cons(mark(x0), x1))
A__SEL(s(2nd(y0)), cons(y1, 2nd(x0))) → A__SEL(a__2nd(mark(y0)), a__2nd(mark(x0)))
A__SEL(s(take(y0, y1)), cons(y2, sel(x0, x1))) → A__SEL(a__take(mark(y0), mark(y1)), a__sel(mark(x0), mark(x1)))
A__SEL(s(2nd(y0)), cons(y1, take(x0, x1))) → A__SEL(a__2nd(mark(y0)), a__take(mark(x0), mark(x1)))
A__SEL(s(sel(y0, y1)), cons(y2, take(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), a__take(mark(x0), mark(x1)))
A__SEL(s(sel(y0, y1)), cons(y2, 2nd(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__2nd(mark(x0)))
A__SEL(s(2nd(y0)), cons(y1, from(x0))) → A__SEL(a__2nd(mark(y0)), a__from(mark(x0)))
A__SEL(s(from(y0)), cons(y1, 2nd(x0))) → A__SEL(a__from(mark(y0)), a__2nd(mark(x0)))
A__SEL(s(head(x0)), cons(y1, y2)) → A__SEL(a__head(mark(x0)), mark(y2))
A__SEL(s(s(y0)), cons(y1, from(x0))) → A__SEL(s(mark(y0)), a__from(mark(x0)))
A__SEL(s(s(y0)), cons(y1, take(x0, x1))) → A__SEL(s(mark(y0)), a__take(mark(x0), mark(x1)))
A__SEL(s(s(y0)), cons(y1, 2nd(x0))) → A__SEL(s(mark(y0)), a__2nd(mark(x0)))
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(2nd(y0)), cons(y1, cons(x0, x1))) → A__SEL(a__2nd(mark(y0)), cons(mark(x0), x1))
A__SEL(s(take(y0, y1)), cons(y2, head(x0))) → A__SEL(a__take(mark(y0), mark(y1)), a__head(mark(x0)))
A__SEL(s(take(y0, y1)), cons(y2, cons(x0, x1))) → A__SEL(a__take(mark(y0), mark(y1)), cons(mark(x0), x1))
A__SEL(s(from(y0)), cons(y1, take(x0, x1))) → A__SEL(a__from(mark(y0)), a__take(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(take(y0, y1)), cons(y2, take(x0, x1))) → A__SEL(a__take(mark(y0), mark(y1)), a__take(mark(x0), mark(x1)))
A__SEL(s(take(y0, y1)), cons(y2, from(x0))) → A__SEL(a__take(mark(y0), mark(y1)), a__from(mark(x0)))
A__SEL(s(take(y0, y1)), cons(y2, 2nd(x0))) → A__SEL(a__take(mark(y0), mark(y1)), a__2nd(mark(x0)))
A__SEL(s(s(y0)), cons(y1, cons(x0, x1))) → A__SEL(s(mark(y0)), cons(mark(x0), x1))
A__SEL(s(from(y0)), cons(y1, sel(x0, x1))) → A__SEL(a__from(mark(y0)), a__sel(mark(x0), mark(x1)))
A__SEL(s(from(y0)), cons(y1, from(x0))) → A__SEL(a__from(mark(y0)), a__from(mark(x0)))

The TRS R consists of the following rules:

a__from(X) → cons(mark(X), from(s(X)))
a__head(cons(X, XS)) → mark(X)
a__2nd(cons(X, XS)) → a__head(mark(XS))
a__take(0, XS) → nil
a__take(s(N), cons(X, XS)) → cons(mark(X), take(N, XS))
a__sel(0, cons(X, XS)) → mark(X)
a__sel(s(N), cons(X, XS)) → a__sel(mark(N), mark(XS))
mark(from(X)) → a__from(mark(X))
mark(head(X)) → a__head(mark(X))
mark(2nd(X)) → a__2nd(mark(X))
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(sel(X1, X2)) → a__sel(mark(X1), mark(X2))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(s(X)) → s(mark(X))
mark(0) → 0
mark(nil) → nil
a__from(X) → from(X)
a__head(X) → head(X)
a__2nd(X) → 2nd(X)
a__take(X1, X2) → take(X1, X2)
a__sel(X1, X2) → sel(X1, X2)

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

A__SEL(s(head(y0)), cons(y1, take(x0, x1))) → A__SEL(a__head(mark(y0)), a__take(mark(x0), mark(x1)))
A__SEL(s(head(y0)), cons(y1, 2nd(x0))) → A__SEL(a__head(mark(y0)), a__2nd(mark(x0)))
A__SEL(s(head(y0)), cons(y1, from(x0))) → A__SEL(a__head(mark(y0)), a__from(mark(x0)))
A__SEL(s(head(y0)), cons(y1, 0)) → A__SEL(a__head(mark(y0)), 0)
A__SEL(s(head(y0)), cons(y1, head(x0))) → A__SEL(a__head(mark(y0)), a__head(mark(x0)))
A__SEL(s(head(y0)), cons(y1, cons(x0, x1))) → A__SEL(a__head(mark(y0)), cons(mark(x0), x1))
A__SEL(s(head(y0)), cons(y1, s(x0))) → A__SEL(a__head(mark(y0)), s(mark(x0)))
A__SEL(s(head(y0)), cons(y1, sel(x0, x1))) → A__SEL(a__head(mark(y0)), a__sel(mark(x0), mark(x1)))
A__SEL(s(head(y0)), cons(y1, nil)) → A__SEL(a__head(mark(y0)), nil)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ QDPOrderProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

                ↳ Narrowing

                  ↳ QDP

                    ↳ DependencyGraphProof

                      ↳ 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(head(y0)), cons(y1, take(x0, x1))) → A__SEL(a__head(mark(y0)), a__take(mark(x0), mark(x1)))
A__SEL(s(sel(y0, y1)), cons(y2, head(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__head(mark(x0)))
A__SEL(s(sel(y0, y1)), cons(y2, from(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__from(mark(x0)))
A__SEL(s(head(y0)), cons(y1, s(x0))) → A__SEL(a__head(mark(y0)), s(mark(x0)))
A__SEL(s(s(y0)), cons(y1, head(x0))) → A__SEL(s(mark(y0)), a__head(mark(x0)))
A__SEL(s(2nd(y0)), cons(y1, sel(x0, x1))) → A__SEL(a__2nd(mark(y0)), a__sel(mark(x0), mark(x1)))
A__SEL(s(head(y0)), cons(y1, 2nd(x0))) → A__SEL(a__head(mark(y0)), a__2nd(mark(x0)))
A__SEL(s(2nd(y0)), cons(y1, head(x0))) → A__SEL(a__2nd(mark(y0)), a__head(mark(x0)))
A__SEL(s(head(y0)), cons(y1, from(x0))) → A__SEL(a__head(mark(y0)), a__from(mark(x0)))
A__SEL(s(from(y0)), cons(y1, cons(x0, x1))) → A__SEL(a__from(mark(y0)), cons(mark(x0), x1))
A__SEL(s(from(y0)), cons(y1, head(x0))) → A__SEL(a__from(mark(y0)), a__head(mark(x0)))
A__SEL(s(2nd(y0)), cons(y1, 2nd(x0))) → A__SEL(a__2nd(mark(y0)), a__2nd(mark(x0)))
A__SEL(s(sel(y0, y1)), cons(y2, take(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), a__take(mark(x0), mark(x1)))
A__SEL(s(2nd(y0)), cons(y1, take(x0, x1))) → A__SEL(a__2nd(mark(y0)), a__take(mark(x0), mark(x1)))
A__SEL(s(take(y0, y1)), cons(y2, sel(x0, x1))) → A__SEL(a__take(mark(y0), mark(y1)), a__sel(mark(x0), mark(x1)))
A__SEL(s(head(y0)), cons(y1, cons(x0, x1))) → A__SEL(a__head(mark(y0)), cons(mark(x0), x1))
A__SEL(s(sel(y0, y1)), cons(y2, 2nd(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__2nd(mark(x0)))
A__SEL(s(from(y0)), cons(y1, 2nd(x0))) → A__SEL(a__from(mark(y0)), a__2nd(mark(x0)))
A__SEL(s(2nd(y0)), cons(y1, from(x0))) → A__SEL(a__2nd(mark(y0)), a__from(mark(x0)))
A__SEL(s(s(y0)), cons(y1, from(x0))) → A__SEL(s(mark(y0)), a__from(mark(x0)))
A__SEL(s(s(y0)), cons(y1, 2nd(x0))) → A__SEL(s(mark(y0)), a__2nd(mark(x0)))
A__SEL(s(s(y0)), cons(y1, take(x0, x1))) → A__SEL(s(mark(y0)), a__take(mark(x0), mark(x1)))
A__SEL(s(sel(y0, y1)), cons(y2, sel(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), a__sel(mark(x0), mark(x1)))
A__SEL(s(s(y0)), cons(y1, sel(x0, x1))) → A__SEL(s(mark(y0)), a__sel(mark(x0), mark(x1)))
A__SEL(s(2nd(y0)), cons(y1, cons(x0, x1))) → A__SEL(a__2nd(mark(y0)), cons(mark(x0), x1))
A__SEL(s(take(y0, y1)), cons(y2, cons(x0, x1))) → A__SEL(a__take(mark(y0), mark(y1)), cons(mark(x0), x1))
A__SEL(s(take(y0, y1)), cons(y2, head(x0))) → A__SEL(a__take(mark(y0), mark(y1)), a__head(mark(x0)))
A__SEL(s(from(y0)), cons(y1, take(x0, x1))) → A__SEL(a__from(mark(y0)), a__take(mark(x0), mark(x1)))
A__SEL(s(head(y0)), cons(y1, sel(x0, x1))) → A__SEL(a__head(mark(y0)), 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(take(y0, y1)), cons(y2, take(x0, x1))) → A__SEL(a__take(mark(y0), mark(y1)), a__take(mark(x0), mark(x1)))
A__SEL(s(s(y0)), cons(y1, cons(x0, x1))) → A__SEL(s(mark(y0)), cons(mark(x0), x1))
A__SEL(s(take(y0, y1)), cons(y2, 2nd(x0))) → A__SEL(a__take(mark(y0), mark(y1)), a__2nd(mark(x0)))
A__SEL(s(take(y0, y1)), cons(y2, from(x0))) → A__SEL(a__take(mark(y0), mark(y1)), a__from(mark(x0)))
A__SEL(s(head(y0)), cons(y1, 0)) → A__SEL(a__head(mark(y0)), 0)
A__SEL(s(head(y0)), cons(y1, head(x0))) → A__SEL(a__head(mark(y0)), a__head(mark(x0)))
A__SEL(s(from(y0)), cons(y1, sel(x0, x1))) → A__SEL(a__from(mark(y0)), a__sel(mark(x0), mark(x1)))
A__SEL(s(from(y0)), cons(y1, from(x0))) → A__SEL(a__from(mark(y0)), a__from(mark(x0)))
A__SEL(s(head(y0)), cons(y1, nil)) → A__SEL(a__head(mark(y0)), nil)

The TRS R consists of the following rules:

a__from(X) → cons(mark(X), from(s(X)))
a__head(cons(X, XS)) → mark(X)
a__2nd(cons(X, XS)) → a__head(mark(XS))
a__take(0, XS) → nil
a__take(s(N), cons(X, XS)) → cons(mark(X), take(N, XS))
a__sel(0, cons(X, XS)) → mark(X)
a__sel(s(N), cons(X, XS)) → a__sel(mark(N), mark(XS))
mark(from(X)) → a__from(mark(X))
mark(head(X)) → a__head(mark(X))
mark(2nd(X)) → a__2nd(mark(X))
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(sel(X1, X2)) → a__sel(mark(X1), mark(X2))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(s(X)) → s(mark(X))
mark(0) → 0
mark(nil) → nil
a__from(X) → from(X)
a__head(X) → head(X)
a__2nd(X) → 2nd(X)
a__take(X1, X2) → take(X1, X2)
a__sel(X1, X2) → sel(X1, X2)

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ QDPOrderProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

                ↳ Narrowing

                  ↳ QDP

                    ↳ DependencyGraphProof

                      ↳ QDP

                        ↳ 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:

A__SEL(s(head(y0)), cons(y1, take(x0, x1))) → A__SEL(a__head(mark(y0)), a__take(mark(x0), mark(x1)))
A__SEL(s(sel(y0, y1)), cons(y2, head(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__head(mark(x0)))
A__SEL(s(sel(y0, y1)), cons(y2, from(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__from(mark(x0)))
A__SEL(s(s(y0)), cons(y1, head(x0))) → A__SEL(s(mark(y0)), a__head(mark(x0)))
A__SEL(s(2nd(y0)), cons(y1, sel(x0, x1))) → A__SEL(a__2nd(mark(y0)), a__sel(mark(x0), mark(x1)))
A__SEL(s(head(y0)), cons(y1, 2nd(x0))) → A__SEL(a__head(mark(y0)), a__2nd(mark(x0)))
A__SEL(s(2nd(y0)), cons(y1, head(x0))) → A__SEL(a__2nd(mark(y0)), a__head(mark(x0)))
A__SEL(s(head(y0)), cons(y1, from(x0))) → A__SEL(a__head(mark(y0)), a__from(mark(x0)))
A__SEL(s(from(y0)), cons(y1, head(x0))) → A__SEL(a__from(mark(y0)), a__head(mark(x0)))
A__SEL(s(from(y0)), cons(y1, cons(x0, x1))) → A__SEL(a__from(mark(y0)), cons(mark(x0), x1))
A__SEL(s(2nd(y0)), cons(y1, 2nd(x0))) → A__SEL(a__2nd(mark(y0)), a__2nd(mark(x0)))
A__SEL(s(take(y0, y1)), cons(y2, sel(x0, x1))) → A__SEL(a__take(mark(y0), mark(y1)), a__sel(mark(x0), mark(x1)))
A__SEL(s(2nd(y0)), cons(y1, take(x0, x1))) → A__SEL(a__2nd(mark(y0)), a__take(mark(x0), mark(x1)))
A__SEL(s(sel(y0, y1)), cons(y2, take(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), a__take(mark(x0), mark(x1)))
A__SEL(s(head(y0)), cons(y1, cons(x0, x1))) → A__SEL(a__head(mark(y0)), cons(mark(x0), x1))
A__SEL(s(sel(y0, y1)), cons(y2, 2nd(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__2nd(mark(x0)))
A__SEL(s(2nd(y0)), cons(y1, from(x0))) → A__SEL(a__2nd(mark(y0)), a__from(mark(x0)))
A__SEL(s(from(y0)), cons(y1, 2nd(x0))) → A__SEL(a__from(mark(y0)), a__2nd(mark(x0)))
A__SEL(s(s(y0)), cons(y1, from(x0))) → A__SEL(s(mark(y0)), a__from(mark(x0)))
A__SEL(s(s(y0)), cons(y1, take(x0, x1))) → A__SEL(s(mark(y0)), a__take(mark(x0), mark(x1)))
A__SEL(s(s(y0)), cons(y1, 2nd(x0))) → A__SEL(s(mark(y0)), a__2nd(mark(x0)))
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(2nd(y0)), cons(y1, cons(x0, x1))) → A__SEL(a__2nd(mark(y0)), cons(mark(x0), x1))
A__SEL(s(take(y0, y1)), cons(y2, head(x0))) → A__SEL(a__take(mark(y0), mark(y1)), a__head(mark(x0)))
A__SEL(s(take(y0, y1)), cons(y2, cons(x0, x1))) → A__SEL(a__take(mark(y0), mark(y1)), cons(mark(x0), x1))
A__SEL(s(from(y0)), cons(y1, take(x0, x1))) → A__SEL(a__from(mark(y0)), a__take(mark(x0), mark(x1)))
A__SEL(s(head(y0)), cons(y1, sel(x0, x1))) → A__SEL(a__head(mark(y0)), 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(take(y0, y1)), cons(y2, take(x0, x1))) → A__SEL(a__take(mark(y0), mark(y1)), a__take(mark(x0), mark(x1)))
A__SEL(s(take(y0, y1)), cons(y2, from(x0))) → A__SEL(a__take(mark(y0), mark(y1)), a__from(mark(x0)))
A__SEL(s(take(y0, y1)), cons(y2, 2nd(x0))) → A__SEL(a__take(mark(y0), mark(y1)), a__2nd(mark(x0)))
A__SEL(s(s(y0)), cons(y1, cons(x0, x1))) → A__SEL(s(mark(y0)), cons(mark(x0), x1))
A__SEL(s(head(y0)), cons(y1, head(x0))) → A__SEL(a__head(mark(y0)), a__head(mark(x0)))
A__SEL(s(from(y0)), cons(y1, sel(x0, x1))) → A__SEL(a__from(mark(y0)), a__sel(mark(x0), mark(x1)))
A__SEL(s(from(y0)), cons(y1, from(x0))) → A__SEL(a__from(mark(y0)), a__from(mark(x0)))

The TRS R consists of the following rules:

a__from(X) → cons(mark(X), from(s(X)))
a__head(cons(X, XS)) → mark(X)
a__2nd(cons(X, XS)) → a__head(mark(XS))
a__take(0, XS) → nil
a__take(s(N), cons(X, XS)) → cons(mark(X), take(N, XS))
a__sel(0, cons(X, XS)) → mark(X)
a__sel(s(N), cons(X, XS)) → a__sel(mark(N), mark(XS))
mark(from(X)) → a__from(mark(X))
mark(head(X)) → a__head(mark(X))
mark(2nd(X)) → a__2nd(mark(X))
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(sel(X1, X2)) → a__sel(mark(X1), mark(X2))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(s(X)) → s(mark(X))
mark(0) → 0
mark(nil) → nil
a__from(X) → from(X)
a__head(X) → head(X)
a__2nd(X) → 2nd(X)
a__take(X1, X2) → take(X1, X2)
a__sel(X1, X2) → sel(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].

The following pairs can be oriented strictly and are deleted.

A__SEL(s(from(y0)), cons(y1, head(x0))) → A__SEL(a__from(mark(y0)), a__head(mark(x0)))
A__SEL(s(from(y0)), cons(y1, cons(x0, x1))) → A__SEL(a__from(mark(y0)), cons(mark(x0), x1))
A__SEL(s(take(y0, y1)), cons(y2, sel(x0, x1))) → A__SEL(a__take(mark(y0), mark(y1)), a__sel(mark(x0), mark(x1)))
A__SEL(s(from(y0)), cons(y1, 2nd(x0))) → A__SEL(a__from(mark(y0)), a__2nd(mark(x0)))
A__SEL(s(take(y0, y1)), cons(y2, head(x0))) → A__SEL(a__take(mark(y0), mark(y1)), a__head(mark(x0)))
A__SEL(s(take(y0, y1)), cons(y2, cons(x0, x1))) → A__SEL(a__take(mark(y0), mark(y1)), cons(mark(x0), x1))
A__SEL(s(from(y0)), cons(y1, take(x0, x1))) → A__SEL(a__from(mark(y0)), a__take(mark(x0), mark(x1)))
A__SEL(s(take(y0, y1)), cons(y2, take(x0, x1))) → A__SEL(a__take(mark(y0), mark(y1)), a__take(mark(x0), mark(x1)))
A__SEL(s(take(y0, y1)), cons(y2, from(x0))) → A__SEL(a__take(mark(y0), mark(y1)), a__from(mark(x0)))
A__SEL(s(take(y0, y1)), cons(y2, 2nd(x0))) → A__SEL(a__take(mark(y0), mark(y1)), a__2nd(mark(x0)))
A__SEL(s(from(y0)), cons(y1, sel(x0, x1))) → A__SEL(a__from(mark(y0)), a__sel(mark(x0), mark(x1)))
A__SEL(s(from(y0)), cons(y1, from(x0))) → A__SEL(a__from(mark(y0)), a__from(mark(x0)))

The remaining pairs can at least be oriented weakly.

A__SEL(s(head(y0)), cons(y1, take(x0, x1))) → A__SEL(a__head(mark(y0)), a__take(mark(x0), mark(x1)))
A__SEL(s(sel(y0, y1)), cons(y2, head(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__head(mark(x0)))
A__SEL(s(sel(y0, y1)), cons(y2, from(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__from(mark(x0)))
A__SEL(s(s(y0)), cons(y1, head(x0))) → A__SEL(s(mark(y0)), a__head(mark(x0)))
A__SEL(s(2nd(y0)), cons(y1, sel(x0, x1))) → A__SEL(a__2nd(mark(y0)), a__sel(mark(x0), mark(x1)))
A__SEL(s(head(y0)), cons(y1, 2nd(x0))) → A__SEL(a__head(mark(y0)), a__2nd(mark(x0)))
A__SEL(s(2nd(y0)), cons(y1, head(x0))) → A__SEL(a__2nd(mark(y0)), a__head(mark(x0)))
A__SEL(s(head(y0)), cons(y1, from(x0))) → A__SEL(a__head(mark(y0)), a__from(mark(x0)))
A__SEL(s(2nd(y0)), cons(y1, 2nd(x0))) → A__SEL(a__2nd(mark(y0)), a__2nd(mark(x0)))
A__SEL(s(2nd(y0)), cons(y1, take(x0, x1))) → A__SEL(a__2nd(mark(y0)), a__take(mark(x0), mark(x1)))
A__SEL(s(sel(y0, y1)), cons(y2, take(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), a__take(mark(x0), mark(x1)))
A__SEL(s(head(y0)), cons(y1, cons(x0, x1))) → A__SEL(a__head(mark(y0)), cons(mark(x0), x1))
A__SEL(s(sel(y0, y1)), cons(y2, 2nd(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__2nd(mark(x0)))
A__SEL(s(2nd(y0)), cons(y1, from(x0))) → A__SEL(a__2nd(mark(y0)), a__from(mark(x0)))
A__SEL(s(s(y0)), cons(y1, from(x0))) → A__SEL(s(mark(y0)), a__from(mark(x0)))
A__SEL(s(s(y0)), cons(y1, take(x0, x1))) → A__SEL(s(mark(y0)), a__take(mark(x0), mark(x1)))
A__SEL(s(s(y0)), cons(y1, 2nd(x0))) → A__SEL(s(mark(y0)), a__2nd(mark(x0)))
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(2nd(y0)), cons(y1, cons(x0, x1))) → A__SEL(a__2nd(mark(y0)), cons(mark(x0), x1))
A__SEL(s(head(y0)), cons(y1, sel(x0, x1))) → A__SEL(a__head(mark(y0)), 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(s(y0)), cons(y1, cons(x0, x1))) → A__SEL(s(mark(y0)), cons(mark(x0), x1))
A__SEL(s(head(y0)), cons(y1, head(x0))) → A__SEL(a__head(mark(y0)), a__head(mark(x0)))

Used ordering: Polynomial interpretation [25]:

POL(0) = 1
POL(2nd(x₁)) = 1
POL(A__SEL(x₁, x₂)) = x₁
POL(a__2nd(x₁)) = 1
POL(a__from(x₁)) = 0
POL(a__head(x₁)) = 1
POL(a__sel(x₁, x₂)) = x₁
POL(a__take(x₁, x₂)) = 0
POL(cons(x₁, x₂)) = 0
POL(from(x₁)) = 0
POL(head(x₁)) = 0
POL(mark(x₁)) = 1
POL(nil) = 0
POL(s(x₁)) = 1
POL(sel(x₁, x₂)) = 0
POL(take(x₁, x₂)) = 0

The following usable rules [17] were oriented:

mark(s(X)) → s(mark(X))
mark(0) → 0
mark(nil) → nil
a__from(X) → from(X)
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(cons(X1, X2)) → cons(mark(X1), X2)
a__head(X) → head(X)
a__2nd(X) → 2nd(X)
a__take(X1, X2) → take(X1, X2)
a__sel(X1, X2) → sel(X1, X2)
a__from(X) → cons(mark(X), from(s(X)))
a__take(s(N), cons(X, XS)) → cons(mark(X), take(N, XS))
a__take(0, XS) → nil
a__sel(s(N), cons(X, XS)) → a__sel(mark(N), mark(XS))
a__2nd(cons(X, XS)) → a__head(mark(XS))
a__sel(0, cons(X, XS)) → mark(X)
mark(2nd(X)) → a__2nd(mark(X))
mark(head(X)) → a__head(mark(X))
mark(sel(X1, X2)) → a__sel(mark(X1), mark(X2))
a__head(cons(X, XS)) → mark(X)
mark(from(X)) → a__from(mark(X))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ QDPOrderProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

                ↳ Narrowing

                  ↳ QDP

                    ↳ DependencyGraphProof

                      ↳ QDP

                        ↳ 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:

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(head(y0)), cons(y1, take(x0, x1))) → A__SEL(a__head(mark(y0)), a__take(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(2nd(y0)), cons(y1, cons(x0, x1))) → A__SEL(a__2nd(mark(y0)), cons(mark(x0), x1))
A__SEL(s(sel(y0, y1)), cons(y2, head(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__head(mark(x0)))
A__SEL(s(sel(y0, y1)), cons(y2, from(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__from(mark(x0)))
A__SEL(s(s(y0)), cons(y1, head(x0))) → A__SEL(s(mark(y0)), a__head(mark(x0)))
A__SEL(s(2nd(y0)), cons(y1, sel(x0, x1))) → A__SEL(a__2nd(mark(y0)), 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(head(y0)), cons(y1, sel(x0, x1))) → A__SEL(a__head(mark(y0)), a__sel(mark(x0), mark(x1)))
A__SEL(s(s(y0)), cons(y1, cons(x0, x1))) → A__SEL(s(mark(y0)), cons(mark(x0), x1))
A__SEL(s(head(y0)), cons(y1, 2nd(x0))) → A__SEL(a__head(mark(y0)), a__2nd(mark(x0)))
A__SEL(s(2nd(y0)), cons(y1, head(x0))) → A__SEL(a__2nd(mark(y0)), a__head(mark(x0)))
A__SEL(s(head(y0)), cons(y1, from(x0))) → A__SEL(a__head(mark(y0)), a__from(mark(x0)))
A__SEL(s(2nd(y0)), cons(y1, 2nd(x0))) → A__SEL(a__2nd(mark(y0)), a__2nd(mark(x0)))
A__SEL(s(head(y0)), cons(y1, head(x0))) → A__SEL(a__head(mark(y0)), a__head(mark(x0)))
A__SEL(s(2nd(y0)), cons(y1, take(x0, x1))) → A__SEL(a__2nd(mark(y0)), a__take(mark(x0), mark(x1)))
A__SEL(s(sel(y0, y1)), cons(y2, take(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), a__take(mark(x0), mark(x1)))
A__SEL(s(head(y0)), cons(y1, cons(x0, x1))) → A__SEL(a__head(mark(y0)), cons(mark(x0), x1))
A__SEL(s(sel(y0, y1)), cons(y2, 2nd(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__2nd(mark(x0)))
A__SEL(s(2nd(y0)), cons(y1, from(x0))) → A__SEL(a__2nd(mark(y0)), a__from(mark(x0)))
A__SEL(s(s(y0)), cons(y1, from(x0))) → A__SEL(s(mark(y0)), a__from(mark(x0)))
A__SEL(s(s(y0)), cons(y1, 2nd(x0))) → A__SEL(s(mark(y0)), a__2nd(mark(x0)))
A__SEL(s(s(y0)), cons(y1, take(x0, x1))) → A__SEL(s(mark(y0)), a__take(mark(x0), mark(x1)))

The TRS R consists of the following rules:

a__from(X) → cons(mark(X), from(s(X)))
a__head(cons(X, XS)) → mark(X)
a__2nd(cons(X, XS)) → a__head(mark(XS))
a__take(0, XS) → nil
a__take(s(N), cons(X, XS)) → cons(mark(X), take(N, XS))
a__sel(0, cons(X, XS)) → mark(X)
a__sel(s(N), cons(X, XS)) → a__sel(mark(N), mark(XS))
mark(from(X)) → a__from(mark(X))
mark(head(X)) → a__head(mark(X))
mark(2nd(X)) → a__2nd(mark(X))
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(sel(X1, X2)) → a__sel(mark(X1), mark(X2))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(s(X)) → s(mark(X))
mark(0) → 0
mark(nil) → nil
a__from(X) → from(X)
a__head(X) → head(X)
a__2nd(X) → 2nd(X)
a__take(X1, X2) → take(X1, X2)
a__sel(X1, X2) → sel(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].

The following pairs can be oriented strictly and are deleted.

A__SEL(s(head(y0)), cons(y1, 2nd(x0))) → A__SEL(a__head(mark(y0)), a__2nd(mark(x0)))
A__SEL(s(2nd(y0)), cons(y1, 2nd(x0))) → A__SEL(a__2nd(mark(y0)), a__2nd(mark(x0)))
A__SEL(s(sel(y0, y1)), cons(y2, 2nd(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__2nd(mark(x0)))
A__SEL(s(s(y0)), cons(y1, 2nd(x0))) → A__SEL(s(mark(y0)), a__2nd(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(head(y0)), cons(y1, take(x0, x1))) → A__SEL(a__head(mark(y0)), a__take(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(2nd(y0)), cons(y1, cons(x0, x1))) → A__SEL(a__2nd(mark(y0)), cons(mark(x0), x1))
A__SEL(s(sel(y0, y1)), cons(y2, head(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__head(mark(x0)))
A__SEL(s(sel(y0, y1)), cons(y2, from(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__from(mark(x0)))
A__SEL(s(s(y0)), cons(y1, head(x0))) → A__SEL(s(mark(y0)), a__head(mark(x0)))
A__SEL(s(2nd(y0)), cons(y1, sel(x0, x1))) → A__SEL(a__2nd(mark(y0)), 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(head(y0)), cons(y1, sel(x0, x1))) → A__SEL(a__head(mark(y0)), a__sel(mark(x0), mark(x1)))
A__SEL(s(s(y0)), cons(y1, cons(x0, x1))) → A__SEL(s(mark(y0)), cons(mark(x0), x1))
A__SEL(s(2nd(y0)), cons(y1, head(x0))) → A__SEL(a__2nd(mark(y0)), a__head(mark(x0)))
A__SEL(s(head(y0)), cons(y1, from(x0))) → A__SEL(a__head(mark(y0)), a__from(mark(x0)))
A__SEL(s(head(y0)), cons(y1, head(x0))) → A__SEL(a__head(mark(y0)), a__head(mark(x0)))
A__SEL(s(2nd(y0)), cons(y1, take(x0, x1))) → A__SEL(a__2nd(mark(y0)), a__take(mark(x0), mark(x1)))
A__SEL(s(sel(y0, y1)), cons(y2, take(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), a__take(mark(x0), mark(x1)))
A__SEL(s(head(y0)), cons(y1, cons(x0, x1))) → A__SEL(a__head(mark(y0)), cons(mark(x0), x1))
A__SEL(s(2nd(y0)), cons(y1, from(x0))) → A__SEL(a__2nd(mark(y0)), a__from(mark(x0)))
A__SEL(s(s(y0)), cons(y1, from(x0))) → A__SEL(s(mark(y0)), a__from(mark(x0)))
A__SEL(s(s(y0)), cons(y1, take(x0, x1))) → A__SEL(s(mark(y0)), a__take(mark(x0), mark(x1)))

Used ordering: Matrix interpretation [3]:
Non-tuple symbols:

M( sel(x₁, x₂) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	1	0	\
\	0	0	/

x₂

M( from(x₁) ) =

/	0	\
\	0	/

/	1	0	\
\	0	0	/

x₁

M( a__head(x₁) ) =

/	0	\
\	1	/

/	1	0	\
\	0	0	/

x₁

M( a__2nd(x₁) ) =

/	1	\
\	1	/

/	1	0	\
\	0	0	/

x₁

M( head(x₁) ) =

/	0	\
\	0	/

/	1	0	\
\	0	0	/

x₁

M( mark(x₁) ) =

/	0	\
\	1	/

/	1	0	\
\	0	0	/

x₁

M( a__from(x₁) ) =

/	0	\
\	0	/

/	1	0	\
\	0	0	/

x₁

M( take(x₁, x₂) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	1	0	\
\	0	0	/

x₂

M( 0 ) =

/	0	\
\	0	/

M( a__take(x₁, x₂) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	1	0	\
\	0	0	/

x₂

M( cons(x₁, x₂) ) =

/	0	\
\	0	/

/	1	0	\
\	0	0	/

x₁

/	1	1	\
\	0	0	/

x₂

M( a__sel(x₁, x₂) ) =

/	0	\
\	1	/

/	0	0	\
\	0	0	/

x₁

/	1	0	\
\	0	0	/

x₂

M( 2nd(x₁) ) =

/	1	\
\	1	/

/	1	0	\
\	0	0	/

x₁

M( s(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

M( nil ) =

/	0	\
\	0	/

Tuple symbols:

M( A__SEL(x₁, x₂) ) =

[

]

x₁

[

]

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:

mark(s(X)) → s(mark(X))
mark(0) → 0
mark(nil) → nil
a__from(X) → from(X)
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(cons(X1, X2)) → cons(mark(X1), X2)
a__head(X) → head(X)
a__2nd(X) → 2nd(X)
a__take(X1, X2) → take(X1, X2)
a__sel(X1, X2) → sel(X1, X2)
a__from(X) → cons(mark(X), from(s(X)))
a__take(s(N), cons(X, XS)) → cons(mark(X), take(N, XS))
a__take(0, XS) → nil
a__sel(s(N), cons(X, XS)) → a__sel(mark(N), mark(XS))
a__2nd(cons(X, XS)) → a__head(mark(XS))
a__sel(0, cons(X, XS)) → mark(X)
mark(2nd(X)) → a__2nd(mark(X))
mark(head(X)) → a__head(mark(X))
mark(sel(X1, X2)) → a__sel(mark(X1), mark(X2))
a__head(cons(X, XS)) → mark(X)
mark(from(X)) → a__from(mark(X))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ QDPOrderProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

                ↳ Narrowing

                  ↳ QDP

                    ↳ DependencyGraphProof

                      ↳ QDP

                        ↳ 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:

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(head(y0)), cons(y1, take(x0, x1))) → A__SEL(a__head(mark(y0)), a__take(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(2nd(y0)), cons(y1, cons(x0, x1))) → A__SEL(a__2nd(mark(y0)), cons(mark(x0), x1))
A__SEL(s(sel(y0, y1)), cons(y2, head(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__head(mark(x0)))
A__SEL(s(sel(y0, y1)), cons(y2, from(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__from(mark(x0)))
A__SEL(s(s(y0)), cons(y1, head(x0))) → A__SEL(s(mark(y0)), a__head(mark(x0)))
A__SEL(s(2nd(y0)), cons(y1, sel(x0, x1))) → A__SEL(a__2nd(mark(y0)), 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(head(y0)), cons(y1, sel(x0, x1))) → A__SEL(a__head(mark(y0)), a__sel(mark(x0), mark(x1)))
A__SEL(s(s(y0)), cons(y1, cons(x0, x1))) → A__SEL(s(mark(y0)), cons(mark(x0), x1))
A__SEL(s(2nd(y0)), cons(y1, head(x0))) → A__SEL(a__2nd(mark(y0)), a__head(mark(x0)))
A__SEL(s(head(y0)), cons(y1, from(x0))) → A__SEL(a__head(mark(y0)), a__from(mark(x0)))
A__SEL(s(sel(y0, y1)), cons(y2, take(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), a__take(mark(x0), mark(x1)))
A__SEL(s(2nd(y0)), cons(y1, take(x0, x1))) → A__SEL(a__2nd(mark(y0)), a__take(mark(x0), mark(x1)))
A__SEL(s(head(y0)), cons(y1, head(x0))) → A__SEL(a__head(mark(y0)), a__head(mark(x0)))
A__SEL(s(head(y0)), cons(y1, cons(x0, x1))) → A__SEL(a__head(mark(y0)), cons(mark(x0), x1))
A__SEL(s(2nd(y0)), cons(y1, from(x0))) → A__SEL(a__2nd(mark(y0)), a__from(mark(x0)))
A__SEL(s(s(y0)), cons(y1, from(x0))) → A__SEL(s(mark(y0)), a__from(mark(x0)))
A__SEL(s(s(y0)), cons(y1, take(x0, x1))) → A__SEL(s(mark(y0)), a__take(mark(x0), mark(x1)))

The TRS R consists of the following rules:

a__from(X) → cons(mark(X), from(s(X)))
a__head(cons(X, XS)) → mark(X)
a__2nd(cons(X, XS)) → a__head(mark(XS))
a__take(0, XS) → nil
a__take(s(N), cons(X, XS)) → cons(mark(X), take(N, XS))
a__sel(0, cons(X, XS)) → mark(X)
a__sel(s(N), cons(X, XS)) → a__sel(mark(N), mark(XS))
mark(from(X)) → a__from(mark(X))
mark(head(X)) → a__head(mark(X))
mark(2nd(X)) → a__2nd(mark(X))
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(sel(X1, X2)) → a__sel(mark(X1), mark(X2))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(s(X)) → s(mark(X))
mark(0) → 0
mark(nil) → nil
a__from(X) → from(X)
a__head(X) → head(X)
a__2nd(X) → 2nd(X)
a__take(X1, X2) → take(X1, X2)
a__sel(X1, X2) → sel(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].

The following pairs can be oriented strictly and are deleted.

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(2nd(y0)), cons(y1, sel(x0, x1))) → A__SEL(a__2nd(mark(y0)), a__sel(mark(x0), mark(x1)))
A__SEL(s(head(y0)), cons(y1, sel(x0, x1))) → A__SEL(a__head(mark(y0)), a__sel(mark(x0), mark(x1)))

The remaining pairs can at least be oriented weakly.

A__SEL(s(head(y0)), cons(y1, take(x0, x1))) → A__SEL(a__head(mark(y0)), a__take(mark(x0), mark(x1)))
A__SEL(s(2nd(y0)), cons(y1, cons(x0, x1))) → A__SEL(a__2nd(mark(y0)), cons(mark(x0), x1))
A__SEL(s(sel(y0, y1)), cons(y2, head(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__head(mark(x0)))
A__SEL(s(sel(y0, y1)), cons(y2, from(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__from(mark(x0)))
A__SEL(s(s(y0)), cons(y1, head(x0))) → A__SEL(s(mark(y0)), a__head(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(2nd(y0)), cons(y1, head(x0))) → A__SEL(a__2nd(mark(y0)), a__head(mark(x0)))
A__SEL(s(head(y0)), cons(y1, from(x0))) → A__SEL(a__head(mark(y0)), a__from(mark(x0)))
A__SEL(s(sel(y0, y1)), cons(y2, take(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), a__take(mark(x0), mark(x1)))
A__SEL(s(2nd(y0)), cons(y1, take(x0, x1))) → A__SEL(a__2nd(mark(y0)), a__take(mark(x0), mark(x1)))
A__SEL(s(head(y0)), cons(y1, head(x0))) → A__SEL(a__head(mark(y0)), a__head(mark(x0)))
A__SEL(s(head(y0)), cons(y1, cons(x0, x1))) → A__SEL(a__head(mark(y0)), cons(mark(x0), x1))
A__SEL(s(2nd(y0)), cons(y1, from(x0))) → A__SEL(a__2nd(mark(y0)), a__from(mark(x0)))
A__SEL(s(s(y0)), cons(y1, from(x0))) → A__SEL(s(mark(y0)), a__from(mark(x0)))
A__SEL(s(s(y0)), cons(y1, take(x0, x1))) → A__SEL(s(mark(y0)), a__take(mark(x0), mark(x1)))

Used ordering: Matrix interpretation [3]:
Non-tuple symbols:

M( sel(x₁, x₂) ) =

/	1	\
\	1	/

/	0	0	\
\	0	0	/

x₁

/	0	1	\
\	1	1	/

x₂

M( from(x₁) ) =

/	0	\
\	0	/

/	0	1	\
\	1	0	/

x₁

M( a__head(x₁) ) =

/	0	\
\	0	/

/	0	1	\
\	1	1	/

x₁

M( a__2nd(x₁) ) =

/	0	\
\	0	/

/	0	1	\
\	0	1	/

x₁

M( head(x₁) ) =

/	0	\
\	0	/

/	0	1	\
\	1	1	/

x₁

M( mark(x₁) ) =

/	0	\
\	0	/

/	1	0	\
\	0	1	/

x₁

M( a__from(x₁) ) =

/	0	\
\	0	/

/	0	1	\
\	1	0	/

x₁

M( take(x₁, x₂) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	1	0	\
\	0	1	/

x₂

M( 0 ) =

/	0	\
\	0	/

M( a__take(x₁, x₂) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	1	0	\
\	0	1	/

x₂

M( cons(x₁, x₂) ) =

/	0	\
\	0	/

/	0	1	\
\	1	0	/

x₁

/	0	0	\
\	1	1	/

x₂

M( a__sel(x₁, x₂) ) =

/	1	\
\	1	/

/	0	0	\
\	0	0	/

x₁

/	0	1	\
\	1	1	/

x₂

M( 2nd(x₁) ) =

/	0	\
\	0	/

/	0	1	\
\	0	1	/

x₁

M( s(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

M( nil ) =

/	0	\
\	0	/

Tuple symbols:

M( A__SEL(x₁, x₂) ) =

[

]

x₁

[

]

x₂

mark(s(X)) → s(mark(X))
mark(0) → 0
mark(nil) → nil
a__from(X) → from(X)
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(cons(X1, X2)) → cons(mark(X1), X2)
a__head(X) → head(X)
a__2nd(X) → 2nd(X)
a__take(X1, X2) → take(X1, X2)
a__sel(X1, X2) → sel(X1, X2)
a__from(X) → cons(mark(X), from(s(X)))
a__take(s(N), cons(X, XS)) → cons(mark(X), take(N, XS))
a__take(0, XS) → nil
a__sel(s(N), cons(X, XS)) → a__sel(mark(N), mark(XS))
a__2nd(cons(X, XS)) → a__head(mark(XS))
a__sel(0, cons(X, XS)) → mark(X)
mark(2nd(X)) → a__2nd(mark(X))
mark(head(X)) → a__head(mark(X))
mark(sel(X1, X2)) → a__sel(mark(X1), mark(X2))
a__head(cons(X, XS)) → mark(X)
mark(from(X)) → a__from(mark(X))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ QDPOrderProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

                ↳ Narrowing

                  ↳ QDP

                    ↳ DependencyGraphProof

                      ↳ QDP

                        ↳ 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

                                                                                    ↳ QDPOrderProof

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

A__SEL(s(head(y0)), cons(y1, take(x0, x1))) → A__SEL(a__head(mark(y0)), a__take(mark(x0), mark(x1)))
A__SEL(s(2nd(y0)), cons(y1, cons(x0, x1))) → A__SEL(a__2nd(mark(y0)), cons(mark(x0), x1))
A__SEL(s(sel(y0, y1)), cons(y2, head(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__head(mark(x0)))
A__SEL(s(sel(y0, y1)), cons(y2, from(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__from(mark(x0)))
A__SEL(s(s(y0)), cons(y1, head(x0))) → A__SEL(s(mark(y0)), a__head(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(2nd(y0)), cons(y1, head(x0))) → A__SEL(a__2nd(mark(y0)), a__head(mark(x0)))
A__SEL(s(head(y0)), cons(y1, from(x0))) → A__SEL(a__head(mark(y0)), a__from(mark(x0)))
A__SEL(s(sel(y0, y1)), cons(y2, take(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), a__take(mark(x0), mark(x1)))
A__SEL(s(2nd(y0)), cons(y1, take(x0, x1))) → A__SEL(a__2nd(mark(y0)), a__take(mark(x0), mark(x1)))
A__SEL(s(head(y0)), cons(y1, head(x0))) → A__SEL(a__head(mark(y0)), a__head(mark(x0)))
A__SEL(s(head(y0)), cons(y1, cons(x0, x1))) → A__SEL(a__head(mark(y0)), cons(mark(x0), x1))
A__SEL(s(2nd(y0)), cons(y1, from(x0))) → A__SEL(a__2nd(mark(y0)), a__from(mark(x0)))
A__SEL(s(s(y0)), cons(y1, from(x0))) → A__SEL(s(mark(y0)), a__from(mark(x0)))
A__SEL(s(s(y0)), cons(y1, take(x0, x1))) → A__SEL(s(mark(y0)), a__take(mark(x0), mark(x1)))

The TRS R consists of the following rules:

a__from(X) → cons(mark(X), from(s(X)))
a__head(cons(X, XS)) → mark(X)
a__2nd(cons(X, XS)) → a__head(mark(XS))
a__take(0, XS) → nil
a__take(s(N), cons(X, XS)) → cons(mark(X), take(N, XS))
a__sel(0, cons(X, XS)) → mark(X)
a__sel(s(N), cons(X, XS)) → a__sel(mark(N), mark(XS))
mark(from(X)) → a__from(mark(X))
mark(head(X)) → a__head(mark(X))
mark(2nd(X)) → a__2nd(mark(X))
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(sel(X1, X2)) → a__sel(mark(X1), mark(X2))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(s(X)) → s(mark(X))
mark(0) → 0
mark(nil) → nil
a__from(X) → from(X)
a__head(X) → head(X)
a__2nd(X) → 2nd(X)
a__take(X1, X2) → take(X1, X2)
a__sel(X1, X2) → sel(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].

The following pairs can be oriented strictly and are deleted.

A__SEL(s(sel(y0, y1)), cons(y2, head(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__head(mark(x0)))
A__SEL(s(s(y0)), cons(y1, head(x0))) → A__SEL(s(mark(y0)), a__head(mark(x0)))
A__SEL(s(2nd(y0)), cons(y1, head(x0))) → A__SEL(a__2nd(mark(y0)), a__head(mark(x0)))
A__SEL(s(head(y0)), cons(y1, head(x0))) → A__SEL(a__head(mark(y0)), a__head(mark(x0)))

The remaining pairs can at least be oriented weakly.

A__SEL(s(head(y0)), cons(y1, take(x0, x1))) → A__SEL(a__head(mark(y0)), a__take(mark(x0), mark(x1)))
A__SEL(s(2nd(y0)), cons(y1, cons(x0, x1))) → A__SEL(a__2nd(mark(y0)), cons(mark(x0), x1))
A__SEL(s(sel(y0, y1)), cons(y2, from(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__from(mark(x0)))
A__SEL(s(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(head(y0)), cons(y1, from(x0))) → A__SEL(a__head(mark(y0)), a__from(mark(x0)))
A__SEL(s(sel(y0, y1)), cons(y2, take(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), a__take(mark(x0), mark(x1)))
A__SEL(s(2nd(y0)), cons(y1, take(x0, x1))) → A__SEL(a__2nd(mark(y0)), a__take(mark(x0), mark(x1)))
A__SEL(s(head(y0)), cons(y1, cons(x0, x1))) → A__SEL(a__head(mark(y0)), cons(mark(x0), x1))
A__SEL(s(2nd(y0)), cons(y1, from(x0))) → A__SEL(a__2nd(mark(y0)), a__from(mark(x0)))
A__SEL(s(s(y0)), cons(y1, from(x0))) → A__SEL(s(mark(y0)), a__from(mark(x0)))
A__SEL(s(s(y0)), cons(y1, take(x0, x1))) → A__SEL(s(mark(y0)), a__take(mark(x0), mark(x1)))

Used ordering: Matrix interpretation [3]:
Non-tuple symbols:

M( sel(x₁, x₂) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	0	1	\
\	1	1	/

x₂

M( from(x₁) ) =

/	0	\
\	0	/

/	0	1	\
\	1	0	/

x₁

M( a__head(x₁) ) =

/	1	\
\	1	/

/	0	1	\
\	1	0	/

x₁

M( a__2nd(x₁) ) =

/	1	\
\	1	/

/	0	1	\
\	0	1	/

x₁

M( head(x₁) ) =

/	1	\
\	1	/

/	0	1	\
\	1	0	/

x₁

M( mark(x₁) ) =

/	0	\
\	0	/

/	1	0	\
\	0	1	/

x₁

M( a__from(x₁) ) =

/	0	\
\	0	/

/	0	1	\
\	1	0	/

x₁

M( take(x₁, x₂) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	1	0	\
\	0	1	/

x₂

M( 0 ) =

/	0	\
\	0	/

M( a__take(x₁, x₂) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	1	0	\
\	0	1	/

x₂

M( cons(x₁, x₂) ) =

/	0	\
\	0	/

/	0	1	\
\	1	0	/

x₁

/	0	0	\
\	1	1	/

x₂

M( a__sel(x₁, x₂) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	0	1	\
\	1	1	/

x₂

M( 2nd(x₁) ) =

/	1	\
\	1	/

/	0	1	\
\	0	1	/

x₁

M( s(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

M( nil ) =

/	0	\
\	0	/

Tuple symbols:

M( A__SEL(x₁, x₂) ) =

[

]

x₁

[

]

x₂

mark(s(X)) → s(mark(X))
mark(0) → 0
mark(nil) → nil
a__from(X) → from(X)
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(cons(X1, X2)) → cons(mark(X1), X2)
a__head(X) → head(X)
a__2nd(X) → 2nd(X)
a__take(X1, X2) → take(X1, X2)
a__sel(X1, X2) → sel(X1, X2)
a__from(X) → cons(mark(X), from(s(X)))
a__take(s(N), cons(X, XS)) → cons(mark(X), take(N, XS))
a__take(0, XS) → nil
a__sel(s(N), cons(X, XS)) → a__sel(mark(N), mark(XS))
a__2nd(cons(X, XS)) → a__head(mark(XS))
a__sel(0, cons(X, XS)) → mark(X)
mark(2nd(X)) → a__2nd(mark(X))
mark(head(X)) → a__head(mark(X))
mark(sel(X1, X2)) → a__sel(mark(X1), mark(X2))
a__head(cons(X, XS)) → mark(X)
mark(from(X)) → a__from(mark(X))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ QDPOrderProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

                ↳ Narrowing

                  ↳ QDP

                    ↳ DependencyGraphProof

                      ↳ QDP

                        ↳ 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

                                                                                    ↳ QDPOrderProof

                                                                                      ↳ QDP

                                                                                        ↳ QDPOrderProof

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

A__SEL(s(head(y0)), cons(y1, take(x0, x1))) → A__SEL(a__head(mark(y0)), a__take(mark(x0), mark(x1)))
A__SEL(s(head(y0)), cons(y1, from(x0))) → A__SEL(a__head(mark(y0)), a__from(mark(x0)))
A__SEL(s(2nd(y0)), cons(y1, cons(x0, x1))) → A__SEL(a__2nd(mark(y0)), cons(mark(x0), x1))
A__SEL(s(2nd(y0)), cons(y1, take(x0, x1))) → A__SEL(a__2nd(mark(y0)), a__take(mark(x0), mark(x1)))
A__SEL(s(sel(y0, y1)), cons(y2, take(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), a__take(mark(x0), mark(x1)))
A__SEL(s(head(y0)), cons(y1, cons(x0, x1))) → A__SEL(a__head(mark(y0)), cons(mark(x0), x1))
A__SEL(s(sel(y0, y1)), cons(y2, from(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__from(mark(x0)))
A__SEL(s(2nd(y0)), cons(y1, from(x0))) → A__SEL(a__2nd(mark(y0)), a__from(mark(x0)))
A__SEL(s(sel(y0, y1)), cons(y2, cons(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), cons(mark(x0), x1))
A__SEL(s(s(y0)), cons(y1, cons(x0, x1))) → A__SEL(s(mark(y0)), cons(mark(x0), x1))
A__SEL(s(s(y0)), cons(y1, from(x0))) → A__SEL(s(mark(y0)), a__from(mark(x0)))
A__SEL(s(s(y0)), cons(y1, take(x0, x1))) → A__SEL(s(mark(y0)), a__take(mark(x0), mark(x1)))

The TRS R consists of the following rules:

a__from(X) → cons(mark(X), from(s(X)))
a__head(cons(X, XS)) → mark(X)
a__2nd(cons(X, XS)) → a__head(mark(XS))
a__take(0, XS) → nil
a__take(s(N), cons(X, XS)) → cons(mark(X), take(N, XS))
a__sel(0, cons(X, XS)) → mark(X)
a__sel(s(N), cons(X, XS)) → a__sel(mark(N), mark(XS))
mark(from(X)) → a__from(mark(X))
mark(head(X)) → a__head(mark(X))
mark(2nd(X)) → a__2nd(mark(X))
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(sel(X1, X2)) → a__sel(mark(X1), mark(X2))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(s(X)) → s(mark(X))
mark(0) → 0
mark(nil) → nil
a__from(X) → from(X)
a__head(X) → head(X)
a__2nd(X) → 2nd(X)
a__take(X1, X2) → take(X1, X2)
a__sel(X1, X2) → sel(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].

The following pairs can be oriented strictly and are deleted.

A__SEL(s(2nd(y0)), cons(y1, cons(x0, x1))) → A__SEL(a__2nd(mark(y0)), cons(mark(x0), x1))
A__SEL(s(head(y0)), cons(y1, cons(x0, x1))) → A__SEL(a__head(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(head(y0)), cons(y1, take(x0, x1))) → A__SEL(a__head(mark(y0)), a__take(mark(x0), mark(x1)))
A__SEL(s(head(y0)), cons(y1, from(x0))) → A__SEL(a__head(mark(y0)), a__from(mark(x0)))
A__SEL(s(2nd(y0)), cons(y1, take(x0, x1))) → A__SEL(a__2nd(mark(y0)), a__take(mark(x0), mark(x1)))
A__SEL(s(sel(y0, y1)), cons(y2, take(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), a__take(mark(x0), mark(x1)))
A__SEL(s(sel(y0, y1)), cons(y2, from(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__from(mark(x0)))
A__SEL(s(2nd(y0)), cons(y1, from(x0))) → A__SEL(a__2nd(mark(y0)), a__from(mark(x0)))
A__SEL(s(s(y0)), cons(y1, from(x0))) → A__SEL(s(mark(y0)), a__from(mark(x0)))
A__SEL(s(s(y0)), cons(y1, take(x0, x1))) → A__SEL(s(mark(y0)), a__take(mark(x0), mark(x1)))

Used ordering: Matrix interpretation [3]:
Non-tuple symbols:

M( sel(x₁, x₂) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	0	0	\
\	0	1	/

x₂

M( from(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	1	/

x₁

M( a__head(x₁) ) =

/	1	\
\	0	/

/	0	0	\
\	0	1	/

x₁

M( a__2nd(x₁) ) =

/	0	\
\	0	/

/	1	0	\
\	0	1	/

x₁

M( head(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	1	/

x₁

M( mark(x₁) ) =

/	1	\
\	0	/

/	0	0	\
\	0	1	/

x₁

M( a__from(x₁) ) =

/	1	\
\	0	/

/	0	0	\
\	0	1	/

x₁

M( take(x₁, x₂) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	0	0	\
\	0	1	/

x₂

M( 0 ) =

/	0	\
\	0	/

M( a__take(x₁, x₂) ) =

/	1	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	0	0	\
\	0	1	/

x₂

M( cons(x₁, x₂) ) =

/	1	\
\	0	/

/	0	0	\
\	0	1	/

x₁

/	0	0	\
\	1	1	/

x₂

M( a__sel(x₁, x₂) ) =

/	1	\
\	0	/

/	0	0	\
\	0	0	/

x₁

/	0	0	\
\	0	1	/

x₂

M( 2nd(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	1	/

x₁

M( s(x₁) ) =

/	0	\
\	0	/

/	0	0	\
\	0	0	/

x₁

M( nil ) =

/	0	\
\	0	/

Tuple symbols:

M( A__SEL(x₁, x₂) ) =

[

]

x₁

[

]

x₂

mark(s(X)) → s(mark(X))
mark(0) → 0
mark(nil) → nil
a__from(X) → from(X)
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(cons(X1, X2)) → cons(mark(X1), X2)
a__head(X) → head(X)
a__2nd(X) → 2nd(X)
a__take(X1, X2) → take(X1, X2)
a__sel(X1, X2) → sel(X1, X2)
a__from(X) → cons(mark(X), from(s(X)))
a__take(s(N), cons(X, XS)) → cons(mark(X), take(N, XS))
a__take(0, XS) → nil
a__sel(s(N), cons(X, XS)) → a__sel(mark(N), mark(XS))
a__2nd(cons(X, XS)) → a__head(mark(XS))
a__sel(0, cons(X, XS)) → mark(X)
mark(2nd(X)) → a__2nd(mark(X))
mark(head(X)) → a__head(mark(X))
mark(sel(X1, X2)) → a__sel(mark(X1), mark(X2))
a__head(cons(X, XS)) → mark(X)
mark(from(X)) → a__from(mark(X))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ QDPOrderProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

                ↳ Narrowing

                  ↳ QDP

                    ↳ DependencyGraphProof

                      ↳ QDP

                        ↳ 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

                                                                                    ↳ QDPOrderProof

                                                                                      ↳ QDP

                                                                                        ↳ QDPOrderProof

                                                                                          ↳ QDP

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

A__SEL(s(head(y0)), cons(y1, take(x0, x1))) → A__SEL(a__head(mark(y0)), a__take(mark(x0), mark(x1)))
A__SEL(s(head(y0)), cons(y1, from(x0))) → A__SEL(a__head(mark(y0)), a__from(mark(x0)))
A__SEL(s(sel(y0, y1)), cons(y2, take(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), a__take(mark(x0), mark(x1)))
A__SEL(s(2nd(y0)), cons(y1, take(x0, x1))) → A__SEL(a__2nd(mark(y0)), a__take(mark(x0), mark(x1)))
A__SEL(s(sel(y0, y1)), cons(y2, from(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__from(mark(x0)))
A__SEL(s(2nd(y0)), cons(y1, from(x0))) → A__SEL(a__2nd(mark(y0)), a__from(mark(x0)))
A__SEL(s(s(y0)), cons(y1, from(x0))) → A__SEL(s(mark(y0)), a__from(mark(x0)))
A__SEL(s(s(y0)), cons(y1, take(x0, x1))) → A__SEL(s(mark(y0)), a__take(mark(x0), mark(x1)))

The TRS R consists of the following rules:

a__from(X) → cons(mark(X), from(s(X)))
a__head(cons(X, XS)) → mark(X)
a__2nd(cons(X, XS)) → a__head(mark(XS))
a__take(0, XS) → nil
a__take(s(N), cons(X, XS)) → cons(mark(X), take(N, XS))
a__sel(0, cons(X, XS)) → mark(X)
a__sel(s(N), cons(X, XS)) → a__sel(mark(N), mark(XS))
mark(from(X)) → a__from(mark(X))
mark(head(X)) → a__head(mark(X))
mark(2nd(X)) → a__2nd(mark(X))
mark(take(X1, X2)) → a__take(mark(X1), mark(X2))
mark(sel(X1, X2)) → a__sel(mark(X1), mark(X2))
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(s(X)) → s(mark(X))
mark(0) → 0
mark(nil) → nil
a__from(X) → from(X)
a__head(X) → head(X)
a__2nd(X) → 2nd(X)
a__take(X1, X2) → take(X1, X2)
a__sel(X1, X2) → sel(X1, X2)

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