Termination w.r.t. Q proof of /tmp/tmpdPoizc/Ex8_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__fib(N) → a__sel(mark(N), a__fib1(s(0), s(0)))
a__fib1(X, Y) → cons(mark(X), fib1(Y, add(X, Y)))
a__add(0, X) → mark(X)
a__add(s(X), Y) → s(a__add(mark(X), mark(Y)))
a__sel(0, cons(X, XS)) → mark(X)
a__sel(s(N), cons(X, XS)) → a__sel(mark(N), mark(XS))
mark(fib(X)) → a__fib(mark(X))
mark(sel(X1, X2)) → a__sel(mark(X1), mark(X2))
mark(fib1(X1, X2)) → a__fib1(mark(X1), mark(X2))
mark(add(X1, X2)) → a__add(mark(X1), mark(X2))
mark(s(X)) → s(mark(X))
mark(0) → 0
mark(cons(X1, X2)) → cons(mark(X1), X2)
a__fib(X) → fib(X)
a__sel(X1, X2) → sel(X1, X2)
a__fib1(X1, X2) → fib1(X1, X2)
a__add(X1, X2) → add(X1, X2)

Q is empty.

↳ QTRS

  ↳ DependencyPairsProof

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

a__fib(N) → a__sel(mark(N), a__fib1(s(0), s(0)))
a__fib1(X, Y) → cons(mark(X), fib1(Y, add(X, Y)))
a__add(0, X) → mark(X)
a__add(s(X), Y) → s(a__add(mark(X), mark(Y)))
a__sel(0, cons(X, XS)) → mark(X)
a__sel(s(N), cons(X, XS)) → a__sel(mark(N), mark(XS))
mark(fib(X)) → a__fib(mark(X))
mark(sel(X1, X2)) → a__sel(mark(X1), mark(X2))
mark(fib1(X1, X2)) → a__fib1(mark(X1), mark(X2))
mark(add(X1, X2)) → a__add(mark(X1), mark(X2))
mark(s(X)) → s(mark(X))
mark(0) → 0
mark(cons(X1, X2)) → cons(mark(X1), X2)
a__fib(X) → fib(X)
a__sel(X1, X2) → sel(X1, X2)
a__fib1(X1, X2) → fib1(X1, X2)
a__add(X1, X2) → add(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(fib(X)) → MARK(X)
A__ADD(0, X) → MARK(X)
A__FIB1(X, Y) → MARK(X)
MARK(fib1(X1, X2)) → MARK(X1)
MARK(s(X)) → MARK(X)
MARK(add(X1, X2)) → MARK(X2)
MARK(fib(X)) → A__FIB(mark(X))
MARK(cons(X1, X2)) → MARK(X1)
MARK(fib1(X1, X2)) → MARK(X2)
A__SEL(0, cons(X, XS)) → MARK(X)
A__FIB(N) → MARK(N)
MARK(add(X1, X2)) → MARK(X1)
A__SEL(s(N), cons(X, XS)) → MARK(XS)
A__ADD(s(X), Y) → A__ADD(mark(X), mark(Y))
A__ADD(s(X), Y) → MARK(Y)
A__FIB(N) → A__SEL(mark(N), a__fib1(s(0), s(0)))
A__SEL(s(N), cons(X, XS)) → MARK(N)
MARK(sel(X1, X2)) → MARK(X1)
MARK(fib1(X1, X2)) → A__FIB1(mark(X1), mark(X2))
MARK(sel(X1, X2)) → A__SEL(mark(X1), mark(X2))
A__SEL(s(N), cons(X, XS)) → A__SEL(mark(N), mark(XS))
A__ADD(s(X), Y) → MARK(X)
A__FIB(N) → A__FIB1(s(0), s(0))
MARK(add(X1, X2)) → A__ADD(mark(X1), mark(X2))
MARK(sel(X1, X2)) → MARK(X2)

The TRS R consists of the following rules:

a__fib(N) → a__sel(mark(N), a__fib1(s(0), s(0)))
a__fib1(X, Y) → cons(mark(X), fib1(Y, add(X, Y)))
a__add(0, X) → mark(X)
a__add(s(X), Y) → s(a__add(mark(X), mark(Y)))
a__sel(0, cons(X, XS)) → mark(X)
a__sel(s(N), cons(X, XS)) → a__sel(mark(N), mark(XS))
mark(fib(X)) → a__fib(mark(X))
mark(sel(X1, X2)) → a__sel(mark(X1), mark(X2))
mark(fib1(X1, X2)) → a__fib1(mark(X1), mark(X2))
mark(add(X1, X2)) → a__add(mark(X1), mark(X2))
mark(s(X)) → s(mark(X))
mark(0) → 0
mark(cons(X1, X2)) → cons(mark(X1), X2)
a__fib(X) → fib(X)
a__sel(X1, X2) → sel(X1, X2)
a__fib1(X1, X2) → fib1(X1, X2)
a__add(X1, X2) → add(X1, X2)

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ Narrowing

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

MARK(fib(X)) → MARK(X)
A__ADD(0, X) → MARK(X)
A__FIB1(X, Y) → MARK(X)
MARK(fib1(X1, X2)) → MARK(X1)
MARK(s(X)) → MARK(X)
MARK(add(X1, X2)) → MARK(X2)
MARK(fib(X)) → A__FIB(mark(X))
MARK(cons(X1, X2)) → MARK(X1)
MARK(fib1(X1, X2)) → MARK(X2)
A__SEL(0, cons(X, XS)) → MARK(X)
A__FIB(N) → MARK(N)
MARK(add(X1, X2)) → MARK(X1)
A__SEL(s(N), cons(X, XS)) → MARK(XS)
A__ADD(s(X), Y) → A__ADD(mark(X), mark(Y))
A__ADD(s(X), Y) → MARK(Y)
A__FIB(N) → A__SEL(mark(N), a__fib1(s(0), s(0)))
A__SEL(s(N), cons(X, XS)) → MARK(N)
MARK(sel(X1, X2)) → MARK(X1)
MARK(fib1(X1, X2)) → A__FIB1(mark(X1), mark(X2))
MARK(sel(X1, X2)) → A__SEL(mark(X1), mark(X2))
A__SEL(s(N), cons(X, XS)) → A__SEL(mark(N), mark(XS))
A__ADD(s(X), Y) → MARK(X)
A__FIB(N) → A__FIB1(s(0), s(0))
MARK(add(X1, X2)) → A__ADD(mark(X1), mark(X2))
MARK(sel(X1, X2)) → MARK(X2)

The TRS R consists of the following rules:

a__fib(N) → a__sel(mark(N), a__fib1(s(0), s(0)))
a__fib1(X, Y) → cons(mark(X), fib1(Y, add(X, Y)))
a__add(0, X) → mark(X)
a__add(s(X), Y) → s(a__add(mark(X), mark(Y)))
a__sel(0, cons(X, XS)) → mark(X)
a__sel(s(N), cons(X, XS)) → a__sel(mark(N), mark(XS))
mark(fib(X)) → a__fib(mark(X))
mark(sel(X1, X2)) → a__sel(mark(X1), mark(X2))
mark(fib1(X1, X2)) → a__fib1(mark(X1), mark(X2))
mark(add(X1, X2)) → a__add(mark(X1), mark(X2))
mark(s(X)) → s(mark(X))
mark(0) → 0
mark(cons(X1, X2)) → cons(mark(X1), X2)
a__fib(X) → fib(X)
a__sel(X1, X2) → sel(X1, X2)
a__fib1(X1, X2) → fib1(X1, X2)
a__add(X1, X2) → add(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule A__FIB(N) → A__SEL(mark(N), a__fib1(s(0), s(0))) at position [0] we obtained the following new rules:

A__FIB(s(x0)) → A__SEL(s(mark(x0)), a__fib1(s(0), s(0)))
A__FIB(fib1(x0, x1)) → A__SEL(a__fib1(mark(x0), mark(x1)), a__fib1(s(0), s(0)))
A__FIB(sel(x0, x1)) → A__SEL(a__sel(mark(x0), mark(x1)), a__fib1(s(0), s(0)))
A__FIB(add(x0, x1)) → A__SEL(a__add(mark(x0), mark(x1)), a__fib1(s(0), s(0)))
A__FIB(0) → A__SEL(0, a__fib1(s(0), s(0)))
A__FIB(cons(x0, x1)) → A__SEL(cons(mark(x0), x1), a__fib1(s(0), s(0)))
A__FIB(fib(x0)) → A__SEL(a__fib(mark(x0)), a__fib1(s(0), s(0)))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ Narrowing

        ↳ QDP

          ↳ DependencyGraphProof

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

MARK(fib(X)) → MARK(X)
MARK(fib1(X1, X2)) → MARK(X1)
A__FIB1(X, Y) → MARK(X)
A__ADD(0, X) → MARK(X)
MARK(s(X)) → MARK(X)
MARK(add(X1, X2)) → MARK(X2)
MARK(fib(X)) → A__FIB(mark(X))
MARK(cons(X1, X2)) → MARK(X1)
MARK(fib1(X1, X2)) → MARK(X2)
A__FIB(cons(x0, x1)) → A__SEL(cons(mark(x0), x1), a__fib1(s(0), s(0)))
A__FIB(0) → A__SEL(0, a__fib1(s(0), s(0)))
A__SEL(0, cons(X, XS)) → MARK(X)
MARK(add(X1, X2)) → MARK(X1)
A__FIB(N) → MARK(N)
A__FIB(s(x0)) → A__SEL(s(mark(x0)), a__fib1(s(0), s(0)))
A__ADD(s(X), Y) → A__ADD(mark(X), mark(Y))
A__SEL(s(N), cons(X, XS)) → MARK(XS)
A__FIB(fib1(x0, x1)) → A__SEL(a__fib1(mark(x0), mark(x1)), a__fib1(s(0), s(0)))
A__ADD(s(X), Y) → MARK(Y)
A__FIB(sel(x0, x1)) → A__SEL(a__sel(mark(x0), mark(x1)), a__fib1(s(0), s(0)))
A__SEL(s(N), cons(X, XS)) → MARK(N)
MARK(fib1(X1, X2)) → A__FIB1(mark(X1), mark(X2))
MARK(sel(X1, X2)) → MARK(X1)
MARK(sel(X1, X2)) → A__SEL(mark(X1), mark(X2))
A__SEL(s(N), cons(X, XS)) → A__SEL(mark(N), mark(XS))
A__ADD(s(X), Y) → MARK(X)
A__FIB(N) → A__FIB1(s(0), s(0))
A__FIB(fib(x0)) → A__SEL(a__fib(mark(x0)), a__fib1(s(0), s(0)))
A__FIB(add(x0, x1)) → A__SEL(a__add(mark(x0), mark(x1)), a__fib1(s(0), s(0)))
MARK(add(X1, X2)) → A__ADD(mark(X1), mark(X2))
MARK(sel(X1, X2)) → MARK(X2)

The TRS R consists of the following rules:

a__fib(N) → a__sel(mark(N), a__fib1(s(0), s(0)))
a__fib1(X, Y) → cons(mark(X), fib1(Y, add(X, Y)))
a__add(0, X) → mark(X)
a__add(s(X), Y) → s(a__add(mark(X), mark(Y)))
a__sel(0, cons(X, XS)) → mark(X)
a__sel(s(N), cons(X, XS)) → a__sel(mark(N), mark(XS))
mark(fib(X)) → a__fib(mark(X))
mark(sel(X1, X2)) → a__sel(mark(X1), mark(X2))
mark(fib1(X1, X2)) → a__fib1(mark(X1), mark(X2))
mark(add(X1, X2)) → a__add(mark(X1), mark(X2))
mark(s(X)) → s(mark(X))
mark(0) → 0
mark(cons(X1, X2)) → cons(mark(X1), X2)
a__fib(X) → fib(X)
a__sel(X1, X2) → sel(X1, X2)
a__fib1(X1, X2) → fib1(X1, X2)
a__add(X1, X2) → add(X1, X2)

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ Narrowing

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ QDP

              ↳ Narrowing

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

MARK(fib(X)) → MARK(X)
A__ADD(0, X) → MARK(X)
MARK(fib1(X1, X2)) → MARK(X1)
A__FIB1(X, Y) → MARK(X)
MARK(s(X)) → MARK(X)
MARK(add(X1, X2)) → MARK(X2)
MARK(fib(X)) → A__FIB(mark(X))
MARK(cons(X1, X2)) → MARK(X1)
MARK(fib1(X1, X2)) → MARK(X2)
A__FIB(0) → A__SEL(0, a__fib1(s(0), s(0)))
A__SEL(0, cons(X, XS)) → MARK(X)
A__FIB(N) → MARK(N)
MARK(add(X1, X2)) → MARK(X1)
A__ADD(s(X), Y) → A__ADD(mark(X), mark(Y))
A__SEL(s(N), cons(X, XS)) → MARK(XS)
A__FIB(s(x0)) → A__SEL(s(mark(x0)), a__fib1(s(0), s(0)))
A__FIB(fib1(x0, x1)) → A__SEL(a__fib1(mark(x0), mark(x1)), a__fib1(s(0), s(0)))
A__ADD(s(X), Y) → MARK(Y)
A__FIB(sel(x0, x1)) → A__SEL(a__sel(mark(x0), mark(x1)), a__fib1(s(0), s(0)))
A__SEL(s(N), cons(X, XS)) → MARK(N)
MARK(fib1(X1, X2)) → A__FIB1(mark(X1), mark(X2))
MARK(sel(X1, X2)) → MARK(X1)
MARK(sel(X1, X2)) → A__SEL(mark(X1), mark(X2))
A__SEL(s(N), cons(X, XS)) → A__SEL(mark(N), mark(XS))
A__ADD(s(X), Y) → MARK(X)
A__FIB(N) → A__FIB1(s(0), s(0))
A__FIB(fib(x0)) → A__SEL(a__fib(mark(x0)), a__fib1(s(0), s(0)))
A__FIB(add(x0, x1)) → A__SEL(a__add(mark(x0), mark(x1)), a__fib1(s(0), s(0)))
MARK(add(X1, X2)) → A__ADD(mark(X1), mark(X2))
MARK(sel(X1, X2)) → MARK(X2)

The TRS R consists of the following rules:

a__fib(N) → a__sel(mark(N), a__fib1(s(0), s(0)))
a__fib1(X, Y) → cons(mark(X), fib1(Y, add(X, Y)))
a__add(0, X) → mark(X)
a__add(s(X), Y) → s(a__add(mark(X), mark(Y)))
a__sel(0, cons(X, XS)) → mark(X)
a__sel(s(N), cons(X, XS)) → a__sel(mark(N), mark(XS))
mark(fib(X)) → a__fib(mark(X))
mark(sel(X1, X2)) → a__sel(mark(X1), mark(X2))
mark(fib1(X1, X2)) → a__fib1(mark(X1), mark(X2))
mark(add(X1, X2)) → a__add(mark(X1), mark(X2))
mark(s(X)) → s(mark(X))
mark(0) → 0
mark(cons(X1, X2)) → cons(mark(X1), X2)
a__fib(X) → fib(X)
a__sel(X1, X2) → sel(X1, X2)
a__fib1(X1, X2) → fib1(X1, X2)
a__add(X1, X2) → add(X1, X2)

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

MARK(sel(fib(x0), y1)) → A__SEL(a__fib(mark(x0)), mark(y1))
MARK(sel(cons(x0, x1), y1)) → A__SEL(cons(mark(x0), x1), mark(y1))
MARK(sel(0, y1)) → A__SEL(0, mark(y1))
MARK(sel(add(x0, x1), y1)) → A__SEL(a__add(mark(x0), mark(x1)), mark(y1))
MARK(sel(fib1(x0, x1), y1)) → A__SEL(a__fib1(mark(x0), mark(x1)), mark(y1))
MARK(sel(sel(x0, x1), y1)) → A__SEL(a__sel(mark(x0), mark(x1)), mark(y1))
MARK(sel(s(x0), y1)) → A__SEL(s(mark(x0)), mark(y1))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ Narrowing

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ QDP

              ↳ Narrowing

                ↳ QDP

                  ↳ DependencyGraphProof

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

MARK(sel(fib(x0), y1)) → A__SEL(a__fib(mark(x0)), mark(y1))
MARK(fib(X)) → MARK(X)
MARK(sel(cons(x0, x1), y1)) → A__SEL(cons(mark(x0), x1), mark(y1))
A__FIB1(X, Y) → MARK(X)
MARK(fib1(X1, X2)) → MARK(X1)
A__ADD(0, X) → MARK(X)
MARK(s(X)) → MARK(X)
MARK(add(X1, X2)) → MARK(X2)
MARK(sel(fib1(x0, x1), y1)) → A__SEL(a__fib1(mark(x0), mark(x1)), mark(y1))
MARK(fib(X)) → A__FIB(mark(X))
MARK(cons(X1, X2)) → MARK(X1)
MARK(fib1(X1, X2)) → MARK(X2)
A__FIB(0) → A__SEL(0, a__fib1(s(0), s(0)))
A__SEL(0, cons(X, XS)) → MARK(X)
MARK(add(X1, X2)) → MARK(X1)
A__FIB(N) → MARK(N)
A__FIB(s(x0)) → A__SEL(s(mark(x0)), a__fib1(s(0), s(0)))
A__SEL(s(N), cons(X, XS)) → MARK(XS)
A__ADD(s(X), Y) → A__ADD(mark(X), mark(Y))
A__ADD(s(X), Y) → MARK(Y)
A__FIB(fib1(x0, x1)) → A__SEL(a__fib1(mark(x0), mark(x1)), a__fib1(s(0), s(0)))
MARK(sel(0, y1)) → A__SEL(0, mark(y1))
A__FIB(sel(x0, x1)) → A__SEL(a__sel(mark(x0), mark(x1)), a__fib1(s(0), s(0)))
A__SEL(s(N), cons(X, XS)) → MARK(N)
MARK(sel(X1, X2)) → MARK(X1)
MARK(fib1(X1, X2)) → A__FIB1(mark(X1), mark(X2))
A__SEL(s(N), cons(X, XS)) → A__SEL(mark(N), mark(XS))
MARK(sel(add(x0, x1), y1)) → A__SEL(a__add(mark(x0), mark(x1)), mark(y1))
A__ADD(s(X), Y) → MARK(X)
MARK(sel(s(x0), y1)) → A__SEL(s(mark(x0)), mark(y1))
A__FIB(N) → A__FIB1(s(0), s(0))
A__FIB(fib(x0)) → A__SEL(a__fib(mark(x0)), a__fib1(s(0), s(0)))
MARK(sel(sel(x0, x1), y1)) → A__SEL(a__sel(mark(x0), mark(x1)), mark(y1))
MARK(add(X1, X2)) → A__ADD(mark(X1), mark(X2))
A__FIB(add(x0, x1)) → A__SEL(a__add(mark(x0), mark(x1)), a__fib1(s(0), s(0)))
MARK(sel(X1, X2)) → MARK(X2)

The TRS R consists of the following rules:

a__fib(N) → a__sel(mark(N), a__fib1(s(0), s(0)))
a__fib1(X, Y) → cons(mark(X), fib1(Y, add(X, Y)))
a__add(0, X) → mark(X)
a__add(s(X), Y) → s(a__add(mark(X), mark(Y)))
a__sel(0, cons(X, XS)) → mark(X)
a__sel(s(N), cons(X, XS)) → a__sel(mark(N), mark(XS))
mark(fib(X)) → a__fib(mark(X))
mark(sel(X1, X2)) → a__sel(mark(X1), mark(X2))
mark(fib1(X1, X2)) → a__fib1(mark(X1), mark(X2))
mark(add(X1, X2)) → a__add(mark(X1), mark(X2))
mark(s(X)) → s(mark(X))
mark(0) → 0
mark(cons(X1, X2)) → cons(mark(X1), X2)
a__fib(X) → fib(X)
a__sel(X1, X2) → sel(X1, X2)
a__fib1(X1, X2) → fib1(X1, X2)
a__add(X1, X2) → add(X1, X2)

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ Narrowing

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ QDP

              ↳ Narrowing

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ QDP

                      ↳ Narrowing

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

MARK(fib(X)) → MARK(X)
MARK(sel(fib(x0), y1)) → A__SEL(a__fib(mark(x0)), mark(y1))
A__ADD(0, X) → MARK(X)
MARK(fib1(X1, X2)) → MARK(X1)
A__FIB1(X, Y) → MARK(X)
MARK(s(X)) → MARK(X)
MARK(add(X1, X2)) → MARK(X2)
MARK(fib(X)) → A__FIB(mark(X))
MARK(sel(fib1(x0, x1), y1)) → A__SEL(a__fib1(mark(x0), mark(x1)), mark(y1))
MARK(cons(X1, X2)) → MARK(X1)
MARK(fib1(X1, X2)) → MARK(X2)
A__FIB(0) → A__SEL(0, a__fib1(s(0), s(0)))
A__SEL(0, cons(X, XS)) → MARK(X)
MARK(add(X1, X2)) → MARK(X1)
A__FIB(N) → MARK(N)
A__FIB(s(x0)) → A__SEL(s(mark(x0)), a__fib1(s(0), s(0)))
A__ADD(s(X), Y) → A__ADD(mark(X), mark(Y))
A__SEL(s(N), cons(X, XS)) → MARK(XS)
A__FIB(fib1(x0, x1)) → A__SEL(a__fib1(mark(x0), mark(x1)), a__fib1(s(0), s(0)))
A__ADD(s(X), Y) → MARK(Y)
MARK(sel(0, y1)) → A__SEL(0, mark(y1))
A__FIB(sel(x0, x1)) → A__SEL(a__sel(mark(x0), mark(x1)), a__fib1(s(0), s(0)))
A__SEL(s(N), cons(X, XS)) → MARK(N)
MARK(fib1(X1, X2)) → A__FIB1(mark(X1), mark(X2))
MARK(sel(X1, X2)) → MARK(X1)
A__SEL(s(N), cons(X, XS)) → A__SEL(mark(N), mark(XS))
MARK(sel(add(x0, x1), y1)) → A__SEL(a__add(mark(x0), mark(x1)), mark(y1))
A__ADD(s(X), Y) → MARK(X)
MARK(sel(s(x0), y1)) → A__SEL(s(mark(x0)), mark(y1))
A__FIB(N) → A__FIB1(s(0), s(0))
A__FIB(fib(x0)) → A__SEL(a__fib(mark(x0)), a__fib1(s(0), s(0)))
MARK(sel(sel(x0, x1), y1)) → A__SEL(a__sel(mark(x0), mark(x1)), mark(y1))
A__FIB(add(x0, x1)) → A__SEL(a__add(mark(x0), mark(x1)), a__fib1(s(0), s(0)))
MARK(add(X1, X2)) → A__ADD(mark(X1), mark(X2))
MARK(sel(X1, X2)) → MARK(X2)

The TRS R consists of the following rules:

a__fib(N) → a__sel(mark(N), a__fib1(s(0), s(0)))
a__fib1(X, Y) → cons(mark(X), fib1(Y, add(X, Y)))
a__add(0, X) → mark(X)
a__add(s(X), Y) → s(a__add(mark(X), mark(Y)))
a__sel(0, cons(X, XS)) → mark(X)
a__sel(s(N), cons(X, XS)) → a__sel(mark(N), mark(XS))
mark(fib(X)) → a__fib(mark(X))
mark(sel(X1, X2)) → a__sel(mark(X1), mark(X2))
mark(fib1(X1, X2)) → a__fib1(mark(X1), mark(X2))
mark(add(X1, X2)) → a__add(mark(X1), mark(X2))
mark(s(X)) → s(mark(X))
mark(0) → 0
mark(cons(X1, X2)) → cons(mark(X1), X2)
a__fib(X) → fib(X)
a__sel(X1, X2) → sel(X1, X2)
a__fib1(X1, X2) → fib1(X1, X2)
a__add(X1, X2) → add(X1, X2)

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

MARK(add(fib1(x0, x1), y1)) → A__ADD(a__fib1(mark(x0), mark(x1)), mark(y1))
MARK(add(add(x0, x1), y1)) → A__ADD(a__add(mark(x0), mark(x1)), mark(y1))
MARK(add(0, y1)) → A__ADD(0, mark(y1))
MARK(add(s(x0), y1)) → A__ADD(s(mark(x0)), mark(y1))
MARK(add(fib(x0), y1)) → A__ADD(a__fib(mark(x0)), mark(y1))
MARK(add(sel(x0, x1), y1)) → A__ADD(a__sel(mark(x0), mark(x1)), mark(y1))
MARK(add(cons(x0, x1), y1)) → A__ADD(cons(mark(x0), x1), mark(y1))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ Narrowing

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ QDP

              ↳ Narrowing

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ QDP

                      ↳ Narrowing

                        ↳ QDP

                          ↳ DependencyGraphProof

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

MARK(sel(fib(x0), y1)) → A__SEL(a__fib(mark(x0)), mark(y1))
MARK(fib(X)) → MARK(X)
A__FIB1(X, Y) → MARK(X)
MARK(fib1(X1, X2)) → MARK(X1)
A__ADD(0, X) → MARK(X)
MARK(s(X)) → MARK(X)
MARK(add(X1, X2)) → MARK(X2)
MARK(sel(fib1(x0, x1), y1)) → A__SEL(a__fib1(mark(x0), mark(x1)), mark(y1))
MARK(fib(X)) → A__FIB(mark(X))
MARK(cons(X1, X2)) → MARK(X1)
MARK(fib1(X1, X2)) → MARK(X2)
A__FIB(0) → A__SEL(0, a__fib1(s(0), s(0)))
MARK(add(cons(x0, x1), y1)) → A__ADD(cons(mark(x0), x1), mark(y1))
A__SEL(0, cons(X, XS)) → MARK(X)
A__FIB(N) → MARK(N)
MARK(add(X1, X2)) → MARK(X1)
MARK(add(fib1(x0, x1), y1)) → A__ADD(a__fib1(mark(x0), mark(x1)), mark(y1))
A__SEL(s(N), cons(X, XS)) → MARK(XS)
A__ADD(s(X), Y) → A__ADD(mark(X), mark(Y))
A__FIB(s(x0)) → A__SEL(s(mark(x0)), a__fib1(s(0), s(0)))
A__ADD(s(X), Y) → MARK(Y)
A__FIB(fib1(x0, x1)) → A__SEL(a__fib1(mark(x0), mark(x1)), a__fib1(s(0), s(0)))
MARK(sel(0, y1)) → A__SEL(0, mark(y1))
MARK(add(0, y1)) → A__ADD(0, mark(y1))
A__FIB(sel(x0, x1)) → A__SEL(a__sel(mark(x0), mark(x1)), a__fib1(s(0), s(0)))
MARK(add(s(x0), y1)) → A__ADD(s(mark(x0)), mark(y1))
MARK(add(sel(x0, x1), y1)) → A__ADD(a__sel(mark(x0), mark(x1)), mark(y1))
A__SEL(s(N), cons(X, XS)) → MARK(N)
MARK(sel(X1, X2)) → MARK(X1)
MARK(fib1(X1, X2)) → A__FIB1(mark(X1), mark(X2))
A__SEL(s(N), cons(X, XS)) → A__SEL(mark(N), mark(XS))
MARK(sel(add(x0, x1), y1)) → A__SEL(a__add(mark(x0), mark(x1)), mark(y1))
MARK(add(add(x0, x1), y1)) → A__ADD(a__add(mark(x0), mark(x1)), mark(y1))
A__ADD(s(X), Y) → MARK(X)
MARK(sel(s(x0), y1)) → A__SEL(s(mark(x0)), mark(y1))
A__FIB(N) → A__FIB1(s(0), s(0))
MARK(add(fib(x0), y1)) → A__ADD(a__fib(mark(x0)), mark(y1))
A__FIB(fib(x0)) → A__SEL(a__fib(mark(x0)), a__fib1(s(0), s(0)))
MARK(sel(sel(x0, x1), y1)) → A__SEL(a__sel(mark(x0), mark(x1)), mark(y1))
A__FIB(add(x0, x1)) → A__SEL(a__add(mark(x0), mark(x1)), a__fib1(s(0), s(0)))
MARK(sel(X1, X2)) → MARK(X2)

The TRS R consists of the following rules:

a__fib(N) → a__sel(mark(N), a__fib1(s(0), s(0)))
a__fib1(X, Y) → cons(mark(X), fib1(Y, add(X, Y)))
a__add(0, X) → mark(X)
a__add(s(X), Y) → s(a__add(mark(X), mark(Y)))
a__sel(0, cons(X, XS)) → mark(X)
a__sel(s(N), cons(X, XS)) → a__sel(mark(N), mark(XS))
mark(fib(X)) → a__fib(mark(X))
mark(sel(X1, X2)) → a__sel(mark(X1), mark(X2))
mark(fib1(X1, X2)) → a__fib1(mark(X1), mark(X2))
mark(add(X1, X2)) → a__add(mark(X1), mark(X2))
mark(s(X)) → s(mark(X))
mark(0) → 0
mark(cons(X1, X2)) → cons(mark(X1), X2)
a__fib(X) → fib(X)
a__sel(X1, X2) → sel(X1, X2)
a__fib1(X1, X2) → fib1(X1, X2)
a__add(X1, X2) → add(X1, X2)

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ Narrowing

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ QDP

              ↳ Narrowing

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ QDP

                      ↳ Narrowing

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ QDP

                              ↳ Narrowing

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

MARK(fib(X)) → MARK(X)
MARK(sel(fib(x0), y1)) → A__SEL(a__fib(mark(x0)), mark(y1))
MARK(fib1(X1, X2)) → MARK(X1)
A__FIB1(X, Y) → MARK(X)
A__ADD(0, X) → MARK(X)
MARK(s(X)) → MARK(X)
MARK(add(X1, X2)) → MARK(X2)
MARK(fib(X)) → A__FIB(mark(X))
MARK(sel(fib1(x0, x1), y1)) → A__SEL(a__fib1(mark(x0), mark(x1)), mark(y1))
MARK(cons(X1, X2)) → MARK(X1)
MARK(fib1(X1, X2)) → MARK(X2)
A__FIB(0) → A__SEL(0, a__fib1(s(0), s(0)))
A__SEL(0, cons(X, XS)) → MARK(X)
MARK(add(X1, X2)) → MARK(X1)
A__FIB(N) → MARK(N)
MARK(add(fib1(x0, x1), y1)) → A__ADD(a__fib1(mark(x0), mark(x1)), mark(y1))
A__FIB(s(x0)) → A__SEL(s(mark(x0)), a__fib1(s(0), s(0)))
A__ADD(s(X), Y) → A__ADD(mark(X), mark(Y))
A__SEL(s(N), cons(X, XS)) → MARK(XS)
A__FIB(fib1(x0, x1)) → A__SEL(a__fib1(mark(x0), mark(x1)), a__fib1(s(0), s(0)))
A__ADD(s(X), Y) → MARK(Y)
MARK(sel(0, y1)) → A__SEL(0, mark(y1))
MARK(add(0, y1)) → A__ADD(0, mark(y1))
A__FIB(sel(x0, x1)) → A__SEL(a__sel(mark(x0), mark(x1)), a__fib1(s(0), s(0)))
MARK(add(s(x0), y1)) → A__ADD(s(mark(x0)), mark(y1))
MARK(add(sel(x0, x1), y1)) → A__ADD(a__sel(mark(x0), mark(x1)), mark(y1))
A__SEL(s(N), cons(X, XS)) → MARK(N)
MARK(fib1(X1, X2)) → A__FIB1(mark(X1), mark(X2))
MARK(sel(X1, X2)) → MARK(X1)
A__SEL(s(N), cons(X, XS)) → A__SEL(mark(N), mark(XS))
MARK(sel(add(x0, x1), y1)) → A__SEL(a__add(mark(x0), mark(x1)), mark(y1))
MARK(add(add(x0, x1), y1)) → A__ADD(a__add(mark(x0), mark(x1)), mark(y1))
A__ADD(s(X), Y) → MARK(X)
MARK(sel(s(x0), y1)) → A__SEL(s(mark(x0)), mark(y1))
A__FIB(N) → A__FIB1(s(0), s(0))
MARK(add(fib(x0), y1)) → A__ADD(a__fib(mark(x0)), mark(y1))
A__FIB(fib(x0)) → A__SEL(a__fib(mark(x0)), a__fib1(s(0), s(0)))
MARK(sel(sel(x0, x1), y1)) → A__SEL(a__sel(mark(x0), mark(x1)), mark(y1))
A__FIB(add(x0, x1)) → A__SEL(a__add(mark(x0), mark(x1)), a__fib1(s(0), s(0)))
MARK(sel(X1, X2)) → MARK(X2)

The TRS R consists of the following rules:

a__fib(N) → a__sel(mark(N), a__fib1(s(0), s(0)))
a__fib1(X, Y) → cons(mark(X), fib1(Y, add(X, Y)))
a__add(0, X) → mark(X)
a__add(s(X), Y) → s(a__add(mark(X), mark(Y)))
a__sel(0, cons(X, XS)) → mark(X)
a__sel(s(N), cons(X, XS)) → a__sel(mark(N), mark(XS))
mark(fib(X)) → a__fib(mark(X))
mark(sel(X1, X2)) → a__sel(mark(X1), mark(X2))
mark(fib1(X1, X2)) → a__fib1(mark(X1), mark(X2))
mark(add(X1, X2)) → a__add(mark(X1), mark(X2))
mark(s(X)) → s(mark(X))
mark(0) → 0
mark(cons(X1, X2)) → cons(mark(X1), X2)
a__fib(X) → fib(X)
a__sel(X1, X2) → sel(X1, X2)
a__fib1(X1, X2) → fib1(X1, X2)
a__add(X1, X2) → add(X1, X2)

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

MARK(sel(fib(y0), add(x0, x1))) → A__SEL(a__fib(mark(y0)), a__add(mark(x0), mark(x1)))
MARK(sel(fib(y0), cons(x0, x1))) → A__SEL(a__fib(mark(y0)), cons(mark(x0), x1))
MARK(sel(fib(y0), fib(x0))) → A__SEL(a__fib(mark(y0)), a__fib(mark(x0)))
MARK(sel(fib(y0), sel(x0, x1))) → A__SEL(a__fib(mark(y0)), a__sel(mark(x0), mark(x1)))
MARK(sel(fib(y0), s(x0))) → A__SEL(a__fib(mark(y0)), s(mark(x0)))
MARK(sel(fib(y0), fib1(x0, x1))) → A__SEL(a__fib(mark(y0)), a__fib1(mark(x0), mark(x1)))
MARK(sel(fib(y0), 0)) → A__SEL(a__fib(mark(y0)), 0)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ Narrowing

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ QDP

              ↳ Narrowing

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ QDP

                      ↳ Narrowing

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ QDP

                              ↳ Narrowing

                                ↳ QDP

                                  ↳ DependencyGraphProof

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

MARK(fib(X)) → MARK(X)
A__ADD(0, X) → MARK(X)
A__FIB1(X, Y) → MARK(X)
MARK(fib1(X1, X2)) → MARK(X1)
MARK(s(X)) → MARK(X)
MARK(add(X1, X2)) → MARK(X2)
MARK(sel(fib1(x0, x1), y1)) → A__SEL(a__fib1(mark(x0), mark(x1)), mark(y1))
MARK(fib(X)) → A__FIB(mark(X))
MARK(cons(X1, X2)) → MARK(X1)
MARK(fib1(X1, X2)) → MARK(X2)
MARK(sel(fib(y0), sel(x0, x1))) → A__SEL(a__fib(mark(y0)), a__sel(mark(x0), mark(x1)))
A__FIB(0) → A__SEL(0, a__fib1(s(0), s(0)))
A__SEL(0, cons(X, XS)) → MARK(X)
MARK(sel(fib(y0), 0)) → A__SEL(a__fib(mark(y0)), 0)
A__FIB(N) → MARK(N)
MARK(add(X1, X2)) → MARK(X1)
MARK(add(fib1(x0, x1), y1)) → A__ADD(a__fib1(mark(x0), mark(x1)), mark(y1))
A__SEL(s(N), cons(X, XS)) → MARK(XS)
A__ADD(s(X), Y) → A__ADD(mark(X), mark(Y))
A__FIB(s(x0)) → A__SEL(s(mark(x0)), a__fib1(s(0), s(0)))
A__ADD(s(X), Y) → MARK(Y)
A__FIB(fib1(x0, x1)) → A__SEL(a__fib1(mark(x0), mark(x1)), a__fib1(s(0), s(0)))
MARK(sel(0, y1)) → A__SEL(0, mark(y1))
MARK(add(0, y1)) → A__ADD(0, mark(y1))
MARK(sel(fib(y0), cons(x0, x1))) → A__SEL(a__fib(mark(y0)), cons(mark(x0), x1))
A__FIB(sel(x0, x1)) → A__SEL(a__sel(mark(x0), mark(x1)), a__fib1(s(0), s(0)))
MARK(add(s(x0), y1)) → A__ADD(s(mark(x0)), mark(y1))
MARK(sel(fib(y0), fib1(x0, x1))) → A__SEL(a__fib(mark(y0)), a__fib1(mark(x0), mark(x1)))
MARK(sel(fib(y0), s(x0))) → A__SEL(a__fib(mark(y0)), s(mark(x0)))
MARK(add(sel(x0, x1), y1)) → A__ADD(a__sel(mark(x0), mark(x1)), mark(y1))
A__SEL(s(N), cons(X, XS)) → MARK(N)
MARK(sel(X1, X2)) → MARK(X1)
MARK(fib1(X1, X2)) → A__FIB1(mark(X1), mark(X2))
A__SEL(s(N), cons(X, XS)) → A__SEL(mark(N), mark(XS))
MARK(sel(fib(y0), add(x0, x1))) → A__SEL(a__fib(mark(y0)), a__add(mark(x0), mark(x1)))
MARK(sel(add(x0, x1), y1)) → A__SEL(a__add(mark(x0), mark(x1)), mark(y1))
MARK(add(add(x0, x1), y1)) → A__ADD(a__add(mark(x0), mark(x1)), mark(y1))
MARK(sel(fib(y0), fib(x0))) → A__SEL(a__fib(mark(y0)), a__fib(mark(x0)))
A__ADD(s(X), Y) → MARK(X)
MARK(sel(s(x0), y1)) → A__SEL(s(mark(x0)), mark(y1))
A__FIB(N) → A__FIB1(s(0), s(0))
MARK(add(fib(x0), y1)) → A__ADD(a__fib(mark(x0)), mark(y1))
A__FIB(fib(x0)) → A__SEL(a__fib(mark(x0)), a__fib1(s(0), s(0)))
MARK(sel(sel(x0, x1), y1)) → A__SEL(a__sel(mark(x0), mark(x1)), mark(y1))
A__FIB(add(x0, x1)) → A__SEL(a__add(mark(x0), mark(x1)), a__fib1(s(0), s(0)))
MARK(sel(X1, X2)) → MARK(X2)

The TRS R consists of the following rules:

a__fib(N) → a__sel(mark(N), a__fib1(s(0), s(0)))
a__fib1(X, Y) → cons(mark(X), fib1(Y, add(X, Y)))
a__add(0, X) → mark(X)
a__add(s(X), Y) → s(a__add(mark(X), mark(Y)))
a__sel(0, cons(X, XS)) → mark(X)
a__sel(s(N), cons(X, XS)) → a__sel(mark(N), mark(XS))
mark(fib(X)) → a__fib(mark(X))
mark(sel(X1, X2)) → a__sel(mark(X1), mark(X2))
mark(fib1(X1, X2)) → a__fib1(mark(X1), mark(X2))
mark(add(X1, X2)) → a__add(mark(X1), mark(X2))
mark(s(X)) → s(mark(X))
mark(0) → 0
mark(cons(X1, X2)) → cons(mark(X1), X2)
a__fib(X) → fib(X)
a__sel(X1, X2) → sel(X1, X2)
a__fib1(X1, X2) → fib1(X1, X2)
a__add(X1, X2) → add(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 1 SCC with 2 less nodes.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ Narrowing

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ QDP

              ↳ Narrowing

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ QDP

                      ↳ Narrowing

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ QDP

                              ↳ Narrowing

                                ↳ QDP

                                  ↳ DependencyGraphProof

                                    ↳ QDP

                                      ↳ Narrowing

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

MARK(fib(X)) → MARK(X)
MARK(fib1(X1, X2)) → MARK(X1)
A__FIB1(X, Y) → MARK(X)
A__ADD(0, X) → MARK(X)
MARK(s(X)) → MARK(X)
MARK(add(X1, X2)) → MARK(X2)
MARK(fib(X)) → A__FIB(mark(X))
MARK(sel(fib1(x0, x1), y1)) → A__SEL(a__fib1(mark(x0), mark(x1)), mark(y1))
MARK(cons(X1, X2)) → MARK(X1)
MARK(fib1(X1, X2)) → MARK(X2)
A__FIB(0) → A__SEL(0, a__fib1(s(0), s(0)))
MARK(sel(fib(y0), sel(x0, x1))) → A__SEL(a__fib(mark(y0)), a__sel(mark(x0), mark(x1)))
A__SEL(0, cons(X, XS)) → MARK(X)
MARK(add(X1, X2)) → MARK(X1)
A__FIB(N) → MARK(N)
MARK(add(fib1(x0, x1), y1)) → A__ADD(a__fib1(mark(x0), mark(x1)), mark(y1))
A__FIB(s(x0)) → A__SEL(s(mark(x0)), a__fib1(s(0), s(0)))
A__ADD(s(X), Y) → A__ADD(mark(X), mark(Y))
A__SEL(s(N), cons(X, XS)) → MARK(XS)
A__FIB(fib1(x0, x1)) → A__SEL(a__fib1(mark(x0), mark(x1)), a__fib1(s(0), s(0)))
A__ADD(s(X), Y) → MARK(Y)
MARK(sel(0, y1)) → A__SEL(0, mark(y1))
MARK(add(0, y1)) → A__ADD(0, mark(y1))
MARK(sel(fib(y0), cons(x0, x1))) → A__SEL(a__fib(mark(y0)), cons(mark(x0), x1))
A__FIB(sel(x0, x1)) → A__SEL(a__sel(mark(x0), mark(x1)), a__fib1(s(0), s(0)))
MARK(add(s(x0), y1)) → A__ADD(s(mark(x0)), mark(y1))
MARK(sel(fib(y0), fib1(x0, x1))) → A__SEL(a__fib(mark(y0)), a__fib1(mark(x0), mark(x1)))
MARK(add(sel(x0, x1), y1)) → A__ADD(a__sel(mark(x0), mark(x1)), mark(y1))
A__SEL(s(N), cons(X, XS)) → MARK(N)
MARK(fib1(X1, X2)) → A__FIB1(mark(X1), mark(X2))
MARK(sel(X1, X2)) → MARK(X1)
A__SEL(s(N), cons(X, XS)) → A__SEL(mark(N), mark(XS))
MARK(sel(fib(y0), add(x0, x1))) → A__SEL(a__fib(mark(y0)), a__add(mark(x0), mark(x1)))
MARK(sel(add(x0, x1), y1)) → A__SEL(a__add(mark(x0), mark(x1)), mark(y1))
MARK(add(add(x0, x1), y1)) → A__ADD(a__add(mark(x0), mark(x1)), mark(y1))
MARK(sel(fib(y0), fib(x0))) → A__SEL(a__fib(mark(y0)), a__fib(mark(x0)))
A__ADD(s(X), Y) → MARK(X)
MARK(sel(s(x0), y1)) → A__SEL(s(mark(x0)), mark(y1))
A__FIB(N) → A__FIB1(s(0), s(0))
MARK(add(fib(x0), y1)) → A__ADD(a__fib(mark(x0)), mark(y1))
A__FIB(fib(x0)) → A__SEL(a__fib(mark(x0)), a__fib1(s(0), s(0)))
MARK(sel(sel(x0, x1), y1)) → A__SEL(a__sel(mark(x0), mark(x1)), mark(y1))
A__FIB(add(x0, x1)) → A__SEL(a__add(mark(x0), mark(x1)), a__fib1(s(0), s(0)))
MARK(sel(X1, X2)) → MARK(X2)

The TRS R consists of the following rules:

a__fib(N) → a__sel(mark(N), a__fib1(s(0), s(0)))
a__fib1(X, Y) → cons(mark(X), fib1(Y, add(X, Y)))
a__add(0, X) → mark(X)
a__add(s(X), Y) → s(a__add(mark(X), mark(Y)))
a__sel(0, cons(X, XS)) → mark(X)
a__sel(s(N), cons(X, XS)) → a__sel(mark(N), mark(XS))
mark(fib(X)) → a__fib(mark(X))
mark(sel(X1, X2)) → a__sel(mark(X1), mark(X2))
mark(fib1(X1, X2)) → a__fib1(mark(X1), mark(X2))
mark(add(X1, X2)) → a__add(mark(X1), mark(X2))
mark(s(X)) → s(mark(X))
mark(0) → 0
mark(cons(X1, X2)) → cons(mark(X1), X2)
a__fib(X) → fib(X)
a__sel(X1, X2) → sel(X1, X2)
a__fib1(X1, X2) → fib1(X1, X2)
a__add(X1, X2) → add(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule MARK(sel(0, y1)) → A__SEL(0, mark(y1)) at position [1] we obtained the following new rules:

MARK(sel(0, fib1(x0, x1))) → A__SEL(0, a__fib1(mark(x0), mark(x1)))
MARK(sel(0, sel(x0, x1))) → A__SEL(0, a__sel(mark(x0), mark(x1)))
MARK(sel(0, cons(x0, x1))) → A__SEL(0, cons(mark(x0), x1))
MARK(sel(0, 0)) → A__SEL(0, 0)
MARK(sel(0, s(x0))) → A__SEL(0, s(mark(x0)))
MARK(sel(0, add(x0, x1))) → A__SEL(0, a__add(mark(x0), mark(x1)))
MARK(sel(0, fib(x0))) → A__SEL(0, a__fib(mark(x0)))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ Narrowing

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ QDP

              ↳ Narrowing

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ QDP

                      ↳ Narrowing

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ QDP

                              ↳ Narrowing

                                ↳ QDP

                                  ↳ DependencyGraphProof

                                    ↳ QDP

                                      ↳ Narrowing

                                        ↳ QDP

                                          ↳ DependencyGraphProof

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

MARK(fib1(X1, X2)) → MARK(X1)
MARK(sel(fib1(x0, x1), y1)) → A__SEL(a__fib1(mark(x0), mark(x1)), mark(y1))
MARK(cons(X1, X2)) → MARK(X1)
MARK(sel(0, add(x0, x1))) → A__SEL(0, a__add(mark(x0), mark(x1)))
MARK(sel(0, fib(x0))) → A__SEL(0, a__fib(mark(x0)))
A__FIB(N) → MARK(N)
A__FIB(s(x0)) → A__SEL(s(mark(x0)), a__fib1(s(0), s(0)))
A__ADD(s(X), Y) → A__ADD(mark(X), mark(Y))
A__SEL(s(N), cons(X, XS)) → MARK(XS)
A__ADD(s(X), Y) → MARK(Y)
MARK(add(0, y1)) → A__ADD(0, mark(y1))
MARK(sel(fib(y0), cons(x0, x1))) → A__SEL(a__fib(mark(y0)), cons(mark(x0), x1))
A__FIB(sel(x0, x1)) → A__SEL(a__sel(mark(x0), mark(x1)), a__fib1(s(0), s(0)))
MARK(add(s(x0), y1)) → A__ADD(s(mark(x0)), mark(y1))
MARK(sel(fib(y0), fib1(x0, x1))) → A__SEL(a__fib(mark(y0)), a__fib1(mark(x0), mark(x1)))
MARK(add(sel(x0, x1), y1)) → A__ADD(a__sel(mark(x0), mark(x1)), mark(y1))
MARK(sel(X1, X2)) → MARK(X1)
MARK(sel(fib(y0), add(x0, x1))) → A__SEL(a__fib(mark(y0)), a__add(mark(x0), mark(x1)))
MARK(add(add(x0, x1), y1)) → A__ADD(a__add(mark(x0), mark(x1)), mark(y1))
A__ADD(s(X), Y) → MARK(X)
MARK(sel(s(x0), y1)) → A__SEL(s(mark(x0)), mark(y1))
A__FIB(N) → A__FIB1(s(0), s(0))
MARK(add(fib(x0), y1)) → A__ADD(a__fib(mark(x0)), mark(y1))
A__FIB(fib(x0)) → A__SEL(a__fib(mark(x0)), a__fib1(s(0), s(0)))
MARK(sel(0, cons(x0, x1))) → A__SEL(0, cons(mark(x0), x1))
MARK(sel(0, 0)) → A__SEL(0, 0)
MARK(sel(sel(x0, x1), y1)) → A__SEL(a__sel(mark(x0), mark(x1)), mark(y1))
MARK(fib(X)) → MARK(X)
A__FIB1(X, Y) → MARK(X)
A__ADD(0, X) → MARK(X)
MARK(s(X)) → MARK(X)
MARK(add(X1, X2)) → MARK(X2)
MARK(fib(X)) → A__FIB(mark(X))
MARK(fib1(X1, X2)) → MARK(X2)
A__FIB(0) → A__SEL(0, a__fib1(s(0), s(0)))
MARK(sel(fib(y0), sel(x0, x1))) → A__SEL(a__fib(mark(y0)), a__sel(mark(x0), mark(x1)))
A__SEL(0, cons(X, XS)) → MARK(X)
MARK(add(X1, X2)) → MARK(X1)
MARK(add(fib1(x0, x1), y1)) → A__ADD(a__fib1(mark(x0), mark(x1)), mark(y1))
MARK(sel(0, fib1(x0, x1))) → A__SEL(0, a__fib1(mark(x0), mark(x1)))
A__FIB(fib1(x0, x1)) → A__SEL(a__fib1(mark(x0), mark(x1)), a__fib1(s(0), s(0)))
A__SEL(s(N), cons(X, XS)) → MARK(N)
MARK(fib1(X1, X2)) → A__FIB1(mark(X1), mark(X2))
A__SEL(s(N), cons(X, XS)) → A__SEL(mark(N), mark(XS))
MARK(sel(add(x0, x1), y1)) → A__SEL(a__add(mark(x0), mark(x1)), mark(y1))
MARK(sel(0, s(x0))) → A__SEL(0, s(mark(x0)))
MARK(sel(fib(y0), fib(x0))) → A__SEL(a__fib(mark(y0)), a__fib(mark(x0)))
MARK(sel(0, sel(x0, x1))) → A__SEL(0, a__sel(mark(x0), mark(x1)))
A__FIB(add(x0, x1)) → A__SEL(a__add(mark(x0), mark(x1)), a__fib1(s(0), s(0)))
MARK(sel(X1, X2)) → MARK(X2)

The TRS R consists of the following rules:

a__fib(N) → a__sel(mark(N), a__fib1(s(0), s(0)))
a__fib1(X, Y) → cons(mark(X), fib1(Y, add(X, Y)))
a__add(0, X) → mark(X)
a__add(s(X), Y) → s(a__add(mark(X), mark(Y)))
a__sel(0, cons(X, XS)) → mark(X)
a__sel(s(N), cons(X, XS)) → a__sel(mark(N), mark(XS))
mark(fib(X)) → a__fib(mark(X))
mark(sel(X1, X2)) → a__sel(mark(X1), mark(X2))
mark(fib1(X1, X2)) → a__fib1(mark(X1), mark(X2))
mark(add(X1, X2)) → a__add(mark(X1), mark(X2))
mark(s(X)) → s(mark(X))
mark(0) → 0
mark(cons(X1, X2)) → cons(mark(X1), X2)
a__fib(X) → fib(X)
a__sel(X1, X2) → sel(X1, X2)
a__fib1(X1, X2) → fib1(X1, X2)
a__add(X1, X2) → add(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 1 SCC with 2 less nodes.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ Narrowing

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ QDP

              ↳ Narrowing

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ QDP

                      ↳ Narrowing

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ QDP

                              ↳ Narrowing

                                ↳ QDP

                                  ↳ DependencyGraphProof

                                    ↳ QDP

                                      ↳ Narrowing

                                        ↳ QDP

                                          ↳ DependencyGraphProof

                                            ↳ QDP

                                              ↳ Narrowing

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

MARK(fib(X)) → MARK(X)
MARK(fib1(X1, X2)) → MARK(X1)
A__FIB1(X, Y) → MARK(X)
A__ADD(0, X) → MARK(X)
MARK(s(X)) → MARK(X)
MARK(add(X1, X2)) → MARK(X2)
MARK(fib(X)) → A__FIB(mark(X))
MARK(sel(fib1(x0, x1), y1)) → A__SEL(a__fib1(mark(x0), mark(x1)), mark(y1))
MARK(cons(X1, X2)) → MARK(X1)
MARK(fib1(X1, X2)) → MARK(X2)
A__FIB(0) → A__SEL(0, a__fib1(s(0), s(0)))
MARK(sel(fib(y0), sel(x0, x1))) → A__SEL(a__fib(mark(y0)), a__sel(mark(x0), mark(x1)))
MARK(sel(0, add(x0, x1))) → A__SEL(0, a__add(mark(x0), mark(x1)))
A__SEL(0, cons(X, XS)) → MARK(X)
MARK(sel(0, fib(x0))) → A__SEL(0, a__fib(mark(x0)))
MARK(add(X1, X2)) → MARK(X1)
A__FIB(N) → MARK(N)
MARK(add(fib1(x0, x1), y1)) → A__ADD(a__fib1(mark(x0), mark(x1)), mark(y1))
MARK(sel(0, fib1(x0, x1))) → A__SEL(0, a__fib1(mark(x0), mark(x1)))
A__FIB(s(x0)) → A__SEL(s(mark(x0)), a__fib1(s(0), s(0)))
A__SEL(s(N), cons(X, XS)) → MARK(XS)
A__ADD(s(X), Y) → A__ADD(mark(X), mark(Y))
A__FIB(fib1(x0, x1)) → A__SEL(a__fib1(mark(x0), mark(x1)), a__fib1(s(0), s(0)))
A__ADD(s(X), Y) → MARK(Y)
MARK(add(0, y1)) → A__ADD(0, mark(y1))
MARK(sel(fib(y0), cons(x0, x1))) → A__SEL(a__fib(mark(y0)), cons(mark(x0), x1))
A__FIB(sel(x0, x1)) → A__SEL(a__sel(mark(x0), mark(x1)), a__fib1(s(0), s(0)))
MARK(add(s(x0), y1)) → A__ADD(s(mark(x0)), mark(y1))
MARK(sel(fib(y0), fib1(x0, x1))) → A__SEL(a__fib(mark(y0)), a__fib1(mark(x0), mark(x1)))
MARK(add(sel(x0, x1), y1)) → A__ADD(a__sel(mark(x0), mark(x1)), mark(y1))
A__SEL(s(N), cons(X, XS)) → MARK(N)
MARK(fib1(X1, X2)) → A__FIB1(mark(X1), mark(X2))
MARK(sel(X1, X2)) → MARK(X1)
A__SEL(s(N), cons(X, XS)) → A__SEL(mark(N), mark(XS))
MARK(sel(fib(y0), add(x0, x1))) → A__SEL(a__fib(mark(y0)), a__add(mark(x0), mark(x1)))
MARK(sel(add(x0, x1), y1)) → A__SEL(a__add(mark(x0), mark(x1)), mark(y1))
MARK(add(add(x0, x1), y1)) → A__ADD(a__add(mark(x0), mark(x1)), mark(y1))
MARK(sel(fib(y0), fib(x0))) → A__SEL(a__fib(mark(y0)), a__fib(mark(x0)))
A__ADD(s(X), Y) → MARK(X)
MARK(sel(s(x0), y1)) → A__SEL(s(mark(x0)), mark(y1))
A__FIB(N) → A__FIB1(s(0), s(0))
MARK(add(fib(x0), y1)) → A__ADD(a__fib(mark(x0)), mark(y1))
A__FIB(fib(x0)) → A__SEL(a__fib(mark(x0)), a__fib1(s(0), s(0)))
MARK(sel(0, sel(x0, x1))) → A__SEL(0, a__sel(mark(x0), mark(x1)))
MARK(sel(0, cons(x0, x1))) → A__SEL(0, cons(mark(x0), x1))
MARK(sel(sel(x0, x1), y1)) → A__SEL(a__sel(mark(x0), mark(x1)), mark(y1))
A__FIB(add(x0, x1)) → A__SEL(a__add(mark(x0), mark(x1)), a__fib1(s(0), s(0)))
MARK(sel(X1, X2)) → MARK(X2)

The TRS R consists of the following rules:

a__fib(N) → a__sel(mark(N), a__fib1(s(0), s(0)))
a__fib1(X, Y) → cons(mark(X), fib1(Y, add(X, Y)))
a__add(0, X) → mark(X)
a__add(s(X), Y) → s(a__add(mark(X), mark(Y)))
a__sel(0, cons(X, XS)) → mark(X)
a__sel(s(N), cons(X, XS)) → a__sel(mark(N), mark(XS))
mark(fib(X)) → a__fib(mark(X))
mark(sel(X1, X2)) → a__sel(mark(X1), mark(X2))
mark(fib1(X1, X2)) → a__fib1(mark(X1), mark(X2))
mark(add(X1, X2)) → a__add(mark(X1), mark(X2))
mark(s(X)) → s(mark(X))
mark(0) → 0
mark(cons(X1, X2)) → cons(mark(X1), X2)
a__fib(X) → fib(X)
a__sel(X1, X2) → sel(X1, X2)
a__fib1(X1, X2) → fib1(X1, X2)
a__add(X1, X2) → add(X1, X2)

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

MARK(sel(add(y0, y1), fib(x0))) → A__SEL(a__add(mark(y0), mark(y1)), a__fib(mark(x0)))
MARK(sel(add(y0, y1), add(x0, x1))) → A__SEL(a__add(mark(y0), mark(y1)), a__add(mark(x0), mark(x1)))
MARK(sel(add(y0, y1), 0)) → A__SEL(a__add(mark(y0), mark(y1)), 0)
MARK(sel(add(y0, y1), s(x0))) → A__SEL(a__add(mark(y0), mark(y1)), s(mark(x0)))
MARK(sel(add(y0, y1), fib1(x0, x1))) → A__SEL(a__add(mark(y0), mark(y1)), a__fib1(mark(x0), mark(x1)))
MARK(sel(add(y0, y1), sel(x0, x1))) → A__SEL(a__add(mark(y0), mark(y1)), a__sel(mark(x0), mark(x1)))
MARK(sel(add(y0, y1), cons(x0, x1))) → A__SEL(a__add(mark(y0), mark(y1)), cons(mark(x0), x1))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ Narrowing

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ QDP

              ↳ Narrowing

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ QDP

                      ↳ Narrowing

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ QDP

                              ↳ Narrowing

                                ↳ QDP

                                  ↳ DependencyGraphProof

                                    ↳ QDP

                                      ↳ Narrowing

                                        ↳ QDP

                                          ↳ DependencyGraphProof

                                            ↳ QDP

                                              ↳ Narrowing

                                                ↳ QDP

                                                  ↳ DependencyGraphProof

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

MARK(fib1(X1, X2)) → MARK(X1)
MARK(sel(fib1(x0, x1), y1)) → A__SEL(a__fib1(mark(x0), mark(x1)), mark(y1))
MARK(cons(X1, X2)) → MARK(X1)
MARK(sel(add(y0, y1), 0)) → A__SEL(a__add(mark(y0), mark(y1)), 0)
MARK(sel(0, add(x0, x1))) → A__SEL(0, a__add(mark(x0), mark(x1)))
A__FIB(N) → MARK(N)
MARK(sel(0, fib(x0))) → A__SEL(0, a__fib(mark(x0)))
MARK(sel(add(y0, y1), fib1(x0, x1))) → A__SEL(a__add(mark(y0), mark(y1)), a__fib1(mark(x0), mark(x1)))
A__ADD(s(X), Y) → A__ADD(mark(X), mark(Y))
A__SEL(s(N), cons(X, XS)) → MARK(XS)
A__FIB(s(x0)) → A__SEL(s(mark(x0)), a__fib1(s(0), s(0)))
A__ADD(s(X), Y) → MARK(Y)
MARK(add(0, y1)) → A__ADD(0, mark(y1))
MARK(sel(fib(y0), cons(x0, x1))) → A__SEL(a__fib(mark(y0)), cons(mark(x0), x1))
A__FIB(sel(x0, x1)) → A__SEL(a__sel(mark(x0), mark(x1)), a__fib1(s(0), s(0)))
MARK(add(s(x0), y1)) → A__ADD(s(mark(x0)), mark(y1))
MARK(sel(add(y0, y1), sel(x0, x1))) → A__SEL(a__add(mark(y0), mark(y1)), a__sel(mark(x0), mark(x1)))
MARK(add(sel(x0, x1), y1)) → A__ADD(a__sel(mark(x0), mark(x1)), mark(y1))
MARK(sel(fib(y0), fib1(x0, x1))) → A__SEL(a__fib(mark(y0)), a__fib1(mark(x0), mark(x1)))
MARK(sel(X1, X2)) → MARK(X1)
MARK(sel(add(y0, y1), fib(x0))) → A__SEL(a__add(mark(y0), mark(y1)), a__fib(mark(x0)))
MARK(sel(fib(y0), add(x0, x1))) → A__SEL(a__fib(mark(y0)), a__add(mark(x0), mark(x1)))
MARK(add(add(x0, x1), y1)) → A__ADD(a__add(mark(x0), mark(x1)), mark(y1))
A__ADD(s(X), Y) → MARK(X)
MARK(sel(s(x0), y1)) → A__SEL(s(mark(x0)), mark(y1))
A__FIB(N) → A__FIB1(s(0), s(0))
MARK(add(fib(x0), y1)) → A__ADD(a__fib(mark(x0)), mark(y1))
A__FIB(fib(x0)) → A__SEL(a__fib(mark(x0)), a__fib1(s(0), s(0)))
MARK(sel(0, cons(x0, x1))) → A__SEL(0, cons(mark(x0), x1))
MARK(sel(sel(x0, x1), y1)) → A__SEL(a__sel(mark(x0), mark(x1)), mark(y1))
MARK(sel(add(y0, y1), cons(x0, x1))) → A__SEL(a__add(mark(y0), mark(y1)), cons(mark(x0), x1))
MARK(fib(X)) → MARK(X)
A__ADD(0, X) → MARK(X)
A__FIB1(X, Y) → MARK(X)
MARK(s(X)) → MARK(X)
MARK(add(X1, X2)) → MARK(X2)
MARK(fib(X)) → A__FIB(mark(X))
MARK(fib1(X1, X2)) → MARK(X2)
MARK(sel(fib(y0), sel(x0, x1))) → A__SEL(a__fib(mark(y0)), a__sel(mark(x0), mark(x1)))
A__FIB(0) → A__SEL(0, a__fib1(s(0), s(0)))
A__SEL(0, cons(X, XS)) → MARK(X)
MARK(add(X1, X2)) → MARK(X1)
MARK(add(fib1(x0, x1), y1)) → A__ADD(a__fib1(mark(x0), mark(x1)), mark(y1))
MARK(sel(0, fib1(x0, x1))) → A__SEL(0, a__fib1(mark(x0), mark(x1)))
A__FIB(fib1(x0, x1)) → A__SEL(a__fib1(mark(x0), mark(x1)), a__fib1(s(0), s(0)))
A__SEL(s(N), cons(X, XS)) → MARK(N)
MARK(fib1(X1, X2)) → A__FIB1(mark(X1), mark(X2))
A__SEL(s(N), cons(X, XS)) → A__SEL(mark(N), mark(XS))
MARK(sel(fib(y0), fib(x0))) → A__SEL(a__fib(mark(y0)), a__fib(mark(x0)))
MARK(sel(add(y0, y1), add(x0, x1))) → A__SEL(a__add(mark(y0), mark(y1)), a__add(mark(x0), mark(x1)))
MARK(sel(add(y0, y1), s(x0))) → A__SEL(a__add(mark(y0), mark(y1)), s(mark(x0)))
MARK(sel(0, sel(x0, x1))) → A__SEL(0, a__sel(mark(x0), mark(x1)))
A__FIB(add(x0, x1)) → A__SEL(a__add(mark(x0), mark(x1)), a__fib1(s(0), s(0)))
MARK(sel(X1, X2)) → MARK(X2)

The TRS R consists of the following rules:

a__fib(N) → a__sel(mark(N), a__fib1(s(0), s(0)))
a__fib1(X, Y) → cons(mark(X), fib1(Y, add(X, Y)))
a__add(0, X) → mark(X)
a__add(s(X), Y) → s(a__add(mark(X), mark(Y)))
a__sel(0, cons(X, XS)) → mark(X)
a__sel(s(N), cons(X, XS)) → a__sel(mark(N), mark(XS))
mark(fib(X)) → a__fib(mark(X))
mark(sel(X1, X2)) → a__sel(mark(X1), mark(X2))
mark(fib1(X1, X2)) → a__fib1(mark(X1), mark(X2))
mark(add(X1, X2)) → a__add(mark(X1), mark(X2))
mark(s(X)) → s(mark(X))
mark(0) → 0
mark(cons(X1, X2)) → cons(mark(X1), X2)
a__fib(X) → fib(X)
a__sel(X1, X2) → sel(X1, X2)
a__fib1(X1, X2) → fib1(X1, X2)
a__add(X1, X2) → add(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 1 SCC with 2 less nodes.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ Narrowing

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ QDP

              ↳ Narrowing

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ QDP

                      ↳ Narrowing

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ QDP

                              ↳ Narrowing

                                ↳ QDP

                                  ↳ DependencyGraphProof

                                    ↳ QDP

                                      ↳ Narrowing

                                        ↳ QDP

                                          ↳ DependencyGraphProof

                                            ↳ QDP

                                              ↳ Narrowing

                                                ↳ QDP

                                                  ↳ DependencyGraphProof

                                                    ↳ QDP

                                                      ↳ Narrowing

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

MARK(fib1(X1, X2)) → MARK(X1)
MARK(sel(fib1(x0, x1), y1)) → A__SEL(a__fib1(mark(x0), mark(x1)), mark(y1))
MARK(cons(X1, X2)) → MARK(X1)
MARK(sel(0, add(x0, x1))) → A__SEL(0, a__add(mark(x0), mark(x1)))
A__FIB(N) → MARK(N)
MARK(sel(0, fib(x0))) → A__SEL(0, a__fib(mark(x0)))
MARK(sel(add(y0, y1), fib1(x0, x1))) → A__SEL(a__add(mark(y0), mark(y1)), a__fib1(mark(x0), mark(x1)))
A__ADD(s(X), Y) → A__ADD(mark(X), mark(Y))
A__SEL(s(N), cons(X, XS)) → MARK(XS)
A__FIB(s(x0)) → A__SEL(s(mark(x0)), a__fib1(s(0), s(0)))
A__ADD(s(X), Y) → MARK(Y)
MARK(add(0, y1)) → A__ADD(0, mark(y1))
MARK(sel(fib(y0), cons(x0, x1))) → A__SEL(a__fib(mark(y0)), cons(mark(x0), x1))
A__FIB(sel(x0, x1)) → A__SEL(a__sel(mark(x0), mark(x1)), a__fib1(s(0), s(0)))
MARK(add(s(x0), y1)) → A__ADD(s(mark(x0)), mark(y1))
MARK(sel(add(y0, y1), sel(x0, x1))) → A__SEL(a__add(mark(y0), mark(y1)), a__sel(mark(x0), mark(x1)))
MARK(add(sel(x0, x1), y1)) → A__ADD(a__sel(mark(x0), mark(x1)), mark(y1))
MARK(sel(fib(y0), fib1(x0, x1))) → A__SEL(a__fib(mark(y0)), a__fib1(mark(x0), mark(x1)))
MARK(sel(X1, X2)) → MARK(X1)
MARK(sel(fib(y0), add(x0, x1))) → A__SEL(a__fib(mark(y0)), a__add(mark(x0), mark(x1)))
MARK(sel(add(y0, y1), fib(x0))) → A__SEL(a__add(mark(y0), mark(y1)), a__fib(mark(x0)))
MARK(add(add(x0, x1), y1)) → A__ADD(a__add(mark(x0), mark(x1)), mark(y1))
A__ADD(s(X), Y) → MARK(X)
MARK(sel(s(x0), y1)) → A__SEL(s(mark(x0)), mark(y1))
A__FIB(N) → A__FIB1(s(0), s(0))
MARK(add(fib(x0), y1)) → A__ADD(a__fib(mark(x0)), mark(y1))
A__FIB(fib(x0)) → A__SEL(a__fib(mark(x0)), a__fib1(s(0), s(0)))
MARK(sel(0, cons(x0, x1))) → A__SEL(0, cons(mark(x0), x1))
MARK(sel(sel(x0, x1), y1)) → A__SEL(a__sel(mark(x0), mark(x1)), mark(y1))
MARK(sel(add(y0, y1), cons(x0, x1))) → A__SEL(a__add(mark(y0), mark(y1)), cons(mark(x0), x1))
MARK(fib(X)) → MARK(X)
A__ADD(0, X) → MARK(X)
A__FIB1(X, Y) → MARK(X)
MARK(s(X)) → MARK(X)
MARK(add(X1, X2)) → MARK(X2)
MARK(fib(X)) → A__FIB(mark(X))
MARK(fib1(X1, X2)) → MARK(X2)
A__FIB(0) → A__SEL(0, a__fib1(s(0), s(0)))
MARK(sel(fib(y0), sel(x0, x1))) → A__SEL(a__fib(mark(y0)), a__sel(mark(x0), mark(x1)))
A__SEL(0, cons(X, XS)) → MARK(X)
MARK(add(X1, X2)) → MARK(X1)
MARK(add(fib1(x0, x1), y1)) → A__ADD(a__fib1(mark(x0), mark(x1)), mark(y1))
MARK(sel(0, fib1(x0, x1))) → A__SEL(0, a__fib1(mark(x0), mark(x1)))
A__FIB(fib1(x0, x1)) → A__SEL(a__fib1(mark(x0), mark(x1)), a__fib1(s(0), s(0)))
A__SEL(s(N), cons(X, XS)) → MARK(N)
MARK(fib1(X1, X2)) → A__FIB1(mark(X1), mark(X2))
A__SEL(s(N), cons(X, XS)) → A__SEL(mark(N), mark(XS))
MARK(sel(fib(y0), fib(x0))) → A__SEL(a__fib(mark(y0)), a__fib(mark(x0)))
MARK(sel(add(y0, y1), add(x0, x1))) → A__SEL(a__add(mark(y0), mark(y1)), a__add(mark(x0), mark(x1)))
MARK(sel(0, sel(x0, x1))) → A__SEL(0, a__sel(mark(x0), mark(x1)))
A__FIB(add(x0, x1)) → A__SEL(a__add(mark(x0), mark(x1)), a__fib1(s(0), s(0)))
MARK(sel(X1, X2)) → MARK(X2)

The TRS R consists of the following rules:

a__fib(N) → a__sel(mark(N), a__fib1(s(0), s(0)))
a__fib1(X, Y) → cons(mark(X), fib1(Y, add(X, Y)))
a__add(0, X) → mark(X)
a__add(s(X), Y) → s(a__add(mark(X), mark(Y)))
a__sel(0, cons(X, XS)) → mark(X)
a__sel(s(N), cons(X, XS)) → a__sel(mark(N), mark(XS))
mark(fib(X)) → a__fib(mark(X))
mark(sel(X1, X2)) → a__sel(mark(X1), mark(X2))
mark(fib1(X1, X2)) → a__fib1(mark(X1), mark(X2))
mark(add(X1, X2)) → a__add(mark(X1), mark(X2))
mark(s(X)) → s(mark(X))
mark(0) → 0
mark(cons(X1, X2)) → cons(mark(X1), X2)
a__fib(X) → fib(X)
a__sel(X1, X2) → sel(X1, X2)
a__fib1(X1, X2) → fib1(X1, X2)
a__add(X1, X2) → add(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule MARK(sel(fib1(x0, x1), y1)) → A__SEL(a__fib1(mark(x0), mark(x1)), mark(y1)) at position [1] we obtained the following new rules:

MARK(sel(fib1(y0, y1), cons(x0, x1))) → A__SEL(a__fib1(mark(y0), mark(y1)), cons(mark(x0), x1))
MARK(sel(fib1(y0, y1), fib1(x0, x1))) → A__SEL(a__fib1(mark(y0), mark(y1)), a__fib1(mark(x0), mark(x1)))
MARK(sel(fib1(y0, y1), s(x0))) → A__SEL(a__fib1(mark(y0), mark(y1)), s(mark(x0)))
MARK(sel(fib1(y0, y1), sel(x0, x1))) → A__SEL(a__fib1(mark(y0), mark(y1)), a__sel(mark(x0), mark(x1)))
MARK(sel(fib1(y0, y1), add(x0, x1))) → A__SEL(a__fib1(mark(y0), mark(y1)), a__add(mark(x0), mark(x1)))
MARK(sel(fib1(y0, y1), 0)) → A__SEL(a__fib1(mark(y0), mark(y1)), 0)
MARK(sel(fib1(y0, y1), fib(x0))) → A__SEL(a__fib1(mark(y0), mark(y1)), a__fib(mark(x0)))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ Narrowing

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ QDP

              ↳ Narrowing

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ QDP

                      ↳ Narrowing

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ QDP

                              ↳ Narrowing

                                ↳ QDP

                                  ↳ DependencyGraphProof

                                    ↳ QDP

                                      ↳ Narrowing

                                        ↳ QDP

                                          ↳ DependencyGraphProof

                                            ↳ QDP

                                              ↳ Narrowing

                                                ↳ QDP

                                                  ↳ DependencyGraphProof

                                                    ↳ QDP

                                                      ↳ Narrowing

                                                        ↳ QDP

                                                          ↳ DependencyGraphProof

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

MARK(sel(fib1(y0, y1), fib1(x0, x1))) → A__SEL(a__fib1(mark(y0), mark(y1)), a__fib1(mark(x0), mark(x1)))
MARK(fib1(X1, X2)) → MARK(X1)
MARK(cons(X1, X2)) → MARK(X1)
MARK(sel(0, add(x0, x1))) → A__SEL(0, a__add(mark(x0), mark(x1)))
MARK(sel(0, fib(x0))) → A__SEL(0, a__fib(mark(x0)))
A__FIB(N) → MARK(N)
MARK(sel(fib1(y0, y1), add(x0, x1))) → A__SEL(a__fib1(mark(y0), mark(y1)), a__add(mark(x0), mark(x1)))
MARK(sel(add(y0, y1), fib1(x0, x1))) → A__SEL(a__add(mark(y0), mark(y1)), a__fib1(mark(x0), mark(x1)))
A__FIB(s(x0)) → A__SEL(s(mark(x0)), a__fib1(s(0), s(0)))
A__SEL(s(N), cons(X, XS)) → MARK(XS)
A__ADD(s(X), Y) → A__ADD(mark(X), mark(Y))
A__ADD(s(X), Y) → MARK(Y)
MARK(add(0, y1)) → A__ADD(0, mark(y1))
MARK(sel(fib(y0), cons(x0, x1))) → A__SEL(a__fib(mark(y0)), cons(mark(x0), x1))
A__FIB(sel(x0, x1)) → A__SEL(a__sel(mark(x0), mark(x1)), a__fib1(s(0), s(0)))
MARK(add(s(x0), y1)) → A__ADD(s(mark(x0)), mark(y1))
MARK(sel(add(y0, y1), sel(x0, x1))) → A__SEL(a__add(mark(y0), mark(y1)), a__sel(mark(x0), mark(x1)))
MARK(sel(fib1(y0, y1), fib(x0))) → A__SEL(a__fib1(mark(y0), mark(y1)), a__fib(mark(x0)))
MARK(sel(fib(y0), fib1(x0, x1))) → A__SEL(a__fib(mark(y0)), a__fib1(mark(x0), mark(x1)))
MARK(add(sel(x0, x1), y1)) → A__ADD(a__sel(mark(x0), mark(x1)), mark(y1))
MARK(sel(fib1(y0, y1), cons(x0, x1))) → A__SEL(a__fib1(mark(y0), mark(y1)), cons(mark(x0), x1))
MARK(sel(X1, X2)) → MARK(X1)
MARK(sel(add(y0, y1), fib(x0))) → A__SEL(a__add(mark(y0), mark(y1)), a__fib(mark(x0)))
MARK(sel(fib(y0), add(x0, x1))) → A__SEL(a__fib(mark(y0)), a__add(mark(x0), mark(x1)))
MARK(add(add(x0, x1), y1)) → A__ADD(a__add(mark(x0), mark(x1)), mark(y1))
A__ADD(s(X), Y) → MARK(X)
MARK(sel(s(x0), y1)) → A__SEL(s(mark(x0)), mark(y1))
A__FIB(N) → A__FIB1(s(0), s(0))
MARK(add(fib(x0), y1)) → A__ADD(a__fib(mark(x0)), mark(y1))
A__FIB(fib(x0)) → A__SEL(a__fib(mark(x0)), a__fib1(s(0), s(0)))
MARK(sel(0, cons(x0, x1))) → A__SEL(0, cons(mark(x0), x1))
MARK(sel(sel(x0, x1), y1)) → A__SEL(a__sel(mark(x0), mark(x1)), mark(y1))
MARK(sel(add(y0, y1), cons(x0, x1))) → A__SEL(a__add(mark(y0), mark(y1)), cons(mark(x0), x1))
MARK(fib(X)) → MARK(X)
A__FIB1(X, Y) → MARK(X)
A__ADD(0, X) → MARK(X)
MARK(s(X)) → MARK(X)
MARK(add(X1, X2)) → MARK(X2)
MARK(fib(X)) → A__FIB(mark(X))
MARK(fib1(X1, X2)) → MARK(X2)
MARK(sel(fib(y0), sel(x0, x1))) → A__SEL(a__fib(mark(y0)), a__sel(mark(x0), mark(x1)))
A__FIB(0) → A__SEL(0, a__fib1(s(0), s(0)))
A__SEL(0, cons(X, XS)) → MARK(X)
MARK(add(X1, X2)) → MARK(X1)
MARK(add(fib1(x0, x1), y1)) → A__ADD(a__fib1(mark(x0), mark(x1)), mark(y1))
MARK(sel(fib1(y0, y1), 0)) → A__SEL(a__fib1(mark(y0), mark(y1)), 0)
MARK(sel(0, fib1(x0, x1))) → A__SEL(0, a__fib1(mark(x0), mark(x1)))
A__FIB(fib1(x0, x1)) → A__SEL(a__fib1(mark(x0), mark(x1)), a__fib1(s(0), s(0)))
A__SEL(s(N), cons(X, XS)) → MARK(N)
MARK(fib1(X1, X2)) → A__FIB1(mark(X1), mark(X2))
MARK(sel(fib1(y0, y1), s(x0))) → A__SEL(a__fib1(mark(y0), mark(y1)), s(mark(x0)))
A__SEL(s(N), cons(X, XS)) → A__SEL(mark(N), mark(XS))
MARK(sel(fib(y0), fib(x0))) → A__SEL(a__fib(mark(y0)), a__fib(mark(x0)))
MARK(sel(fib1(y0, y1), sel(x0, x1))) → A__SEL(a__fib1(mark(y0), mark(y1)), a__sel(mark(x0), mark(x1)))
MARK(sel(add(y0, y1), add(x0, x1))) → A__SEL(a__add(mark(y0), mark(y1)), a__add(mark(x0), mark(x1)))
MARK(sel(0, sel(x0, x1))) → A__SEL(0, a__sel(mark(x0), mark(x1)))
A__FIB(add(x0, x1)) → A__SEL(a__add(mark(x0), mark(x1)), a__fib1(s(0), s(0)))
MARK(sel(X1, X2)) → MARK(X2)

The TRS R consists of the following rules:

a__fib(N) → a__sel(mark(N), a__fib1(s(0), s(0)))
a__fib1(X, Y) → cons(mark(X), fib1(Y, add(X, Y)))
a__add(0, X) → mark(X)
a__add(s(X), Y) → s(a__add(mark(X), mark(Y)))
a__sel(0, cons(X, XS)) → mark(X)
a__sel(s(N), cons(X, XS)) → a__sel(mark(N), mark(XS))
mark(fib(X)) → a__fib(mark(X))
mark(sel(X1, X2)) → a__sel(mark(X1), mark(X2))
mark(fib1(X1, X2)) → a__fib1(mark(X1), mark(X2))
mark(add(X1, X2)) → a__add(mark(X1), mark(X2))
mark(s(X)) → s(mark(X))
mark(0) → 0
mark(cons(X1, X2)) → cons(mark(X1), X2)
a__fib(X) → fib(X)
a__sel(X1, X2) → sel(X1, X2)
a__fib1(X1, X2) → fib1(X1, X2)
a__add(X1, X2) → add(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 1 SCC with 2 less nodes.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ Narrowing

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ QDP

              ↳ Narrowing

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ QDP

                      ↳ Narrowing

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ QDP

                              ↳ Narrowing

                                ↳ QDP

                                  ↳ DependencyGraphProof

                                    ↳ QDP

                                      ↳ Narrowing

                                        ↳ QDP

                                          ↳ DependencyGraphProof

                                            ↳ QDP

                                              ↳ Narrowing

                                                ↳ QDP

                                                  ↳ DependencyGraphProof

                                                    ↳ QDP

                                                      ↳ Narrowing

                                                        ↳ QDP

                                                          ↳ DependencyGraphProof

                                                            ↳ QDP

                                                              ↳ Narrowing

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

MARK(sel(fib1(y0, y1), fib1(x0, x1))) → A__SEL(a__fib1(mark(y0), mark(y1)), a__fib1(mark(x0), mark(x1)))
MARK(fib1(X1, X2)) → MARK(X1)
MARK(cons(X1, X2)) → MARK(X1)
MARK(sel(0, add(x0, x1))) → A__SEL(0, a__add(mark(x0), mark(x1)))
A__FIB(N) → MARK(N)
MARK(sel(0, fib(x0))) → A__SEL(0, a__fib(mark(x0)))
MARK(sel(add(y0, y1), fib1(x0, x1))) → A__SEL(a__add(mark(y0), mark(y1)), a__fib1(mark(x0), mark(x1)))
MARK(sel(fib1(y0, y1), add(x0, x1))) → A__SEL(a__fib1(mark(y0), mark(y1)), a__add(mark(x0), mark(x1)))
A__FIB(s(x0)) → A__SEL(s(mark(x0)), a__fib1(s(0), s(0)))
A__ADD(s(X), Y) → A__ADD(mark(X), mark(Y))
A__SEL(s(N), cons(X, XS)) → MARK(XS)
A__ADD(s(X), Y) → MARK(Y)
MARK(add(0, y1)) → A__ADD(0, mark(y1))
MARK(sel(fib(y0), cons(x0, x1))) → A__SEL(a__fib(mark(y0)), cons(mark(x0), x1))
A__FIB(sel(x0, x1)) → A__SEL(a__sel(mark(x0), mark(x1)), a__fib1(s(0), s(0)))
MARK(add(s(x0), y1)) → A__ADD(s(mark(x0)), mark(y1))
MARK(sel(add(y0, y1), sel(x0, x1))) → A__SEL(a__add(mark(y0), mark(y1)), a__sel(mark(x0), mark(x1)))
MARK(add(sel(x0, x1), y1)) → A__ADD(a__sel(mark(x0), mark(x1)), mark(y1))
MARK(sel(fib(y0), fib1(x0, x1))) → A__SEL(a__fib(mark(y0)), a__fib1(mark(x0), mark(x1)))
MARK(sel(fib1(y0, y1), fib(x0))) → A__SEL(a__fib1(mark(y0), mark(y1)), a__fib(mark(x0)))
MARK(sel(fib1(y0, y1), cons(x0, x1))) → A__SEL(a__fib1(mark(y0), mark(y1)), cons(mark(x0), x1))
MARK(sel(X1, X2)) → MARK(X1)
MARK(sel(fib(y0), add(x0, x1))) → A__SEL(a__fib(mark(y0)), a__add(mark(x0), mark(x1)))
MARK(sel(add(y0, y1), fib(x0))) → A__SEL(a__add(mark(y0), mark(y1)), a__fib(mark(x0)))
MARK(add(add(x0, x1), y1)) → A__ADD(a__add(mark(x0), mark(x1)), mark(y1))
A__ADD(s(X), Y) → MARK(X)
MARK(sel(s(x0), y1)) → A__SEL(s(mark(x0)), mark(y1))
A__FIB(N) → A__FIB1(s(0), s(0))
MARK(add(fib(x0), y1)) → A__ADD(a__fib(mark(x0)), mark(y1))
A__FIB(fib(x0)) → A__SEL(a__fib(mark(x0)), a__fib1(s(0), s(0)))
MARK(sel(0, cons(x0, x1))) → A__SEL(0, cons(mark(x0), x1))
MARK(sel(sel(x0, x1), y1)) → A__SEL(a__sel(mark(x0), mark(x1)), mark(y1))
MARK(sel(add(y0, y1), cons(x0, x1))) → A__SEL(a__add(mark(y0), mark(y1)), cons(mark(x0), x1))
MARK(fib(X)) → MARK(X)
A__ADD(0, X) → MARK(X)
A__FIB1(X, Y) → MARK(X)
MARK(s(X)) → MARK(X)
MARK(add(X1, X2)) → MARK(X2)
MARK(fib(X)) → A__FIB(mark(X))
MARK(fib1(X1, X2)) → MARK(X2)
A__FIB(0) → A__SEL(0, a__fib1(s(0), s(0)))
MARK(sel(fib(y0), sel(x0, x1))) → A__SEL(a__fib(mark(y0)), a__sel(mark(x0), mark(x1)))
A__SEL(0, cons(X, XS)) → MARK(X)
MARK(add(X1, X2)) → MARK(X1)
MARK(add(fib1(x0, x1), y1)) → A__ADD(a__fib1(mark(x0), mark(x1)), mark(y1))
MARK(sel(0, fib1(x0, x1))) → A__SEL(0, a__fib1(mark(x0), mark(x1)))
A__FIB(fib1(x0, x1)) → A__SEL(a__fib1(mark(x0), mark(x1)), a__fib1(s(0), s(0)))
A__SEL(s(N), cons(X, XS)) → MARK(N)
MARK(fib1(X1, X2)) → A__FIB1(mark(X1), mark(X2))
A__SEL(s(N), cons(X, XS)) → A__SEL(mark(N), mark(XS))
MARK(sel(fib(y0), fib(x0))) → A__SEL(a__fib(mark(y0)), a__fib(mark(x0)))
MARK(sel(add(y0, y1), add(x0, x1))) → A__SEL(a__add(mark(y0), mark(y1)), a__add(mark(x0), mark(x1)))
MARK(sel(fib1(y0, y1), sel(x0, x1))) → A__SEL(a__fib1(mark(y0), mark(y1)), a__sel(mark(x0), mark(x1)))
MARK(sel(0, sel(x0, x1))) → A__SEL(0, a__sel(mark(x0), mark(x1)))
A__FIB(add(x0, x1)) → A__SEL(a__add(mark(x0), mark(x1)), a__fib1(s(0), s(0)))
MARK(sel(X1, X2)) → MARK(X2)

The TRS R consists of the following rules:

a__fib(N) → a__sel(mark(N), a__fib1(s(0), s(0)))
a__fib1(X, Y) → cons(mark(X), fib1(Y, add(X, Y)))
a__add(0, X) → mark(X)
a__add(s(X), Y) → s(a__add(mark(X), mark(Y)))
a__sel(0, cons(X, XS)) → mark(X)
a__sel(s(N), cons(X, XS)) → a__sel(mark(N), mark(XS))
mark(fib(X)) → a__fib(mark(X))
mark(sel(X1, X2)) → a__sel(mark(X1), mark(X2))
mark(fib1(X1, X2)) → a__fib1(mark(X1), mark(X2))
mark(add(X1, X2)) → a__add(mark(X1), mark(X2))
mark(s(X)) → s(mark(X))
mark(0) → 0
mark(cons(X1, X2)) → cons(mark(X1), X2)
a__fib(X) → fib(X)
a__sel(X1, X2) → sel(X1, X2)
a__fib1(X1, X2) → fib1(X1, X2)
a__add(X1, X2) → add(X1, X2)

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

MARK(sel(sel(y0, y1), sel(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), a__sel(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), cons(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), cons(mark(x0), x1))
MARK(sel(sel(y0, y1), s(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), s(mark(x0)))
MARK(sel(sel(y0, y1), fib1(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), a__fib1(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), fib(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__fib(mark(x0)))
MARK(sel(sel(y0, y1), 0)) → A__SEL(a__sel(mark(y0), mark(y1)), 0)
MARK(sel(sel(y0, y1), add(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), a__add(mark(x0), mark(x1)))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ Narrowing

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ QDP

              ↳ Narrowing

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ QDP

                      ↳ Narrowing

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ QDP

                              ↳ Narrowing

                                ↳ QDP

                                  ↳ DependencyGraphProof

                                    ↳ QDP

                                      ↳ Narrowing

                                        ↳ QDP

                                          ↳ DependencyGraphProof

                                            ↳ QDP

                                              ↳ Narrowing

                                                ↳ QDP

                                                  ↳ DependencyGraphProof

                                                    ↳ QDP

                                                      ↳ Narrowing

                                                        ↳ QDP

                                                          ↳ DependencyGraphProof

                                                            ↳ QDP

                                                              ↳ Narrowing

                                                                ↳ QDP

                                                                  ↳ DependencyGraphProof

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

MARK(sel(sel(y0, y1), sel(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), a__sel(mark(x0), mark(x1)))
MARK(sel(fib1(y0, y1), fib1(x0, x1))) → A__SEL(a__fib1(mark(y0), mark(y1)), a__fib1(mark(x0), mark(x1)))
MARK(fib1(X1, X2)) → MARK(X1)
MARK(cons(X1, X2)) → MARK(X1)
MARK(sel(0, add(x0, x1))) → A__SEL(0, a__add(mark(x0), mark(x1)))
MARK(sel(0, fib(x0))) → A__SEL(0, a__fib(mark(x0)))
A__FIB(N) → MARK(N)
MARK(sel(fib1(y0, y1), add(x0, x1))) → A__SEL(a__fib1(mark(y0), mark(y1)), a__add(mark(x0), mark(x1)))
MARK(sel(add(y0, y1), fib1(x0, x1))) → A__SEL(a__add(mark(y0), mark(y1)), a__fib1(mark(x0), mark(x1)))
A__SEL(s(N), cons(X, XS)) → MARK(XS)
A__ADD(s(X), Y) → A__ADD(mark(X), mark(Y))
A__FIB(s(x0)) → A__SEL(s(mark(x0)), a__fib1(s(0), s(0)))
A__ADD(s(X), Y) → MARK(Y)
MARK(add(0, y1)) → A__ADD(0, mark(y1))
MARK(sel(fib(y0), cons(x0, x1))) → A__SEL(a__fib(mark(y0)), cons(mark(x0), x1))
A__FIB(sel(x0, x1)) → A__SEL(a__sel(mark(x0), mark(x1)), a__fib1(s(0), s(0)))
MARK(add(s(x0), y1)) → A__ADD(s(mark(x0)), mark(y1))
MARK(sel(sel(y0, y1), add(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), a__add(mark(x0), mark(x1)))
MARK(sel(add(y0, y1), sel(x0, x1))) → A__SEL(a__add(mark(y0), mark(y1)), a__sel(mark(x0), mark(x1)))
MARK(sel(fib1(y0, y1), fib(x0))) → A__SEL(a__fib1(mark(y0), mark(y1)), a__fib(mark(x0)))
MARK(sel(fib(y0), fib1(x0, x1))) → A__SEL(a__fib(mark(y0)), a__fib1(mark(x0), mark(x1)))
MARK(add(sel(x0, x1), y1)) → A__ADD(a__sel(mark(x0), mark(x1)), mark(y1))
MARK(sel(fib1(y0, y1), cons(x0, x1))) → A__SEL(a__fib1(mark(y0), mark(y1)), cons(mark(x0), x1))
MARK(sel(X1, X2)) → MARK(X1)
MARK(sel(sel(y0, y1), cons(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), cons(mark(x0), x1))
MARK(sel(add(y0, y1), fib(x0))) → A__SEL(a__add(mark(y0), mark(y1)), a__fib(mark(x0)))
MARK(sel(fib(y0), add(x0, x1))) → A__SEL(a__fib(mark(y0)), a__add(mark(x0), mark(x1)))
MARK(add(add(x0, x1), y1)) → A__ADD(a__add(mark(x0), mark(x1)), mark(y1))
A__ADD(s(X), Y) → MARK(X)
MARK(sel(s(x0), y1)) → A__SEL(s(mark(x0)), mark(y1))
A__FIB(N) → A__FIB1(s(0), s(0))
MARK(add(fib(x0), y1)) → A__ADD(a__fib(mark(x0)), mark(y1))
A__FIB(fib(x0)) → A__SEL(a__fib(mark(x0)), a__fib1(s(0), s(0)))
MARK(sel(0, cons(x0, x1))) → A__SEL(0, cons(mark(x0), x1))
MARK(sel(add(y0, y1), cons(x0, x1))) → A__SEL(a__add(mark(y0), mark(y1)), cons(mark(x0), x1))
MARK(fib(X)) → MARK(X)
A__FIB1(X, Y) → MARK(X)
A__ADD(0, X) → MARK(X)
MARK(s(X)) → MARK(X)
MARK(add(X1, X2)) → MARK(X2)
MARK(sel(sel(y0, y1), s(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), s(mark(x0)))
MARK(fib(X)) → A__FIB(mark(X))
MARK(fib1(X1, X2)) → MARK(X2)
MARK(sel(sel(y0, y1), fib(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__fib(mark(x0)))
MARK(sel(fib(y0), sel(x0, x1))) → A__SEL(a__fib(mark(y0)), a__sel(mark(x0), mark(x1)))
A__FIB(0) → A__SEL(0, a__fib1(s(0), s(0)))
A__SEL(0, cons(X, XS)) → MARK(X)
MARK(add(X1, X2)) → MARK(X1)
MARK(add(fib1(x0, x1), y1)) → A__ADD(a__fib1(mark(x0), mark(x1)), mark(y1))
MARK(sel(0, fib1(x0, x1))) → A__SEL(0, a__fib1(mark(x0), mark(x1)))
A__FIB(fib1(x0, x1)) → A__SEL(a__fib1(mark(x0), mark(x1)), a__fib1(s(0), s(0)))
A__SEL(s(N), cons(X, XS)) → MARK(N)
MARK(fib1(X1, X2)) → A__FIB1(mark(X1), mark(X2))
A__SEL(s(N), cons(X, XS)) → A__SEL(mark(N), mark(XS))
MARK(sel(fib(y0), fib(x0))) → A__SEL(a__fib(mark(y0)), a__fib(mark(x0)))
MARK(sel(sel(y0, y1), fib1(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), a__fib1(mark(x0), mark(x1)))
MARK(sel(fib1(y0, y1), sel(x0, x1))) → A__SEL(a__fib1(mark(y0), mark(y1)), a__sel(mark(x0), mark(x1)))
MARK(sel(add(y0, y1), add(x0, x1))) → A__SEL(a__add(mark(y0), mark(y1)), a__add(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), 0)) → A__SEL(a__sel(mark(y0), mark(y1)), 0)
MARK(sel(0, sel(x0, x1))) → A__SEL(0, a__sel(mark(x0), mark(x1)))
A__FIB(add(x0, x1)) → A__SEL(a__add(mark(x0), mark(x1)), a__fib1(s(0), s(0)))
MARK(sel(X1, X2)) → MARK(X2)

The TRS R consists of the following rules:

a__fib(N) → a__sel(mark(N), a__fib1(s(0), s(0)))
a__fib1(X, Y) → cons(mark(X), fib1(Y, add(X, Y)))
a__add(0, X) → mark(X)
a__add(s(X), Y) → s(a__add(mark(X), mark(Y)))
a__sel(0, cons(X, XS)) → mark(X)
a__sel(s(N), cons(X, XS)) → a__sel(mark(N), mark(XS))
mark(fib(X)) → a__fib(mark(X))
mark(sel(X1, X2)) → a__sel(mark(X1), mark(X2))
mark(fib1(X1, X2)) → a__fib1(mark(X1), mark(X2))
mark(add(X1, X2)) → a__add(mark(X1), mark(X2))
mark(s(X)) → s(mark(X))
mark(0) → 0
mark(cons(X1, X2)) → cons(mark(X1), X2)
a__fib(X) → fib(X)
a__sel(X1, X2) → sel(X1, X2)
a__fib1(X1, X2) → fib1(X1, X2)
a__add(X1, X2) → add(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 1 SCC with 2 less nodes.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ Narrowing

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ QDP

              ↳ Narrowing

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ QDP

                      ↳ Narrowing

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ QDP

                              ↳ Narrowing

                                ↳ QDP

                                  ↳ DependencyGraphProof

                                    ↳ QDP

                                      ↳ Narrowing

                                        ↳ QDP

                                          ↳ DependencyGraphProof

                                            ↳ QDP

                                              ↳ Narrowing

                                                ↳ QDP

                                                  ↳ DependencyGraphProof

                                                    ↳ QDP

                                                      ↳ Narrowing

                                                        ↳ QDP

                                                          ↳ DependencyGraphProof

                                                            ↳ QDP

                                                              ↳ Narrowing

                                                                ↳ QDP

                                                                  ↳ DependencyGraphProof

                                                                    ↳ QDP

                                                                      ↳ Narrowing

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

MARK(sel(sel(y0, y1), sel(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), a__sel(mark(x0), mark(x1)))
MARK(sel(fib1(y0, y1), fib1(x0, x1))) → A__SEL(a__fib1(mark(y0), mark(y1)), a__fib1(mark(x0), mark(x1)))
MARK(fib1(X1, X2)) → MARK(X1)
MARK(cons(X1, X2)) → MARK(X1)
MARK(sel(0, add(x0, x1))) → A__SEL(0, a__add(mark(x0), mark(x1)))
A__FIB(N) → MARK(N)
MARK(sel(0, fib(x0))) → A__SEL(0, a__fib(mark(x0)))
MARK(sel(add(y0, y1), fib1(x0, x1))) → A__SEL(a__add(mark(y0), mark(y1)), a__fib1(mark(x0), mark(x1)))
MARK(sel(fib1(y0, y1), add(x0, x1))) → A__SEL(a__fib1(mark(y0), mark(y1)), a__add(mark(x0), mark(x1)))
A__FIB(s(x0)) → A__SEL(s(mark(x0)), a__fib1(s(0), s(0)))
A__ADD(s(X), Y) → A__ADD(mark(X), mark(Y))
A__SEL(s(N), cons(X, XS)) → MARK(XS)
A__ADD(s(X), Y) → MARK(Y)
MARK(add(0, y1)) → A__ADD(0, mark(y1))
MARK(sel(fib(y0), cons(x0, x1))) → A__SEL(a__fib(mark(y0)), cons(mark(x0), x1))
A__FIB(sel(x0, x1)) → A__SEL(a__sel(mark(x0), mark(x1)), a__fib1(s(0), s(0)))
MARK(add(s(x0), y1)) → A__ADD(s(mark(x0)), mark(y1))
MARK(sel(sel(y0, y1), add(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), a__add(mark(x0), mark(x1)))
MARK(sel(add(y0, y1), sel(x0, x1))) → A__SEL(a__add(mark(y0), mark(y1)), a__sel(mark(x0), mark(x1)))
MARK(add(sel(x0, x1), y1)) → A__ADD(a__sel(mark(x0), mark(x1)), mark(y1))
MARK(sel(fib(y0), fib1(x0, x1))) → A__SEL(a__fib(mark(y0)), a__fib1(mark(x0), mark(x1)))
MARK(sel(fib1(y0, y1), fib(x0))) → A__SEL(a__fib1(mark(y0), mark(y1)), a__fib(mark(x0)))
MARK(sel(fib1(y0, y1), cons(x0, x1))) → A__SEL(a__fib1(mark(y0), mark(y1)), cons(mark(x0), x1))
MARK(sel(X1, X2)) → MARK(X1)
MARK(sel(sel(y0, y1), cons(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), cons(mark(x0), x1))
MARK(sel(fib(y0), add(x0, x1))) → A__SEL(a__fib(mark(y0)), a__add(mark(x0), mark(x1)))
MARK(sel(add(y0, y1), fib(x0))) → A__SEL(a__add(mark(y0), mark(y1)), a__fib(mark(x0)))
MARK(add(add(x0, x1), y1)) → A__ADD(a__add(mark(x0), mark(x1)), mark(y1))
A__ADD(s(X), Y) → MARK(X)
MARK(sel(s(x0), y1)) → A__SEL(s(mark(x0)), mark(y1))
A__FIB(N) → A__FIB1(s(0), s(0))
MARK(add(fib(x0), y1)) → A__ADD(a__fib(mark(x0)), mark(y1))
A__FIB(fib(x0)) → A__SEL(a__fib(mark(x0)), a__fib1(s(0), s(0)))
MARK(sel(0, cons(x0, x1))) → A__SEL(0, cons(mark(x0), x1))
MARK(sel(add(y0, y1), cons(x0, x1))) → A__SEL(a__add(mark(y0), mark(y1)), cons(mark(x0), x1))
MARK(fib(X)) → MARK(X)
A__ADD(0, X) → MARK(X)
A__FIB1(X, Y) → MARK(X)
MARK(s(X)) → MARK(X)
MARK(add(X1, X2)) → MARK(X2)
MARK(fib(X)) → A__FIB(mark(X))
MARK(fib1(X1, X2)) → MARK(X2)
A__FIB(0) → A__SEL(0, a__fib1(s(0), s(0)))
MARK(sel(sel(y0, y1), fib(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__fib(mark(x0)))
MARK(sel(fib(y0), sel(x0, x1))) → A__SEL(a__fib(mark(y0)), a__sel(mark(x0), mark(x1)))
A__SEL(0, cons(X, XS)) → MARK(X)
MARK(add(X1, X2)) → MARK(X1)
MARK(add(fib1(x0, x1), y1)) → A__ADD(a__fib1(mark(x0), mark(x1)), mark(y1))
MARK(sel(0, fib1(x0, x1))) → A__SEL(0, a__fib1(mark(x0), mark(x1)))
A__FIB(fib1(x0, x1)) → A__SEL(a__fib1(mark(x0), mark(x1)), a__fib1(s(0), s(0)))
A__SEL(s(N), cons(X, XS)) → MARK(N)
MARK(fib1(X1, X2)) → A__FIB1(mark(X1), mark(X2))
A__SEL(s(N), cons(X, XS)) → A__SEL(mark(N), mark(XS))
MARK(sel(fib(y0), fib(x0))) → A__SEL(a__fib(mark(y0)), a__fib(mark(x0)))
MARK(sel(sel(y0, y1), fib1(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), a__fib1(mark(x0), mark(x1)))
MARK(sel(add(y0, y1), add(x0, x1))) → A__SEL(a__add(mark(y0), mark(y1)), a__add(mark(x0), mark(x1)))
MARK(sel(fib1(y0, y1), sel(x0, x1))) → A__SEL(a__fib1(mark(y0), mark(y1)), a__sel(mark(x0), mark(x1)))
MARK(sel(0, sel(x0, x1))) → A__SEL(0, a__sel(mark(x0), mark(x1)))
A__FIB(add(x0, x1)) → A__SEL(a__add(mark(x0), mark(x1)), a__fib1(s(0), s(0)))
MARK(sel(X1, X2)) → MARK(X2)

The TRS R consists of the following rules:

a__fib(N) → a__sel(mark(N), a__fib1(s(0), s(0)))
a__fib1(X, Y) → cons(mark(X), fib1(Y, add(X, Y)))
a__add(0, X) → mark(X)
a__add(s(X), Y) → s(a__add(mark(X), mark(Y)))
a__sel(0, cons(X, XS)) → mark(X)
a__sel(s(N), cons(X, XS)) → a__sel(mark(N), mark(XS))
mark(fib(X)) → a__fib(mark(X))
mark(sel(X1, X2)) → a__sel(mark(X1), mark(X2))
mark(fib1(X1, X2)) → a__fib1(mark(X1), mark(X2))
mark(add(X1, X2)) → a__add(mark(X1), mark(X2))
mark(s(X)) → s(mark(X))
mark(0) → 0
mark(cons(X1, X2)) → cons(mark(X1), X2)
a__fib(X) → fib(X)
a__sel(X1, X2) → sel(X1, X2)
a__fib1(X1, X2) → fib1(X1, X2)
a__add(X1, X2) → add(X1, X2)

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

MARK(sel(s(y0), add(x0, x1))) → A__SEL(s(mark(y0)), a__add(mark(x0), mark(x1)))
MARK(sel(s(y0), fib1(x0, x1))) → A__SEL(s(mark(y0)), a__fib1(mark(x0), mark(x1)))
MARK(sel(s(y0), cons(x0, x1))) → A__SEL(s(mark(y0)), cons(mark(x0), x1))
MARK(sel(s(y0), sel(x0, x1))) → A__SEL(s(mark(y0)), a__sel(mark(x0), mark(x1)))
MARK(sel(s(y0), 0)) → A__SEL(s(mark(y0)), 0)
MARK(sel(s(y0), s(x0))) → A__SEL(s(mark(y0)), s(mark(x0)))
MARK(sel(s(y0), fib(x0))) → A__SEL(s(mark(y0)), a__fib(mark(x0)))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ Narrowing

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ QDP

              ↳ Narrowing

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ QDP

                      ↳ Narrowing

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ QDP

                              ↳ Narrowing

                                ↳ QDP

                                  ↳ DependencyGraphProof

                                    ↳ QDP

                                      ↳ Narrowing

                                        ↳ QDP

                                          ↳ DependencyGraphProof

                                            ↳ QDP

                                              ↳ Narrowing

                                                ↳ QDP

                                                  ↳ DependencyGraphProof

                                                    ↳ QDP

                                                      ↳ Narrowing

                                                        ↳ QDP

                                                          ↳ DependencyGraphProof

                                                            ↳ QDP

                                                              ↳ Narrowing

                                                                ↳ QDP

                                                                  ↳ DependencyGraphProof

                                                                    ↳ QDP

                                                                      ↳ Narrowing

                                                                        ↳ QDP

                                                                          ↳ DependencyGraphProof

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

MARK(sel(sel(y0, y1), sel(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), a__sel(mark(x0), mark(x1)))
MARK(sel(fib1(y0, y1), fib1(x0, x1))) → A__SEL(a__fib1(mark(y0), mark(y1)), a__fib1(mark(x0), mark(x1)))
MARK(fib1(X1, X2)) → MARK(X1)
MARK(sel(s(y0), cons(x0, x1))) → A__SEL(s(mark(y0)), cons(mark(x0), x1))
MARK(cons(X1, X2)) → MARK(X1)
MARK(sel(0, add(x0, x1))) → A__SEL(0, a__add(mark(x0), mark(x1)))
MARK(sel(0, fib(x0))) → A__SEL(0, a__fib(mark(x0)))
A__FIB(N) → MARK(N)
MARK(sel(fib1(y0, y1), add(x0, x1))) → A__SEL(a__fib1(mark(y0), mark(y1)), a__add(mark(x0), mark(x1)))
MARK(sel(add(y0, y1), fib1(x0, x1))) → A__SEL(a__add(mark(y0), mark(y1)), a__fib1(mark(x0), mark(x1)))
A__SEL(s(N), cons(X, XS)) → MARK(XS)
A__ADD(s(X), Y) → A__ADD(mark(X), mark(Y))
A__FIB(s(x0)) → A__SEL(s(mark(x0)), a__fib1(s(0), s(0)))
A__ADD(s(X), Y) → MARK(Y)
MARK(add(0, y1)) → A__ADD(0, mark(y1))
MARK(sel(fib(y0), cons(x0, x1))) → A__SEL(a__fib(mark(y0)), cons(mark(x0), x1))
A__FIB(sel(x0, x1)) → A__SEL(a__sel(mark(x0), mark(x1)), a__fib1(s(0), s(0)))
MARK(add(s(x0), y1)) → A__ADD(s(mark(x0)), mark(y1))
MARK(sel(add(y0, y1), sel(x0, x1))) → A__SEL(a__add(mark(y0), mark(y1)), a__sel(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), add(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), a__add(mark(x0), mark(x1)))
MARK(sel(s(y0), fib(x0))) → A__SEL(s(mark(y0)), a__fib(mark(x0)))
MARK(sel(fib1(y0, y1), fib(x0))) → A__SEL(a__fib1(mark(y0), mark(y1)), a__fib(mark(x0)))
MARK(sel(fib(y0), fib1(x0, x1))) → A__SEL(a__fib(mark(y0)), a__fib1(mark(x0), mark(x1)))
MARK(add(sel(x0, x1), y1)) → A__ADD(a__sel(mark(x0), mark(x1)), mark(y1))
MARK(sel(fib1(y0, y1), cons(x0, x1))) → A__SEL(a__fib1(mark(y0), mark(y1)), cons(mark(x0), x1))
MARK(sel(X1, X2)) → MARK(X1)
MARK(sel(sel(y0, y1), cons(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), cons(mark(x0), x1))
MARK(sel(add(y0, y1), fib(x0))) → A__SEL(a__add(mark(y0), mark(y1)), a__fib(mark(x0)))
MARK(sel(fib(y0), add(x0, x1))) → A__SEL(a__fib(mark(y0)), a__add(mark(x0), mark(x1)))
MARK(add(add(x0, x1), y1)) → A__ADD(a__add(mark(x0), mark(x1)), mark(y1))
A__ADD(s(X), Y) → MARK(X)
A__FIB(N) → A__FIB1(s(0), s(0))
MARK(add(fib(x0), y1)) → A__ADD(a__fib(mark(x0)), mark(y1))
A__FIB(fib(x0)) → A__SEL(a__fib(mark(x0)), a__fib1(s(0), s(0)))
MARK(sel(0, cons(x0, x1))) → A__SEL(0, cons(mark(x0), x1))
MARK(sel(add(y0, y1), cons(x0, x1))) → A__SEL(a__add(mark(y0), mark(y1)), cons(mark(x0), x1))
MARK(fib(X)) → MARK(X)
A__FIB1(X, Y) → MARK(X)
A__ADD(0, X) → MARK(X)
MARK(s(X)) → MARK(X)
MARK(add(X1, X2)) → MARK(X2)
MARK(sel(s(y0), sel(x0, x1))) → A__SEL(s(mark(y0)), a__sel(mark(x0), mark(x1)))
MARK(fib(X)) → A__FIB(mark(X))
MARK(fib1(X1, X2)) → MARK(X2)
MARK(sel(fib(y0), sel(x0, x1))) → A__SEL(a__fib(mark(y0)), a__sel(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), fib(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__fib(mark(x0)))
A__FIB(0) → A__SEL(0, a__fib1(s(0), s(0)))
A__SEL(0, cons(X, XS)) → MARK(X)
MARK(add(X1, X2)) → MARK(X1)
MARK(add(fib1(x0, x1), y1)) → A__ADD(a__fib1(mark(x0), mark(x1)), mark(y1))
MARK(sel(0, fib1(x0, x1))) → A__SEL(0, a__fib1(mark(x0), mark(x1)))
A__FIB(fib1(x0, x1)) → A__SEL(a__fib1(mark(x0), mark(x1)), a__fib1(s(0), s(0)))
MARK(sel(s(y0), s(x0))) → A__SEL(s(mark(y0)), s(mark(x0)))
A__SEL(s(N), cons(X, XS)) → MARK(N)
MARK(fib1(X1, X2)) → A__FIB1(mark(X1), mark(X2))
MARK(sel(s(y0), fib1(x0, x1))) → A__SEL(s(mark(y0)), a__fib1(mark(x0), mark(x1)))
A__SEL(s(N), cons(X, XS)) → A__SEL(mark(N), mark(XS))
MARK(sel(s(y0), 0)) → A__SEL(s(mark(y0)), 0)
MARK(sel(fib(y0), fib(x0))) → A__SEL(a__fib(mark(y0)), a__fib(mark(x0)))
MARK(sel(fib1(y0, y1), sel(x0, x1))) → A__SEL(a__fib1(mark(y0), mark(y1)), a__sel(mark(x0), mark(x1)))
MARK(sel(add(y0, y1), add(x0, x1))) → A__SEL(a__add(mark(y0), mark(y1)), a__add(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), fib1(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), a__fib1(mark(x0), mark(x1)))
MARK(sel(s(y0), add(x0, x1))) → A__SEL(s(mark(y0)), a__add(mark(x0), mark(x1)))
MARK(sel(0, sel(x0, x1))) → A__SEL(0, a__sel(mark(x0), mark(x1)))
A__FIB(add(x0, x1)) → A__SEL(a__add(mark(x0), mark(x1)), a__fib1(s(0), s(0)))
MARK(sel(X1, X2)) → MARK(X2)

The TRS R consists of the following rules:

a__fib(N) → a__sel(mark(N), a__fib1(s(0), s(0)))
a__fib1(X, Y) → cons(mark(X), fib1(Y, add(X, Y)))
a__add(0, X) → mark(X)
a__add(s(X), Y) → s(a__add(mark(X), mark(Y)))
a__sel(0, cons(X, XS)) → mark(X)
a__sel(s(N), cons(X, XS)) → a__sel(mark(N), mark(XS))
mark(fib(X)) → a__fib(mark(X))
mark(sel(X1, X2)) → a__sel(mark(X1), mark(X2))
mark(fib1(X1, X2)) → a__fib1(mark(X1), mark(X2))
mark(add(X1, X2)) → a__add(mark(X1), mark(X2))
mark(s(X)) → s(mark(X))
mark(0) → 0
mark(cons(X1, X2)) → cons(mark(X1), X2)
a__fib(X) → fib(X)
a__sel(X1, X2) → sel(X1, X2)
a__fib1(X1, X2) → fib1(X1, X2)
a__add(X1, X2) → add(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 1 SCC with 2 less nodes.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ Narrowing

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ QDP

              ↳ Narrowing

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ QDP

                      ↳ Narrowing

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ QDP

                              ↳ Narrowing

                                ↳ QDP

                                  ↳ DependencyGraphProof

                                    ↳ QDP

                                      ↳ Narrowing

                                        ↳ QDP

                                          ↳ DependencyGraphProof

                                            ↳ QDP

                                              ↳ Narrowing

                                                ↳ QDP

                                                  ↳ DependencyGraphProof

                                                    ↳ QDP

                                                      ↳ Narrowing

                                                        ↳ QDP

                                                          ↳ DependencyGraphProof

                                                            ↳ QDP

                                                              ↳ Narrowing

                                                                ↳ QDP

                                                                  ↳ DependencyGraphProof

                                                                    ↳ QDP

                                                                      ↳ Narrowing

                                                                        ↳ QDP

                                                                          ↳ DependencyGraphProof

                                                                            ↳ QDP

                                                                              ↳ Narrowing

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

MARK(sel(sel(y0, y1), sel(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), a__sel(mark(x0), mark(x1)))
MARK(sel(fib1(y0, y1), fib1(x0, x1))) → A__SEL(a__fib1(mark(y0), mark(y1)), a__fib1(mark(x0), mark(x1)))
MARK(fib1(X1, X2)) → MARK(X1)
MARK(sel(s(y0), cons(x0, x1))) → A__SEL(s(mark(y0)), cons(mark(x0), x1))
MARK(cons(X1, X2)) → MARK(X1)
MARK(sel(0, add(x0, x1))) → A__SEL(0, a__add(mark(x0), mark(x1)))
A__FIB(N) → MARK(N)
MARK(sel(0, fib(x0))) → A__SEL(0, a__fib(mark(x0)))
MARK(sel(add(y0, y1), fib1(x0, x1))) → A__SEL(a__add(mark(y0), mark(y1)), a__fib1(mark(x0), mark(x1)))
MARK(sel(fib1(y0, y1), add(x0, x1))) → A__SEL(a__fib1(mark(y0), mark(y1)), a__add(mark(x0), mark(x1)))
A__FIB(s(x0)) → A__SEL(s(mark(x0)), a__fib1(s(0), s(0)))
A__ADD(s(X), Y) → A__ADD(mark(X), mark(Y))
A__SEL(s(N), cons(X, XS)) → MARK(XS)
A__ADD(s(X), Y) → MARK(Y)
MARK(add(0, y1)) → A__ADD(0, mark(y1))
MARK(sel(fib(y0), cons(x0, x1))) → A__SEL(a__fib(mark(y0)), cons(mark(x0), x1))
A__FIB(sel(x0, x1)) → A__SEL(a__sel(mark(x0), mark(x1)), a__fib1(s(0), s(0)))
MARK(add(s(x0), y1)) → A__ADD(s(mark(x0)), mark(y1))
MARK(sel(sel(y0, y1), add(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), a__add(mark(x0), mark(x1)))
MARK(sel(add(y0, y1), sel(x0, x1))) → A__SEL(a__add(mark(y0), mark(y1)), a__sel(mark(x0), mark(x1)))
MARK(sel(s(y0), fib(x0))) → A__SEL(s(mark(y0)), a__fib(mark(x0)))
MARK(add(sel(x0, x1), y1)) → A__ADD(a__sel(mark(x0), mark(x1)), mark(y1))
MARK(sel(fib(y0), fib1(x0, x1))) → A__SEL(a__fib(mark(y0)), a__fib1(mark(x0), mark(x1)))
MARK(sel(fib1(y0, y1), fib(x0))) → A__SEL(a__fib1(mark(y0), mark(y1)), a__fib(mark(x0)))
MARK(sel(fib1(y0, y1), cons(x0, x1))) → A__SEL(a__fib1(mark(y0), mark(y1)), cons(mark(x0), x1))
MARK(sel(X1, X2)) → MARK(X1)
MARK(sel(sel(y0, y1), cons(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), cons(mark(x0), x1))
MARK(sel(fib(y0), add(x0, x1))) → A__SEL(a__fib(mark(y0)), a__add(mark(x0), mark(x1)))
MARK(sel(add(y0, y1), fib(x0))) → A__SEL(a__add(mark(y0), mark(y1)), a__fib(mark(x0)))
MARK(add(add(x0, x1), y1)) → A__ADD(a__add(mark(x0), mark(x1)), mark(y1))
A__ADD(s(X), Y) → MARK(X)
A__FIB(N) → A__FIB1(s(0), s(0))
MARK(add(fib(x0), y1)) → A__ADD(a__fib(mark(x0)), mark(y1))
A__FIB(fib(x0)) → A__SEL(a__fib(mark(x0)), a__fib1(s(0), s(0)))
MARK(sel(0, cons(x0, x1))) → A__SEL(0, cons(mark(x0), x1))
MARK(sel(add(y0, y1), cons(x0, x1))) → A__SEL(a__add(mark(y0), mark(y1)), cons(mark(x0), x1))
MARK(fib(X)) → MARK(X)
A__ADD(0, X) → MARK(X)
A__FIB1(X, Y) → MARK(X)
MARK(s(X)) → MARK(X)
MARK(add(X1, X2)) → MARK(X2)
MARK(sel(s(y0), sel(x0, x1))) → A__SEL(s(mark(y0)), a__sel(mark(x0), mark(x1)))
MARK(fib(X)) → A__FIB(mark(X))
MARK(fib1(X1, X2)) → MARK(X2)
A__FIB(0) → A__SEL(0, a__fib1(s(0), s(0)))
MARK(sel(sel(y0, y1), fib(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__fib(mark(x0)))
MARK(sel(fib(y0), sel(x0, x1))) → A__SEL(a__fib(mark(y0)), a__sel(mark(x0), mark(x1)))
A__SEL(0, cons(X, XS)) → MARK(X)
MARK(add(X1, X2)) → MARK(X1)
MARK(add(fib1(x0, x1), y1)) → A__ADD(a__fib1(mark(x0), mark(x1)), mark(y1))
MARK(sel(0, fib1(x0, x1))) → A__SEL(0, a__fib1(mark(x0), mark(x1)))
A__FIB(fib1(x0, x1)) → A__SEL(a__fib1(mark(x0), mark(x1)), a__fib1(s(0), s(0)))
A__SEL(s(N), cons(X, XS)) → MARK(N)
MARK(fib1(X1, X2)) → A__FIB1(mark(X1), mark(X2))
MARK(sel(s(y0), fib1(x0, x1))) → A__SEL(s(mark(y0)), a__fib1(mark(x0), mark(x1)))
A__SEL(s(N), cons(X, XS)) → A__SEL(mark(N), mark(XS))
MARK(sel(fib(y0), fib(x0))) → A__SEL(a__fib(mark(y0)), a__fib(mark(x0)))
MARK(sel(sel(y0, y1), fib1(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), a__fib1(mark(x0), mark(x1)))
MARK(sel(add(y0, y1), add(x0, x1))) → A__SEL(a__add(mark(y0), mark(y1)), a__add(mark(x0), mark(x1)))
MARK(sel(fib1(y0, y1), sel(x0, x1))) → A__SEL(a__fib1(mark(y0), mark(y1)), a__sel(mark(x0), mark(x1)))
MARK(sel(s(y0), add(x0, x1))) → A__SEL(s(mark(y0)), a__add(mark(x0), mark(x1)))
MARK(sel(0, sel(x0, x1))) → A__SEL(0, a__sel(mark(x0), mark(x1)))
A__FIB(add(x0, x1)) → A__SEL(a__add(mark(x0), mark(x1)), a__fib1(s(0), s(0)))
MARK(sel(X1, X2)) → MARK(X2)

The TRS R consists of the following rules:

a__fib(N) → a__sel(mark(N), a__fib1(s(0), s(0)))
a__fib1(X, Y) → cons(mark(X), fib1(Y, add(X, Y)))
a__add(0, X) → mark(X)
a__add(s(X), Y) → s(a__add(mark(X), mark(Y)))
a__sel(0, cons(X, XS)) → mark(X)
a__sel(s(N), cons(X, XS)) → a__sel(mark(N), mark(XS))
mark(fib(X)) → a__fib(mark(X))
mark(sel(X1, X2)) → a__sel(mark(X1), mark(X2))
mark(fib1(X1, X2)) → a__fib1(mark(X1), mark(X2))
mark(add(X1, X2)) → a__add(mark(X1), mark(X2))
mark(s(X)) → s(mark(X))
mark(0) → 0
mark(cons(X1, X2)) → cons(mark(X1), X2)
a__fib(X) → fib(X)
a__sel(X1, X2) → sel(X1, X2)
a__fib1(X1, X2) → fib1(X1, X2)
a__add(X1, X2) → add(X1, X2)

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

A__ADD(s(fib(x0)), y1) → A__ADD(a__fib(mark(x0)), mark(y1))
A__ADD(s(s(x0)), y1) → A__ADD(s(mark(x0)), mark(y1))
A__ADD(s(0), y1) → A__ADD(0, mark(y1))
A__ADD(s(cons(x0, x1)), y1) → A__ADD(cons(mark(x0), x1), mark(y1))
A__ADD(s(add(x0, x1)), y1) → A__ADD(a__add(mark(x0), mark(x1)), mark(y1))
A__ADD(s(sel(x0, x1)), y1) → A__ADD(a__sel(mark(x0), mark(x1)), mark(y1))
A__ADD(s(fib1(x0, x1)), y1) → A__ADD(a__fib1(mark(x0), mark(x1)), mark(y1))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ Narrowing

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ QDP

              ↳ Narrowing

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ QDP

                      ↳ Narrowing

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ QDP

                              ↳ Narrowing

                                ↳ QDP

                                  ↳ DependencyGraphProof

                                    ↳ QDP

                                      ↳ Narrowing

                                        ↳ QDP

                                          ↳ DependencyGraphProof

                                            ↳ QDP

                                              ↳ Narrowing

                                                ↳ QDP

                                                  ↳ DependencyGraphProof

                                                    ↳ QDP

                                                      ↳ Narrowing

                                                        ↳ QDP

                                                          ↳ DependencyGraphProof

                                                            ↳ QDP

                                                              ↳ Narrowing

                                                                ↳ QDP

                                                                  ↳ DependencyGraphProof

                                                                    ↳ QDP

                                                                      ↳ Narrowing

                                                                        ↳ QDP

                                                                          ↳ DependencyGraphProof

                                                                            ↳ QDP

                                                                              ↳ Narrowing

                                                                                ↳ QDP

                                                                                  ↳ DependencyGraphProof

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

MARK(sel(sel(y0, y1), sel(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), a__sel(mark(x0), mark(x1)))
MARK(sel(fib1(y0, y1), fib1(x0, x1))) → A__SEL(a__fib1(mark(y0), mark(y1)), a__fib1(mark(x0), mark(x1)))
MARK(fib1(X1, X2)) → MARK(X1)
MARK(sel(s(y0), cons(x0, x1))) → A__SEL(s(mark(y0)), cons(mark(x0), x1))
MARK(cons(X1, X2)) → MARK(X1)
A__ADD(s(add(x0, x1)), y1) → A__ADD(a__add(mark(x0), mark(x1)), mark(y1))
MARK(sel(0, add(x0, x1))) → A__SEL(0, a__add(mark(x0), mark(x1)))
MARK(sel(0, fib(x0))) → A__SEL(0, a__fib(mark(x0)))
A__FIB(N) → MARK(N)
MARK(sel(fib1(y0, y1), add(x0, x1))) → A__SEL(a__fib1(mark(y0), mark(y1)), a__add(mark(x0), mark(x1)))
MARK(sel(add(y0, y1), fib1(x0, x1))) → A__SEL(a__add(mark(y0), mark(y1)), a__fib1(mark(x0), mark(x1)))
A__SEL(s(N), cons(X, XS)) → MARK(XS)
A__FIB(s(x0)) → A__SEL(s(mark(x0)), a__fib1(s(0), s(0)))
A__ADD(s(X), Y) → MARK(Y)
A__ADD(s(0), y1) → A__ADD(0, mark(y1))
MARK(add(0, y1)) → A__ADD(0, mark(y1))
MARK(sel(fib(y0), cons(x0, x1))) → A__SEL(a__fib(mark(y0)), cons(mark(x0), x1))
A__FIB(sel(x0, x1)) → A__SEL(a__sel(mark(x0), mark(x1)), a__fib1(s(0), s(0)))
MARK(add(s(x0), y1)) → A__ADD(s(mark(x0)), mark(y1))
MARK(sel(add(y0, y1), sel(x0, x1))) → A__SEL(a__add(mark(y0), mark(y1)), a__sel(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), add(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), a__add(mark(x0), mark(x1)))
MARK(sel(fib1(y0, y1), fib(x0))) → A__SEL(a__fib1(mark(y0), mark(y1)), a__fib(mark(x0)))
MARK(sel(fib(y0), fib1(x0, x1))) → A__SEL(a__fib(mark(y0)), a__fib1(mark(x0), mark(x1)))
MARK(add(sel(x0, x1), y1)) → A__ADD(a__sel(mark(x0), mark(x1)), mark(y1))
MARK(sel(s(y0), fib(x0))) → A__SEL(s(mark(y0)), a__fib(mark(x0)))
MARK(sel(fib1(y0, y1), cons(x0, x1))) → A__SEL(a__fib1(mark(y0), mark(y1)), cons(mark(x0), x1))
MARK(sel(X1, X2)) → MARK(X1)
MARK(sel(sel(y0, y1), cons(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), cons(mark(x0), x1))
MARK(sel(add(y0, y1), fib(x0))) → A__SEL(a__add(mark(y0), mark(y1)), a__fib(mark(x0)))
MARK(sel(fib(y0), add(x0, x1))) → A__SEL(a__fib(mark(y0)), a__add(mark(x0), mark(x1)))
MARK(add(add(x0, x1), y1)) → A__ADD(a__add(mark(x0), mark(x1)), mark(y1))
A__ADD(s(X), Y) → MARK(X)
A__FIB(N) → A__FIB1(s(0), s(0))
MARK(add(fib(x0), y1)) → A__ADD(a__fib(mark(x0)), mark(y1))
A__FIB(fib(x0)) → A__SEL(a__fib(mark(x0)), a__fib1(s(0), s(0)))
A__ADD(s(fib(x0)), y1) → A__ADD(a__fib(mark(x0)), mark(y1))
MARK(sel(0, cons(x0, x1))) → A__SEL(0, cons(mark(x0), x1))
A__ADD(s(cons(x0, x1)), y1) → A__ADD(cons(mark(x0), x1), mark(y1))
MARK(sel(add(y0, y1), cons(x0, x1))) → A__SEL(a__add(mark(y0), mark(y1)), cons(mark(x0), x1))
A__ADD(s(fib1(x0, x1)), y1) → A__ADD(a__fib1(mark(x0), mark(x1)), mark(y1))
MARK(fib(X)) → MARK(X)
A__FIB1(X, Y) → MARK(X)
A__ADD(0, X) → MARK(X)
MARK(s(X)) → MARK(X)
MARK(add(X1, X2)) → MARK(X2)
MARK(fib(X)) → A__FIB(mark(X))
MARK(sel(s(y0), sel(x0, x1))) → A__SEL(s(mark(y0)), a__sel(mark(x0), mark(x1)))
MARK(fib1(X1, X2)) → MARK(X2)
MARK(sel(fib(y0), sel(x0, x1))) → A__SEL(a__fib(mark(y0)), a__sel(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), fib(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__fib(mark(x0)))
A__FIB(0) → A__SEL(0, a__fib1(s(0), s(0)))
A__ADD(s(sel(x0, x1)), y1) → A__ADD(a__sel(mark(x0), mark(x1)), mark(y1))
A__SEL(0, cons(X, XS)) → MARK(X)
MARK(add(X1, X2)) → MARK(X1)
MARK(add(fib1(x0, x1), y1)) → A__ADD(a__fib1(mark(x0), mark(x1)), mark(y1))
MARK(sel(0, fib1(x0, x1))) → A__SEL(0, a__fib1(mark(x0), mark(x1)))
A__FIB(fib1(x0, x1)) → A__SEL(a__fib1(mark(x0), mark(x1)), a__fib1(s(0), s(0)))
A__SEL(s(N), cons(X, XS)) → MARK(N)
A__ADD(s(s(x0)), y1) → A__ADD(s(mark(x0)), mark(y1))
MARK(fib1(X1, X2)) → A__FIB1(mark(X1), mark(X2))
MARK(sel(s(y0), fib1(x0, x1))) → A__SEL(s(mark(y0)), a__fib1(mark(x0), mark(x1)))
A__SEL(s(N), cons(X, XS)) → A__SEL(mark(N), mark(XS))
MARK(sel(fib(y0), fib(x0))) → A__SEL(a__fib(mark(y0)), a__fib(mark(x0)))
MARK(sel(fib1(y0, y1), sel(x0, x1))) → A__SEL(a__fib1(mark(y0), mark(y1)), a__sel(mark(x0), mark(x1)))
MARK(sel(add(y0, y1), add(x0, x1))) → A__SEL(a__add(mark(y0), mark(y1)), a__add(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), fib1(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), a__fib1(mark(x0), mark(x1)))
MARK(sel(s(y0), add(x0, x1))) → A__SEL(s(mark(y0)), a__add(mark(x0), mark(x1)))
MARK(sel(0, sel(x0, x1))) → A__SEL(0, a__sel(mark(x0), mark(x1)))
A__FIB(add(x0, x1)) → A__SEL(a__add(mark(x0), mark(x1)), a__fib1(s(0), s(0)))
MARK(sel(X1, X2)) → MARK(X2)

The TRS R consists of the following rules:

a__fib(N) → a__sel(mark(N), a__fib1(s(0), s(0)))
a__fib1(X, Y) → cons(mark(X), fib1(Y, add(X, Y)))
a__add(0, X) → mark(X)
a__add(s(X), Y) → s(a__add(mark(X), mark(Y)))
a__sel(0, cons(X, XS)) → mark(X)
a__sel(s(N), cons(X, XS)) → a__sel(mark(N), mark(XS))
mark(fib(X)) → a__fib(mark(X))
mark(sel(X1, X2)) → a__sel(mark(X1), mark(X2))
mark(fib1(X1, X2)) → a__fib1(mark(X1), mark(X2))
mark(add(X1, X2)) → a__add(mark(X1), mark(X2))
mark(s(X)) → s(mark(X))
mark(0) → 0
mark(cons(X1, X2)) → cons(mark(X1), X2)
a__fib(X) → fib(X)
a__sel(X1, X2) → sel(X1, X2)
a__fib1(X1, X2) → fib1(X1, X2)
a__add(X1, X2) → add(X1, X2)

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ Narrowing

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ QDP

              ↳ Narrowing

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ QDP

                      ↳ Narrowing

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ QDP

                              ↳ Narrowing

                                ↳ QDP

                                  ↳ DependencyGraphProof

                                    ↳ QDP

                                      ↳ Narrowing

                                        ↳ QDP

                                          ↳ DependencyGraphProof

                                            ↳ QDP

                                              ↳ Narrowing

                                                ↳ QDP

                                                  ↳ DependencyGraphProof

                                                    ↳ QDP

                                                      ↳ Narrowing

                                                        ↳ QDP

                                                          ↳ DependencyGraphProof

                                                            ↳ QDP

                                                              ↳ Narrowing

                                                                ↳ QDP

                                                                  ↳ DependencyGraphProof

                                                                    ↳ QDP

                                                                      ↳ Narrowing

                                                                        ↳ QDP

                                                                          ↳ DependencyGraphProof

                                                                            ↳ QDP

                                                                              ↳ Narrowing

                                                                                ↳ QDP

                                                                                  ↳ DependencyGraphProof

                                                                                    ↳ QDP

                                                                                      ↳ Narrowing

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

MARK(sel(sel(y0, y1), sel(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), a__sel(mark(x0), mark(x1)))
MARK(sel(fib1(y0, y1), fib1(x0, x1))) → A__SEL(a__fib1(mark(y0), mark(y1)), a__fib1(mark(x0), mark(x1)))
MARK(fib1(X1, X2)) → MARK(X1)
MARK(sel(s(y0), cons(x0, x1))) → A__SEL(s(mark(y0)), cons(mark(x0), x1))
MARK(cons(X1, X2)) → MARK(X1)
A__ADD(s(add(x0, x1)), y1) → A__ADD(a__add(mark(x0), mark(x1)), mark(y1))
MARK(sel(0, add(x0, x1))) → A__SEL(0, a__add(mark(x0), mark(x1)))
A__FIB(N) → MARK(N)
MARK(sel(0, fib(x0))) → A__SEL(0, a__fib(mark(x0)))
MARK(sel(fib1(y0, y1), add(x0, x1))) → A__SEL(a__fib1(mark(y0), mark(y1)), a__add(mark(x0), mark(x1)))
MARK(sel(add(y0, y1), fib1(x0, x1))) → A__SEL(a__add(mark(y0), mark(y1)), a__fib1(mark(x0), mark(x1)))
A__FIB(s(x0)) → A__SEL(s(mark(x0)), a__fib1(s(0), s(0)))
A__SEL(s(N), cons(X, XS)) → MARK(XS)
A__ADD(s(X), Y) → MARK(Y)
A__ADD(s(0), y1) → A__ADD(0, mark(y1))
MARK(add(0, y1)) → A__ADD(0, mark(y1))
MARK(sel(fib(y0), cons(x0, x1))) → A__SEL(a__fib(mark(y0)), cons(mark(x0), x1))
A__FIB(sel(x0, x1)) → A__SEL(a__sel(mark(x0), mark(x1)), a__fib1(s(0), s(0)))
MARK(add(s(x0), y1)) → A__ADD(s(mark(x0)), mark(y1))
MARK(sel(sel(y0, y1), add(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), a__add(mark(x0), mark(x1)))
MARK(sel(add(y0, y1), sel(x0, x1))) → A__SEL(a__add(mark(y0), mark(y1)), a__sel(mark(x0), mark(x1)))
MARK(sel(s(y0), fib(x0))) → A__SEL(s(mark(y0)), a__fib(mark(x0)))
MARK(sel(fib1(y0, y1), fib(x0))) → A__SEL(a__fib1(mark(y0), mark(y1)), a__fib(mark(x0)))
MARK(add(sel(x0, x1), y1)) → A__ADD(a__sel(mark(x0), mark(x1)), mark(y1))
MARK(sel(fib(y0), fib1(x0, x1))) → A__SEL(a__fib(mark(y0)), a__fib1(mark(x0), mark(x1)))
MARK(sel(fib1(y0, y1), cons(x0, x1))) → A__SEL(a__fib1(mark(y0), mark(y1)), cons(mark(x0), x1))
MARK(sel(X1, X2)) → MARK(X1)
MARK(sel(sel(y0, y1), cons(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), cons(mark(x0), x1))
MARK(sel(fib(y0), add(x0, x1))) → A__SEL(a__fib(mark(y0)), a__add(mark(x0), mark(x1)))
MARK(sel(add(y0, y1), fib(x0))) → A__SEL(a__add(mark(y0), mark(y1)), a__fib(mark(x0)))
MARK(add(add(x0, x1), y1)) → A__ADD(a__add(mark(x0), mark(x1)), mark(y1))
A__ADD(s(X), Y) → MARK(X)
A__FIB(N) → A__FIB1(s(0), s(0))
MARK(add(fib(x0), y1)) → A__ADD(a__fib(mark(x0)), mark(y1))
A__FIB(fib(x0)) → A__SEL(a__fib(mark(x0)), a__fib1(s(0), s(0)))
A__ADD(s(fib(x0)), y1) → A__ADD(a__fib(mark(x0)), mark(y1))
MARK(sel(0, cons(x0, x1))) → A__SEL(0, cons(mark(x0), x1))
MARK(sel(add(y0, y1), cons(x0, x1))) → A__SEL(a__add(mark(y0), mark(y1)), cons(mark(x0), x1))
A__ADD(s(fib1(x0, x1)), y1) → A__ADD(a__fib1(mark(x0), mark(x1)), mark(y1))
MARK(fib(X)) → MARK(X)
A__ADD(0, X) → MARK(X)
A__FIB1(X, Y) → MARK(X)
MARK(s(X)) → MARK(X)
MARK(add(X1, X2)) → MARK(X2)
MARK(sel(s(y0), sel(x0, x1))) → A__SEL(s(mark(y0)), a__sel(mark(x0), mark(x1)))
MARK(fib(X)) → A__FIB(mark(X))
MARK(fib1(X1, X2)) → MARK(X2)
A__FIB(0) → A__SEL(0, a__fib1(s(0), s(0)))
MARK(sel(sel(y0, y1), fib(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__fib(mark(x0)))
MARK(sel(fib(y0), sel(x0, x1))) → A__SEL(a__fib(mark(y0)), a__sel(mark(x0), mark(x1)))
A__ADD(s(sel(x0, x1)), y1) → A__ADD(a__sel(mark(x0), mark(x1)), mark(y1))
A__SEL(0, cons(X, XS)) → MARK(X)
MARK(add(X1, X2)) → MARK(X1)
MARK(add(fib1(x0, x1), y1)) → A__ADD(a__fib1(mark(x0), mark(x1)), mark(y1))
MARK(sel(0, fib1(x0, x1))) → A__SEL(0, a__fib1(mark(x0), mark(x1)))
A__FIB(fib1(x0, x1)) → A__SEL(a__fib1(mark(x0), mark(x1)), a__fib1(s(0), s(0)))
A__SEL(s(N), cons(X, XS)) → MARK(N)
A__ADD(s(s(x0)), y1) → A__ADD(s(mark(x0)), mark(y1))
MARK(fib1(X1, X2)) → A__FIB1(mark(X1), mark(X2))
MARK(sel(s(y0), fib1(x0, x1))) → A__SEL(s(mark(y0)), a__fib1(mark(x0), mark(x1)))
A__SEL(s(N), cons(X, XS)) → A__SEL(mark(N), mark(XS))
MARK(sel(fib(y0), fib(x0))) → A__SEL(a__fib(mark(y0)), a__fib(mark(x0)))
MARK(sel(sel(y0, y1), fib1(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), a__fib1(mark(x0), mark(x1)))
MARK(sel(fib1(y0, y1), sel(x0, x1))) → A__SEL(a__fib1(mark(y0), mark(y1)), a__sel(mark(x0), mark(x1)))
MARK(sel(add(y0, y1), add(x0, x1))) → A__SEL(a__add(mark(y0), mark(y1)), a__add(mark(x0), mark(x1)))
MARK(sel(s(y0), add(x0, x1))) → A__SEL(s(mark(y0)), a__add(mark(x0), mark(x1)))
MARK(sel(0, sel(x0, x1))) → A__SEL(0, a__sel(mark(x0), mark(x1)))
A__FIB(add(x0, x1)) → A__SEL(a__add(mark(x0), mark(x1)), a__fib1(s(0), s(0)))
MARK(sel(X1, X2)) → MARK(X2)

The TRS R consists of the following rules:

a__fib(N) → a__sel(mark(N), a__fib1(s(0), s(0)))
a__fib1(X, Y) → cons(mark(X), fib1(Y, add(X, Y)))
a__add(0, X) → mark(X)
a__add(s(X), Y) → s(a__add(mark(X), mark(Y)))
a__sel(0, cons(X, XS)) → mark(X)
a__sel(s(N), cons(X, XS)) → a__sel(mark(N), mark(XS))
mark(fib(X)) → a__fib(mark(X))
mark(sel(X1, X2)) → a__sel(mark(X1), mark(X2))
mark(fib1(X1, X2)) → a__fib1(mark(X1), mark(X2))
mark(add(X1, X2)) → a__add(mark(X1), mark(X2))
mark(s(X)) → s(mark(X))
mark(0) → 0
mark(cons(X1, X2)) → cons(mark(X1), X2)
a__fib(X) → fib(X)
a__sel(X1, X2) → sel(X1, X2)
a__fib1(X1, X2) → fib1(X1, X2)
a__add(X1, X2) → add(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(add(x0, x1)), cons(y1, y2)) → A__SEL(a__add(mark(x0), mark(x1)), mark(y2))
A__SEL(s(sel(x0, x1)), cons(y1, y2)) → A__SEL(a__sel(mark(x0), mark(x1)), mark(y2))
A__SEL(s(fib1(x0, x1)), cons(y1, y2)) → A__SEL(a__fib1(mark(x0), mark(x1)), mark(y2))
A__SEL(s(0), cons(y1, y2)) → A__SEL(0, mark(y2))
A__SEL(s(s(x0)), cons(y1, y2)) → A__SEL(s(mark(x0)), mark(y2))
A__SEL(s(fib(x0)), cons(y1, y2)) → A__SEL(a__fib(mark(x0)), mark(y2))
A__SEL(s(cons(x0, x1)), cons(y1, y2)) → A__SEL(cons(mark(x0), x1), mark(y2))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ Narrowing

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ QDP

              ↳ Narrowing

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ QDP

                      ↳ Narrowing

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ QDP

                              ↳ Narrowing

                                ↳ QDP

                                  ↳ DependencyGraphProof

                                    ↳ QDP

                                      ↳ Narrowing

                                        ↳ QDP

                                          ↳ DependencyGraphProof

                                            ↳ QDP

                                              ↳ Narrowing

                                                ↳ QDP

                                                  ↳ DependencyGraphProof

                                                    ↳ QDP

                                                      ↳ Narrowing

                                                        ↳ QDP

                                                          ↳ DependencyGraphProof

                                                            ↳ QDP

                                                              ↳ Narrowing

                                                                ↳ QDP

                                                                  ↳ DependencyGraphProof

                                                                    ↳ QDP

                                                                      ↳ Narrowing

                                                                        ↳ QDP

                                                                          ↳ DependencyGraphProof

                                                                            ↳ QDP

                                                                              ↳ Narrowing

                                                                                ↳ QDP

                                                                                  ↳ DependencyGraphProof

                                                                                    ↳ QDP

                                                                                      ↳ Narrowing

                                                                                        ↳ QDP

                                                                                          ↳ DependencyGraphProof

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

MARK(sel(sel(y0, y1), sel(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), a__sel(mark(x0), mark(x1)))
MARK(sel(fib1(y0, y1), fib1(x0, x1))) → A__SEL(a__fib1(mark(y0), mark(y1)), a__fib1(mark(x0), mark(x1)))
MARK(fib1(X1, X2)) → MARK(X1)
MARK(sel(s(y0), cons(x0, x1))) → A__SEL(s(mark(y0)), cons(mark(x0), x1))
MARK(cons(X1, X2)) → MARK(X1)
A__ADD(s(add(x0, x1)), y1) → A__ADD(a__add(mark(x0), mark(x1)), mark(y1))
MARK(sel(0, add(x0, x1))) → A__SEL(0, a__add(mark(x0), mark(x1)))
MARK(sel(0, fib(x0))) → A__SEL(0, a__fib(mark(x0)))
A__FIB(N) → MARK(N)
MARK(sel(add(y0, y1), fib1(x0, x1))) → A__SEL(a__add(mark(y0), mark(y1)), a__fib1(mark(x0), mark(x1)))
MARK(sel(fib1(y0, y1), add(x0, x1))) → A__SEL(a__fib1(mark(y0), mark(y1)), a__add(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(N), cons(X, XS)) → MARK(XS)
A__FIB(s(x0)) → A__SEL(s(mark(x0)), a__fib1(s(0), s(0)))
A__ADD(s(X), Y) → MARK(Y)
A__ADD(s(0), y1) → A__ADD(0, mark(y1))
MARK(add(0, y1)) → A__ADD(0, mark(y1))
MARK(sel(fib(y0), cons(x0, x1))) → A__SEL(a__fib(mark(y0)), cons(mark(x0), x1))
A__FIB(sel(x0, x1)) → A__SEL(a__sel(mark(x0), mark(x1)), a__fib1(s(0), s(0)))
MARK(add(s(x0), y1)) → A__ADD(s(mark(x0)), mark(y1))
MARK(sel(add(y0, y1), sel(x0, x1))) → A__SEL(a__add(mark(y0), mark(y1)), a__sel(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), add(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), a__add(mark(x0), mark(x1)))
MARK(sel(fib(y0), fib1(x0, x1))) → A__SEL(a__fib(mark(y0)), a__fib1(mark(x0), mark(x1)))
MARK(add(sel(x0, x1), y1)) → A__ADD(a__sel(mark(x0), mark(x1)), mark(y1))
MARK(sel(fib1(y0, y1), fib(x0))) → A__SEL(a__fib1(mark(y0), mark(y1)), a__fib(mark(x0)))
MARK(sel(s(y0), fib(x0))) → A__SEL(s(mark(y0)), a__fib(mark(x0)))
A__SEL(s(add(x0, x1)), cons(y1, y2)) → A__SEL(a__add(mark(x0), mark(x1)), mark(y2))
MARK(sel(fib1(y0, y1), cons(x0, x1))) → A__SEL(a__fib1(mark(y0), mark(y1)), cons(mark(x0), x1))
MARK(sel(X1, X2)) → MARK(X1)
MARK(sel(sel(y0, y1), cons(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), cons(mark(x0), x1))
MARK(sel(add(y0, y1), fib(x0))) → A__SEL(a__add(mark(y0), mark(y1)), a__fib(mark(x0)))
MARK(sel(fib(y0), add(x0, x1))) → A__SEL(a__fib(mark(y0)), a__add(mark(x0), mark(x1)))
MARK(add(add(x0, x1), y1)) → A__ADD(a__add(mark(x0), mark(x1)), mark(y1))
A__ADD(s(X), Y) → MARK(X)
A__FIB(N) → A__FIB1(s(0), s(0))
MARK(add(fib(x0), y1)) → A__ADD(a__fib(mark(x0)), mark(y1))
A__FIB(fib(x0)) → A__SEL(a__fib(mark(x0)), a__fib1(s(0), s(0)))
A__ADD(s(fib(x0)), y1) → A__ADD(a__fib(mark(x0)), mark(y1))
A__SEL(s(0), cons(y1, y2)) → A__SEL(0, mark(y2))
MARK(sel(0, cons(x0, x1))) → A__SEL(0, cons(mark(x0), x1))
MARK(sel(add(y0, y1), cons(x0, x1))) → A__SEL(a__add(mark(y0), mark(y1)), cons(mark(x0), x1))
A__ADD(s(fib1(x0, x1)), y1) → A__ADD(a__fib1(mark(x0), mark(x1)), mark(y1))
MARK(fib(X)) → MARK(X)
A__SEL(s(fib1(x0, x1)), cons(y1, y2)) → A__SEL(a__fib1(mark(x0), mark(x1)), mark(y2))
A__FIB1(X, Y) → MARK(X)
A__ADD(0, X) → MARK(X)
MARK(s(X)) → MARK(X)
MARK(add(X1, X2)) → MARK(X2)
MARK(fib(X)) → A__FIB(mark(X))
MARK(sel(s(y0), sel(x0, x1))) → A__SEL(s(mark(y0)), a__sel(mark(x0), mark(x1)))
MARK(fib1(X1, X2)) → MARK(X2)
MARK(sel(fib(y0), sel(x0, x1))) → A__SEL(a__fib(mark(y0)), a__sel(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), fib(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__fib(mark(x0)))
A__FIB(0) → A__SEL(0, a__fib1(s(0), s(0)))
A__SEL(0, cons(X, XS)) → MARK(X)
A__ADD(s(sel(x0, x1)), y1) → A__ADD(a__sel(mark(x0), mark(x1)), mark(y1))
A__SEL(s(cons(x0, x1)), cons(y1, y2)) → A__SEL(cons(mark(x0), x1), mark(y2))
MARK(add(X1, X2)) → MARK(X1)
MARK(add(fib1(x0, x1), y1)) → A__ADD(a__fib1(mark(x0), mark(x1)), mark(y1))
MARK(sel(0, fib1(x0, x1))) → A__SEL(0, a__fib1(mark(x0), mark(x1)))
A__FIB(fib1(x0, x1)) → A__SEL(a__fib1(mark(x0), mark(x1)), a__fib1(s(0), s(0)))
A__SEL(s(fib(x0)), cons(y1, y2)) → A__SEL(a__fib(mark(x0)), mark(y2))
A__SEL(s(N), cons(X, XS)) → MARK(N)
A__ADD(s(s(x0)), y1) → A__ADD(s(mark(x0)), mark(y1))
MARK(fib1(X1, X2)) → A__FIB1(mark(X1), mark(X2))
A__SEL(s(s(x0)), cons(y1, y2)) → A__SEL(s(mark(x0)), mark(y2))
MARK(sel(s(y0), fib1(x0, x1))) → A__SEL(s(mark(y0)), a__fib1(mark(x0), mark(x1)))
MARK(sel(fib(y0), fib(x0))) → A__SEL(a__fib(mark(y0)), a__fib(mark(x0)))
MARK(sel(add(y0, y1), add(x0, x1))) → A__SEL(a__add(mark(y0), mark(y1)), a__add(mark(x0), mark(x1)))
MARK(sel(fib1(y0, y1), sel(x0, x1))) → A__SEL(a__fib1(mark(y0), mark(y1)), a__sel(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), fib1(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), a__fib1(mark(x0), mark(x1)))
MARK(sel(s(y0), add(x0, x1))) → A__SEL(s(mark(y0)), a__add(mark(x0), mark(x1)))
MARK(sel(0, sel(x0, x1))) → A__SEL(0, a__sel(mark(x0), mark(x1)))
A__FIB(add(x0, x1)) → A__SEL(a__add(mark(x0), mark(x1)), a__fib1(s(0), s(0)))
MARK(sel(X1, X2)) → MARK(X2)

The TRS R consists of the following rules:

a__fib(N) → a__sel(mark(N), a__fib1(s(0), s(0)))
a__fib1(X, Y) → cons(mark(X), fib1(Y, add(X, Y)))
a__add(0, X) → mark(X)
a__add(s(X), Y) → s(a__add(mark(X), mark(Y)))
a__sel(0, cons(X, XS)) → mark(X)
a__sel(s(N), cons(X, XS)) → a__sel(mark(N), mark(XS))
mark(fib(X)) → a__fib(mark(X))
mark(sel(X1, X2)) → a__sel(mark(X1), mark(X2))
mark(fib1(X1, X2)) → a__fib1(mark(X1), mark(X2))
mark(add(X1, X2)) → a__add(mark(X1), mark(X2))
mark(s(X)) → s(mark(X))
mark(0) → 0
mark(cons(X1, X2)) → cons(mark(X1), X2)
a__fib(X) → fib(X)
a__sel(X1, X2) → sel(X1, X2)
a__fib1(X1, X2) → fib1(X1, X2)
a__add(X1, X2) → add(X1, X2)

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ Narrowing

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ QDP

              ↳ Narrowing

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ QDP

                      ↳ Narrowing

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ QDP

                              ↳ Narrowing

                                ↳ QDP

                                  ↳ DependencyGraphProof

                                    ↳ QDP

                                      ↳ Narrowing

                                        ↳ QDP

                                          ↳ DependencyGraphProof

                                            ↳ QDP

                                              ↳ Narrowing

                                                ↳ QDP

                                                  ↳ DependencyGraphProof

                                                    ↳ QDP

                                                      ↳ Narrowing

                                                        ↳ QDP

                                                          ↳ DependencyGraphProof

                                                            ↳ QDP

                                                              ↳ Narrowing

                                                                ↳ QDP

                                                                  ↳ DependencyGraphProof

                                                                    ↳ QDP

                                                                      ↳ Narrowing

                                                                        ↳ QDP

                                                                          ↳ DependencyGraphProof

                                                                            ↳ QDP

                                                                              ↳ Narrowing

                                                                                ↳ QDP

                                                                                  ↳ DependencyGraphProof

                                                                                    ↳ QDP

                                                                                      ↳ Narrowing

                                                                                        ↳ QDP

                                                                                          ↳ DependencyGraphProof

                                                                                            ↳ QDP

                                                                                              ↳ Narrowing

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

MARK(sel(sel(y0, y1), sel(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), a__sel(mark(x0), mark(x1)))
MARK(sel(fib1(y0, y1), fib1(x0, x1))) → A__SEL(a__fib1(mark(y0), mark(y1)), a__fib1(mark(x0), mark(x1)))
MARK(fib1(X1, X2)) → MARK(X1)
MARK(sel(s(y0), cons(x0, x1))) → A__SEL(s(mark(y0)), cons(mark(x0), x1))
MARK(cons(X1, X2)) → MARK(X1)
A__ADD(s(add(x0, x1)), y1) → A__ADD(a__add(mark(x0), mark(x1)), mark(y1))
MARK(sel(0, add(x0, x1))) → A__SEL(0, a__add(mark(x0), mark(x1)))
A__FIB(N) → MARK(N)
MARK(sel(0, fib(x0))) → A__SEL(0, a__fib(mark(x0)))
MARK(sel(fib1(y0, y1), add(x0, x1))) → A__SEL(a__fib1(mark(y0), mark(y1)), a__add(mark(x0), mark(x1)))
MARK(sel(add(y0, y1), fib1(x0, x1))) → A__SEL(a__add(mark(y0), mark(y1)), a__fib1(mark(x0), mark(x1)))
A__FIB(s(x0)) → A__SEL(s(mark(x0)), a__fib1(s(0), s(0)))
A__SEL(s(N), cons(X, XS)) → MARK(XS)
A__SEL(s(sel(x0, x1)), cons(y1, y2)) → A__SEL(a__sel(mark(x0), mark(x1)), mark(y2))
A__ADD(s(X), Y) → MARK(Y)
A__ADD(s(0), y1) → A__ADD(0, mark(y1))
MARK(add(0, y1)) → A__ADD(0, mark(y1))
MARK(sel(fib(y0), cons(x0, x1))) → A__SEL(a__fib(mark(y0)), cons(mark(x0), x1))
A__FIB(sel(x0, x1)) → A__SEL(a__sel(mark(x0), mark(x1)), a__fib1(s(0), s(0)))
MARK(add(s(x0), y1)) → A__ADD(s(mark(x0)), mark(y1))
MARK(sel(sel(y0, y1), add(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), a__add(mark(x0), mark(x1)))
MARK(sel(add(y0, y1), sel(x0, x1))) → A__SEL(a__add(mark(y0), mark(y1)), a__sel(mark(x0), mark(x1)))
MARK(sel(s(y0), fib(x0))) → A__SEL(s(mark(y0)), a__fib(mark(x0)))
MARK(sel(fib1(y0, y1), fib(x0))) → A__SEL(a__fib1(mark(y0), mark(y1)), a__fib(mark(x0)))
MARK(add(sel(x0, x1), y1)) → A__ADD(a__sel(mark(x0), mark(x1)), mark(y1))
MARK(sel(fib(y0), fib1(x0, x1))) → A__SEL(a__fib(mark(y0)), a__fib1(mark(x0), mark(x1)))
MARK(sel(fib1(y0, y1), cons(x0, x1))) → A__SEL(a__fib1(mark(y0), mark(y1)), cons(mark(x0), x1))
A__SEL(s(add(x0, x1)), cons(y1, y2)) → A__SEL(a__add(mark(x0), mark(x1)), mark(y2))
MARK(sel(X1, X2)) → MARK(X1)
MARK(sel(sel(y0, y1), cons(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), cons(mark(x0), x1))
MARK(sel(fib(y0), add(x0, x1))) → A__SEL(a__fib(mark(y0)), a__add(mark(x0), mark(x1)))
MARK(sel(add(y0, y1), fib(x0))) → A__SEL(a__add(mark(y0), mark(y1)), a__fib(mark(x0)))
MARK(add(add(x0, x1), y1)) → A__ADD(a__add(mark(x0), mark(x1)), mark(y1))
A__ADD(s(X), Y) → MARK(X)
A__FIB(N) → A__FIB1(s(0), s(0))
MARK(add(fib(x0), y1)) → A__ADD(a__fib(mark(x0)), mark(y1))
A__FIB(fib(x0)) → A__SEL(a__fib(mark(x0)), a__fib1(s(0), s(0)))
A__ADD(s(fib(x0)), y1) → A__ADD(a__fib(mark(x0)), mark(y1))
A__SEL(s(0), cons(y1, y2)) → A__SEL(0, mark(y2))
MARK(sel(0, cons(x0, x1))) → A__SEL(0, cons(mark(x0), x1))
MARK(sel(add(y0, y1), cons(x0, x1))) → A__SEL(a__add(mark(y0), mark(y1)), cons(mark(x0), x1))
A__ADD(s(fib1(x0, x1)), y1) → A__ADD(a__fib1(mark(x0), mark(x1)), mark(y1))
MARK(fib(X)) → MARK(X)
A__SEL(s(fib1(x0, x1)), cons(y1, y2)) → A__SEL(a__fib1(mark(x0), mark(x1)), mark(y2))
A__ADD(0, X) → MARK(X)
A__FIB1(X, Y) → MARK(X)
MARK(s(X)) → MARK(X)
MARK(add(X1, X2)) → MARK(X2)
MARK(sel(s(y0), sel(x0, x1))) → A__SEL(s(mark(y0)), a__sel(mark(x0), mark(x1)))
MARK(fib(X)) → A__FIB(mark(X))
MARK(fib1(X1, X2)) → MARK(X2)
A__FIB(0) → A__SEL(0, a__fib1(s(0), s(0)))
MARK(sel(sel(y0, y1), fib(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__fib(mark(x0)))
MARK(sel(fib(y0), sel(x0, x1))) → A__SEL(a__fib(mark(y0)), a__sel(mark(x0), mark(x1)))
A__ADD(s(sel(x0, x1)), y1) → A__ADD(a__sel(mark(x0), mark(x1)), mark(y1))
A__SEL(0, cons(X, XS)) → MARK(X)
MARK(add(X1, X2)) → MARK(X1)
MARK(add(fib1(x0, x1), y1)) → A__ADD(a__fib1(mark(x0), mark(x1)), mark(y1))
MARK(sel(0, fib1(x0, x1))) → A__SEL(0, a__fib1(mark(x0), mark(x1)))
A__FIB(fib1(x0, x1)) → A__SEL(a__fib1(mark(x0), mark(x1)), a__fib1(s(0), s(0)))
A__SEL(s(fib(x0)), cons(y1, y2)) → A__SEL(a__fib(mark(x0)), mark(y2))
A__SEL(s(N), cons(X, XS)) → MARK(N)
A__ADD(s(s(x0)), y1) → A__ADD(s(mark(x0)), mark(y1))
MARK(fib1(X1, X2)) → A__FIB1(mark(X1), mark(X2))
MARK(sel(s(y0), fib1(x0, x1))) → A__SEL(s(mark(y0)), a__fib1(mark(x0), mark(x1)))
A__SEL(s(s(x0)), cons(y1, y2)) → A__SEL(s(mark(x0)), mark(y2))
MARK(sel(fib(y0), fib(x0))) → A__SEL(a__fib(mark(y0)), a__fib(mark(x0)))
MARK(sel(sel(y0, y1), fib1(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), a__fib1(mark(x0), mark(x1)))
MARK(sel(fib1(y0, y1), sel(x0, x1))) → A__SEL(a__fib1(mark(y0), mark(y1)), a__sel(mark(x0), mark(x1)))
MARK(sel(add(y0, y1), add(x0, x1))) → A__SEL(a__add(mark(y0), mark(y1)), a__add(mark(x0), mark(x1)))
MARK(sel(s(y0), add(x0, x1))) → A__SEL(s(mark(y0)), a__add(mark(x0), mark(x1)))
MARK(sel(0, sel(x0, x1))) → A__SEL(0, a__sel(mark(x0), mark(x1)))
A__FIB(add(x0, x1)) → A__SEL(a__add(mark(x0), mark(x1)), a__fib1(s(0), s(0)))
MARK(sel(X1, X2)) → MARK(X2)

The TRS R consists of the following rules:

a__fib(N) → a__sel(mark(N), a__fib1(s(0), s(0)))
a__fib1(X, Y) → cons(mark(X), fib1(Y, add(X, Y)))
a__add(0, X) → mark(X)
a__add(s(X), Y) → s(a__add(mark(X), mark(Y)))
a__sel(0, cons(X, XS)) → mark(X)
a__sel(s(N), cons(X, XS)) → a__sel(mark(N), mark(XS))
mark(fib(X)) → a__fib(mark(X))
mark(sel(X1, X2)) → a__sel(mark(X1), mark(X2))
mark(fib1(X1, X2)) → a__fib1(mark(X1), mark(X2))
mark(add(X1, X2)) → a__add(mark(X1), mark(X2))
mark(s(X)) → s(mark(X))
mark(0) → 0
mark(cons(X1, X2)) → cons(mark(X1), X2)
a__fib(X) → fib(X)
a__sel(X1, X2) → sel(X1, X2)
a__fib1(X1, X2) → fib1(X1, X2)
a__add(X1, X2) → add(X1, X2)

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

A__SEL(s(add(y0, y1)), cons(y2, add(x0, x1))) → A__SEL(a__add(mark(y0), mark(y1)), a__add(mark(x0), mark(x1)))
A__SEL(s(add(y0, y1)), cons(y2, fib(x0))) → A__SEL(a__add(mark(y0), mark(y1)), a__fib(mark(x0)))
A__SEL(s(add(y0, y1)), cons(y2, 0)) → A__SEL(a__add(mark(y0), mark(y1)), 0)
A__SEL(s(add(y0, y1)), cons(y2, s(x0))) → A__SEL(a__add(mark(y0), mark(y1)), s(mark(x0)))
A__SEL(s(add(y0, y1)), cons(y2, cons(x0, x1))) → A__SEL(a__add(mark(y0), mark(y1)), cons(mark(x0), x1))
A__SEL(s(add(y0, y1)), cons(y2, sel(x0, x1))) → A__SEL(a__add(mark(y0), mark(y1)), a__sel(mark(x0), mark(x1)))
A__SEL(s(add(y0, y1)), cons(y2, fib1(x0, x1))) → A__SEL(a__add(mark(y0), mark(y1)), a__fib1(mark(x0), mark(x1)))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ Narrowing

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ QDP

              ↳ Narrowing

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ QDP

                      ↳ Narrowing

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ QDP

                              ↳ Narrowing

                                ↳ QDP

                                  ↳ DependencyGraphProof

                                    ↳ QDP

                                      ↳ Narrowing

                                        ↳ QDP

                                          ↳ DependencyGraphProof

                                            ↳ QDP

                                              ↳ Narrowing

                                                ↳ QDP

                                                  ↳ DependencyGraphProof

                                                    ↳ QDP

                                                      ↳ Narrowing

                                                        ↳ QDP

                                                          ↳ DependencyGraphProof

                                                            ↳ QDP

                                                              ↳ Narrowing

                                                                ↳ QDP

                                                                  ↳ DependencyGraphProof

                                                                    ↳ QDP

                                                                      ↳ Narrowing

                                                                        ↳ QDP

                                                                          ↳ DependencyGraphProof

                                                                            ↳ QDP

                                                                              ↳ Narrowing

                                                                                ↳ QDP

                                                                                  ↳ DependencyGraphProof

                                                                                    ↳ QDP

                                                                                      ↳ Narrowing

                                                                                        ↳ QDP

                                                                                          ↳ DependencyGraphProof

                                                                                            ↳ QDP

                                                                                              ↳ Narrowing

                                                                                                ↳ QDP

                                                                                                  ↳ DependencyGraphProof

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

MARK(sel(sel(y0, y1), sel(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), a__sel(mark(x0), mark(x1)))
MARK(sel(fib1(y0, y1), fib1(x0, x1))) → A__SEL(a__fib1(mark(y0), mark(y1)), a__fib1(mark(x0), mark(x1)))
MARK(fib1(X1, X2)) → MARK(X1)
MARK(sel(s(y0), cons(x0, x1))) → A__SEL(s(mark(y0)), cons(mark(x0), x1))
MARK(cons(X1, X2)) → MARK(X1)
A__ADD(s(add(x0, x1)), y1) → A__ADD(a__add(mark(x0), mark(x1)), mark(y1))
MARK(sel(0, add(x0, x1))) → A__SEL(0, a__add(mark(x0), mark(x1)))
MARK(sel(0, fib(x0))) → A__SEL(0, a__fib(mark(x0)))
A__FIB(N) → MARK(N)
MARK(sel(add(y0, y1), fib1(x0, x1))) → A__SEL(a__add(mark(y0), mark(y1)), a__fib1(mark(x0), mark(x1)))
MARK(sel(fib1(y0, y1), add(x0, x1))) → A__SEL(a__fib1(mark(y0), mark(y1)), a__add(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(N), cons(X, XS)) → MARK(XS)
A__FIB(s(x0)) → A__SEL(s(mark(x0)), a__fib1(s(0), s(0)))
A__ADD(s(X), Y) → MARK(Y)
A__ADD(s(0), y1) → A__ADD(0, mark(y1))
MARK(add(0, y1)) → A__ADD(0, mark(y1))
MARK(sel(fib(y0), cons(x0, x1))) → A__SEL(a__fib(mark(y0)), cons(mark(x0), x1))
A__FIB(sel(x0, x1)) → A__SEL(a__sel(mark(x0), mark(x1)), a__fib1(s(0), s(0)))
MARK(add(s(x0), y1)) → A__ADD(s(mark(x0)), mark(y1))
MARK(sel(add(y0, y1), sel(x0, x1))) → A__SEL(a__add(mark(y0), mark(y1)), a__sel(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), add(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), a__add(mark(x0), mark(x1)))
MARK(sel(fib(y0), fib1(x0, x1))) → A__SEL(a__fib(mark(y0)), a__fib1(mark(x0), mark(x1)))
MARK(add(sel(x0, x1), y1)) → A__ADD(a__sel(mark(x0), mark(x1)), mark(y1))
MARK(sel(fib1(y0, y1), fib(x0))) → A__SEL(a__fib1(mark(y0), mark(y1)), a__fib(mark(x0)))
MARK(sel(s(y0), fib(x0))) → A__SEL(s(mark(y0)), a__fib(mark(x0)))
MARK(sel(fib1(y0, y1), cons(x0, x1))) → A__SEL(a__fib1(mark(y0), mark(y1)), cons(mark(x0), x1))
A__SEL(s(add(y0, y1)), cons(y2, add(x0, x1))) → A__SEL(a__add(mark(y0), mark(y1)), a__add(mark(x0), mark(x1)))
MARK(sel(X1, X2)) → MARK(X1)
MARK(sel(sel(y0, y1), cons(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), cons(mark(x0), x1))
MARK(sel(add(y0, y1), fib(x0))) → A__SEL(a__add(mark(y0), mark(y1)), a__fib(mark(x0)))
MARK(sel(fib(y0), add(x0, x1))) → A__SEL(a__fib(mark(y0)), a__add(mark(x0), mark(x1)))
MARK(add(add(x0, x1), y1)) → A__ADD(a__add(mark(x0), mark(x1)), mark(y1))
A__ADD(s(X), Y) → MARK(X)
A__FIB(N) → A__FIB1(s(0), s(0))
MARK(add(fib(x0), y1)) → A__ADD(a__fib(mark(x0)), mark(y1))
A__SEL(s(add(y0, y1)), cons(y2, 0)) → A__SEL(a__add(mark(y0), mark(y1)), 0)
A__FIB(fib(x0)) → A__SEL(a__fib(mark(x0)), a__fib1(s(0), s(0)))
A__ADD(s(fib(x0)), y1) → A__ADD(a__fib(mark(x0)), mark(y1))
A__SEL(s(0), cons(y1, y2)) → A__SEL(0, mark(y2))
A__SEL(s(add(y0, y1)), cons(y2, cons(x0, x1))) → A__SEL(a__add(mark(y0), mark(y1)), cons(mark(x0), x1))
MARK(sel(0, cons(x0, x1))) → A__SEL(0, cons(mark(x0), x1))
MARK(sel(add(y0, y1), cons(x0, x1))) → A__SEL(a__add(mark(y0), mark(y1)), cons(mark(x0), x1))
A__ADD(s(fib1(x0, x1)), y1) → A__ADD(a__fib1(mark(x0), mark(x1)), mark(y1))
MARK(fib(X)) → MARK(X)
A__SEL(s(add(y0, y1)), cons(y2, fib(x0))) → A__SEL(a__add(mark(y0), mark(y1)), a__fib(mark(x0)))
A__SEL(s(fib1(x0, x1)), cons(y1, y2)) → A__SEL(a__fib1(mark(x0), mark(x1)), mark(y2))
A__FIB1(X, Y) → MARK(X)
A__ADD(0, X) → MARK(X)
MARK(s(X)) → MARK(X)
MARK(add(X1, X2)) → MARK(X2)
MARK(fib(X)) → A__FIB(mark(X))
MARK(sel(s(y0), sel(x0, x1))) → A__SEL(s(mark(y0)), a__sel(mark(x0), mark(x1)))
MARK(fib1(X1, X2)) → MARK(X2)
MARK(sel(fib(y0), sel(x0, x1))) → A__SEL(a__fib(mark(y0)), a__sel(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), fib(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__fib(mark(x0)))
A__FIB(0) → A__SEL(0, a__fib1(s(0), s(0)))
A__SEL(0, cons(X, XS)) → MARK(X)
A__ADD(s(sel(x0, x1)), y1) → A__ADD(a__sel(mark(x0), mark(x1)), mark(y1))
MARK(add(X1, X2)) → MARK(X1)
MARK(add(fib1(x0, x1), y1)) → A__ADD(a__fib1(mark(x0), mark(x1)), mark(y1))
MARK(sel(0, fib1(x0, x1))) → A__SEL(0, a__fib1(mark(x0), mark(x1)))
A__FIB(fib1(x0, x1)) → A__SEL(a__fib1(mark(x0), mark(x1)), a__fib1(s(0), s(0)))
A__SEL(s(add(y0, y1)), cons(y2, fib1(x0, x1))) → A__SEL(a__add(mark(y0), mark(y1)), a__fib1(mark(x0), mark(x1)))
A__SEL(s(fib(x0)), cons(y1, y2)) → A__SEL(a__fib(mark(x0)), mark(y2))
A__SEL(s(N), cons(X, XS)) → MARK(N)
A__ADD(s(s(x0)), y1) → A__ADD(s(mark(x0)), mark(y1))
MARK(fib1(X1, X2)) → A__FIB1(mark(X1), mark(X2))
A__SEL(s(s(x0)), cons(y1, y2)) → A__SEL(s(mark(x0)), mark(y2))
MARK(sel(s(y0), fib1(x0, x1))) → A__SEL(s(mark(y0)), a__fib1(mark(x0), mark(x1)))
MARK(sel(fib(y0), fib(x0))) → A__SEL(a__fib(mark(y0)), a__fib(mark(x0)))
MARK(sel(add(y0, y1), add(x0, x1))) → A__SEL(a__add(mark(y0), mark(y1)), a__add(mark(x0), mark(x1)))
MARK(sel(fib1(y0, y1), sel(x0, x1))) → A__SEL(a__fib1(mark(y0), mark(y1)), a__sel(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), fib1(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), a__fib1(mark(x0), mark(x1)))
MARK(sel(s(y0), add(x0, x1))) → A__SEL(s(mark(y0)), a__add(mark(x0), mark(x1)))
A__SEL(s(add(y0, y1)), cons(y2, s(x0))) → A__SEL(a__add(mark(y0), mark(y1)), s(mark(x0)))
MARK(sel(0, sel(x0, x1))) → A__SEL(0, a__sel(mark(x0), mark(x1)))
A__SEL(s(add(y0, y1)), cons(y2, sel(x0, x1))) → A__SEL(a__add(mark(y0), mark(y1)), a__sel(mark(x0), mark(x1)))
A__FIB(add(x0, x1)) → A__SEL(a__add(mark(x0), mark(x1)), a__fib1(s(0), s(0)))
MARK(sel(X1, X2)) → MARK(X2)

The TRS R consists of the following rules:

a__fib(N) → a__sel(mark(N), a__fib1(s(0), s(0)))
a__fib1(X, Y) → cons(mark(X), fib1(Y, add(X, Y)))
a__add(0, X) → mark(X)
a__add(s(X), Y) → s(a__add(mark(X), mark(Y)))
a__sel(0, cons(X, XS)) → mark(X)
a__sel(s(N), cons(X, XS)) → a__sel(mark(N), mark(XS))
mark(fib(X)) → a__fib(mark(X))
mark(sel(X1, X2)) → a__sel(mark(X1), mark(X2))
mark(fib1(X1, X2)) → a__fib1(mark(X1), mark(X2))
mark(add(X1, X2)) → a__add(mark(X1), mark(X2))
mark(s(X)) → s(mark(X))
mark(0) → 0
mark(cons(X1, X2)) → cons(mark(X1), X2)
a__fib(X) → fib(X)
a__sel(X1, X2) → sel(X1, X2)
a__fib1(X1, X2) → fib1(X1, X2)
a__add(X1, X2) → add(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 1 SCC with 2 less nodes.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ Narrowing

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ QDP

              ↳ Narrowing

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ QDP

                      ↳ Narrowing

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ QDP

                              ↳ Narrowing

                                ↳ QDP

                                  ↳ DependencyGraphProof

                                    ↳ QDP

                                      ↳ Narrowing

                                        ↳ QDP

                                          ↳ DependencyGraphProof

                                            ↳ QDP

                                              ↳ Narrowing

                                                ↳ QDP

                                                  ↳ DependencyGraphProof

                                                    ↳ QDP

                                                      ↳ Narrowing

                                                        ↳ QDP

                                                          ↳ DependencyGraphProof

                                                            ↳ QDP

                                                              ↳ Narrowing

                                                                ↳ QDP

                                                                  ↳ DependencyGraphProof

                                                                    ↳ QDP

                                                                      ↳ Narrowing

                                                                        ↳ QDP

                                                                          ↳ DependencyGraphProof

                                                                            ↳ QDP

                                                                              ↳ Narrowing

                                                                                ↳ QDP

                                                                                  ↳ DependencyGraphProof

                                                                                    ↳ QDP

                                                                                      ↳ Narrowing

                                                                                        ↳ QDP

                                                                                          ↳ DependencyGraphProof

                                                                                            ↳ QDP

                                                                                              ↳ Narrowing

                                                                                                ↳ QDP

                                                                                                  ↳ DependencyGraphProof

                                                                                                    ↳ QDP

                                                                                                      ↳ Narrowing

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

MARK(sel(sel(y0, y1), sel(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), a__sel(mark(x0), mark(x1)))
MARK(sel(fib1(y0, y1), fib1(x0, x1))) → A__SEL(a__fib1(mark(y0), mark(y1)), a__fib1(mark(x0), mark(x1)))
MARK(fib1(X1, X2)) → MARK(X1)
MARK(sel(s(y0), cons(x0, x1))) → A__SEL(s(mark(y0)), cons(mark(x0), x1))
MARK(cons(X1, X2)) → MARK(X1)
A__ADD(s(add(x0, x1)), y1) → A__ADD(a__add(mark(x0), mark(x1)), mark(y1))
MARK(sel(0, add(x0, x1))) → A__SEL(0, a__add(mark(x0), mark(x1)))
MARK(sel(0, fib(x0))) → A__SEL(0, a__fib(mark(x0)))
A__FIB(N) → MARK(N)
MARK(sel(fib1(y0, y1), add(x0, x1))) → A__SEL(a__fib1(mark(y0), mark(y1)), a__add(mark(x0), mark(x1)))
MARK(sel(add(y0, y1), fib1(x0, x1))) → A__SEL(a__add(mark(y0), mark(y1)), a__fib1(mark(x0), mark(x1)))
A__FIB(s(x0)) → A__SEL(s(mark(x0)), a__fib1(s(0), s(0)))
A__SEL(s(N), cons(X, XS)) → MARK(XS)
A__SEL(s(sel(x0, x1)), cons(y1, y2)) → A__SEL(a__sel(mark(x0), mark(x1)), mark(y2))
A__ADD(s(X), Y) → MARK(Y)
A__ADD(s(0), y1) → A__ADD(0, mark(y1))
MARK(add(0, y1)) → A__ADD(0, mark(y1))
MARK(sel(fib(y0), cons(x0, x1))) → A__SEL(a__fib(mark(y0)), cons(mark(x0), x1))
A__FIB(sel(x0, x1)) → A__SEL(a__sel(mark(x0), mark(x1)), a__fib1(s(0), s(0)))
MARK(add(s(x0), y1)) → A__ADD(s(mark(x0)), mark(y1))
MARK(sel(sel(y0, y1), add(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), a__add(mark(x0), mark(x1)))
MARK(sel(add(y0, y1), sel(x0, x1))) → A__SEL(a__add(mark(y0), mark(y1)), a__sel(mark(x0), mark(x1)))
MARK(sel(s(y0), fib(x0))) → A__SEL(s(mark(y0)), a__fib(mark(x0)))
MARK(sel(fib1(y0, y1), fib(x0))) → A__SEL(a__fib1(mark(y0), mark(y1)), a__fib(mark(x0)))
MARK(add(sel(x0, x1), y1)) → A__ADD(a__sel(mark(x0), mark(x1)), mark(y1))
MARK(sel(fib(y0), fib1(x0, x1))) → A__SEL(a__fib(mark(y0)), a__fib1(mark(x0), mark(x1)))
MARK(sel(fib1(y0, y1), cons(x0, x1))) → A__SEL(a__fib1(mark(y0), mark(y1)), cons(mark(x0), x1))
MARK(sel(X1, X2)) → MARK(X1)
A__SEL(s(add(y0, y1)), cons(y2, add(x0, x1))) → A__SEL(a__add(mark(y0), mark(y1)), a__add(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), cons(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), cons(mark(x0), x1))
MARK(sel(add(y0, y1), fib(x0))) → A__SEL(a__add(mark(y0), mark(y1)), a__fib(mark(x0)))
MARK(sel(fib(y0), add(x0, x1))) → A__SEL(a__fib(mark(y0)), a__add(mark(x0), mark(x1)))
MARK(add(add(x0, x1), y1)) → A__ADD(a__add(mark(x0), mark(x1)), mark(y1))
A__ADD(s(X), Y) → MARK(X)
A__FIB(N) → A__FIB1(s(0), s(0))
MARK(add(fib(x0), y1)) → A__ADD(a__fib(mark(x0)), mark(y1))
A__FIB(fib(x0)) → A__SEL(a__fib(mark(x0)), a__fib1(s(0), s(0)))
A__ADD(s(fib(x0)), y1) → A__ADD(a__fib(mark(x0)), mark(y1))
A__SEL(s(0), cons(y1, y2)) → A__SEL(0, mark(y2))
A__SEL(s(add(y0, y1)), cons(y2, cons(x0, x1))) → A__SEL(a__add(mark(y0), mark(y1)), cons(mark(x0), x1))
MARK(sel(0, cons(x0, x1))) → A__SEL(0, cons(mark(x0), x1))
MARK(sel(add(y0, y1), cons(x0, x1))) → A__SEL(a__add(mark(y0), mark(y1)), cons(mark(x0), x1))
A__ADD(s(fib1(x0, x1)), y1) → A__ADD(a__fib1(mark(x0), mark(x1)), mark(y1))
MARK(fib(X)) → MARK(X)
A__SEL(s(fib1(x0, x1)), cons(y1, y2)) → A__SEL(a__fib1(mark(x0), mark(x1)), mark(y2))
A__SEL(s(add(y0, y1)), cons(y2, fib(x0))) → A__SEL(a__add(mark(y0), mark(y1)), a__fib(mark(x0)))
A__ADD(0, X) → MARK(X)
A__FIB1(X, Y) → MARK(X)
MARK(s(X)) → MARK(X)
MARK(add(X1, X2)) → MARK(X2)
MARK(sel(s(y0), sel(x0, x1))) → A__SEL(s(mark(y0)), a__sel(mark(x0), mark(x1)))
MARK(fib(X)) → A__FIB(mark(X))
MARK(fib1(X1, X2)) → MARK(X2)
A__FIB(0) → A__SEL(0, a__fib1(s(0), s(0)))
MARK(sel(sel(y0, y1), fib(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__fib(mark(x0)))
MARK(sel(fib(y0), sel(x0, x1))) → A__SEL(a__fib(mark(y0)), a__sel(mark(x0), mark(x1)))
A__ADD(s(sel(x0, x1)), y1) → A__ADD(a__sel(mark(x0), mark(x1)), mark(y1))
A__SEL(0, cons(X, XS)) → MARK(X)
MARK(add(X1, X2)) → MARK(X1)
MARK(add(fib1(x0, x1), y1)) → A__ADD(a__fib1(mark(x0), mark(x1)), mark(y1))
MARK(sel(0, fib1(x0, x1))) → A__SEL(0, a__fib1(mark(x0), mark(x1)))
A__FIB(fib1(x0, x1)) → A__SEL(a__fib1(mark(x0), mark(x1)), a__fib1(s(0), s(0)))
A__SEL(s(fib(x0)), cons(y1, y2)) → A__SEL(a__fib(mark(x0)), mark(y2))
A__SEL(s(add(y0, y1)), cons(y2, fib1(x0, x1))) → A__SEL(a__add(mark(y0), mark(y1)), a__fib1(mark(x0), mark(x1)))
A__SEL(s(N), cons(X, XS)) → MARK(N)
A__ADD(s(s(x0)), y1) → A__ADD(s(mark(x0)), mark(y1))
MARK(fib1(X1, X2)) → A__FIB1(mark(X1), mark(X2))
MARK(sel(s(y0), fib1(x0, x1))) → A__SEL(s(mark(y0)), a__fib1(mark(x0), mark(x1)))
A__SEL(s(s(x0)), cons(y1, y2)) → A__SEL(s(mark(x0)), mark(y2))
MARK(sel(fib(y0), fib(x0))) → A__SEL(a__fib(mark(y0)), a__fib(mark(x0)))
MARK(sel(sel(y0, y1), fib1(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), a__fib1(mark(x0), mark(x1)))
MARK(sel(fib1(y0, y1), sel(x0, x1))) → A__SEL(a__fib1(mark(y0), mark(y1)), a__sel(mark(x0), mark(x1)))
MARK(sel(add(y0, y1), add(x0, x1))) → A__SEL(a__add(mark(y0), mark(y1)), a__add(mark(x0), mark(x1)))
MARK(sel(s(y0), add(x0, x1))) → A__SEL(s(mark(y0)), a__add(mark(x0), mark(x1)))
MARK(sel(0, sel(x0, x1))) → A__SEL(0, a__sel(mark(x0), mark(x1)))
A__SEL(s(add(y0, y1)), cons(y2, sel(x0, x1))) → A__SEL(a__add(mark(y0), mark(y1)), a__sel(mark(x0), mark(x1)))
A__FIB(add(x0, x1)) → A__SEL(a__add(mark(x0), mark(x1)), a__fib1(s(0), s(0)))
MARK(sel(X1, X2)) → MARK(X2)

The TRS R consists of the following rules:

a__fib(N) → a__sel(mark(N), a__fib1(s(0), s(0)))
a__fib1(X, Y) → cons(mark(X), fib1(Y, add(X, Y)))
a__add(0, X) → mark(X)
a__add(s(X), Y) → s(a__add(mark(X), mark(Y)))
a__sel(0, cons(X, XS)) → mark(X)
a__sel(s(N), cons(X, XS)) → a__sel(mark(N), mark(XS))
mark(fib(X)) → a__fib(mark(X))
mark(sel(X1, X2)) → a__sel(mark(X1), mark(X2))
mark(fib1(X1, X2)) → a__fib1(mark(X1), mark(X2))
mark(add(X1, X2)) → a__add(mark(X1), mark(X2))
mark(s(X)) → s(mark(X))
mark(0) → 0
mark(cons(X1, X2)) → cons(mark(X1), X2)
a__fib(X) → fib(X)
a__sel(X1, X2) → sel(X1, X2)
a__fib1(X1, X2) → fib1(X1, X2)
a__add(X1, X2) → add(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, fib(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__fib(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, add(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), a__add(mark(x0), mark(x1)))
A__SEL(s(sel(y0, y1)), cons(y2, fib1(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), a__fib1(mark(x0), mark(x1)))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ Narrowing

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ QDP

              ↳ Narrowing

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ QDP

                      ↳ Narrowing

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ QDP

                              ↳ Narrowing

                                ↳ QDP

                                  ↳ DependencyGraphProof

                                    ↳ QDP

                                      ↳ Narrowing

                                        ↳ QDP

                                          ↳ DependencyGraphProof

                                            ↳ QDP

                                              ↳ Narrowing

                                                ↳ QDP

                                                  ↳ DependencyGraphProof

                                                    ↳ QDP

                                                      ↳ Narrowing

                                                        ↳ QDP

                                                          ↳ DependencyGraphProof

                                                            ↳ QDP

                                                              ↳ Narrowing

                                                                ↳ QDP

                                                                  ↳ DependencyGraphProof

                                                                    ↳ QDP

                                                                      ↳ Narrowing

                                                                        ↳ QDP

                                                                          ↳ DependencyGraphProof

                                                                            ↳ QDP

                                                                              ↳ Narrowing

                                                                                ↳ QDP

                                                                                  ↳ DependencyGraphProof

                                                                                    ↳ QDP

                                                                                      ↳ Narrowing

                                                                                        ↳ QDP

                                                                                          ↳ DependencyGraphProof

                                                                                            ↳ QDP

                                                                                              ↳ Narrowing

                                                                                                ↳ QDP

                                                                                                  ↳ DependencyGraphProof

                                                                                                    ↳ QDP

                                                                                                      ↳ Narrowing

                                                                                                        ↳ QDP

                                                                                                          ↳ DependencyGraphProof

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

MARK(sel(sel(y0, y1), sel(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), a__sel(mark(x0), mark(x1)))
MARK(sel(fib1(y0, y1), fib1(x0, x1))) → A__SEL(a__fib1(mark(y0), mark(y1)), a__fib1(mark(x0), mark(x1)))
MARK(fib1(X1, X2)) → MARK(X1)
MARK(sel(s(y0), cons(x0, x1))) → A__SEL(s(mark(y0)), cons(mark(x0), x1))
MARK(cons(X1, X2)) → MARK(X1)
A__ADD(s(add(x0, x1)), y1) → A__ADD(a__add(mark(x0), mark(x1)), mark(y1))
MARK(sel(0, add(x0, x1))) → A__SEL(0, a__add(mark(x0), mark(x1)))
A__FIB(N) → MARK(N)
MARK(sel(0, fib(x0))) → A__SEL(0, a__fib(mark(x0)))
A__SEL(s(sel(y0, y1)), cons(y2, 0)) → A__SEL(a__sel(mark(y0), mark(y1)), 0)
MARK(sel(add(y0, y1), fib1(x0, x1))) → A__SEL(a__add(mark(y0), mark(y1)), a__fib1(mark(x0), mark(x1)))
MARK(sel(fib1(y0, y1), add(x0, x1))) → A__SEL(a__fib1(mark(y0), mark(y1)), a__add(mark(x0), mark(x1)))
A__SEL(s(N), cons(X, XS)) → MARK(XS)
A__FIB(s(x0)) → A__SEL(s(mark(x0)), a__fib1(s(0), s(0)))
A__ADD(s(X), Y) → MARK(Y)
A__ADD(s(0), y1) → A__ADD(0, mark(y1))
MARK(add(0, y1)) → A__ADD(0, mark(y1))
MARK(sel(fib(y0), cons(x0, x1))) → A__SEL(a__fib(mark(y0)), cons(mark(x0), x1))
A__FIB(sel(x0, x1)) → A__SEL(a__sel(mark(x0), mark(x1)), a__fib1(s(0), s(0)))
MARK(add(s(x0), y1)) → A__ADD(s(mark(x0)), mark(y1))
MARK(sel(add(y0, y1), sel(x0, x1))) → A__SEL(a__add(mark(y0), mark(y1)), a__sel(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), add(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), a__add(mark(x0), mark(x1)))
MARK(sel(fib(y0), fib1(x0, x1))) → A__SEL(a__fib(mark(y0)), a__fib1(mark(x0), mark(x1)))
MARK(add(sel(x0, x1), y1)) → A__ADD(a__sel(mark(x0), mark(x1)), mark(y1))
MARK(sel(fib1(y0, y1), fib(x0))) → A__SEL(a__fib1(mark(y0), mark(y1)), a__fib(mark(x0)))
MARK(sel(s(y0), fib(x0))) → A__SEL(s(mark(y0)), a__fib(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)))
MARK(sel(fib1(y0, y1), cons(x0, x1))) → A__SEL(a__fib1(mark(y0), mark(y1)), cons(mark(x0), x1))
A__SEL(s(add(y0, y1)), cons(y2, add(x0, x1))) → A__SEL(a__add(mark(y0), mark(y1)), a__add(mark(x0), mark(x1)))
MARK(sel(X1, X2)) → MARK(X1)
MARK(sel(sel(y0, y1), cons(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), cons(mark(x0), x1))
MARK(sel(fib(y0), add(x0, x1))) → A__SEL(a__fib(mark(y0)), a__add(mark(x0), mark(x1)))
MARK(sel(add(y0, y1), fib(x0))) → A__SEL(a__add(mark(y0), mark(y1)), a__fib(mark(x0)))
MARK(add(add(x0, x1), y1)) → A__ADD(a__add(mark(x0), mark(x1)), mark(y1))
A__ADD(s(X), Y) → MARK(X)
A__SEL(s(sel(y0, y1)), cons(y2, cons(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), cons(mark(x0), x1))
A__FIB(N) → A__FIB1(s(0), s(0))
MARK(add(fib(x0), y1)) → A__ADD(a__fib(mark(x0)), mark(y1))
A__FIB(fib(x0)) → A__SEL(a__fib(mark(x0)), a__fib1(s(0), s(0)))
A__ADD(s(fib(x0)), y1) → A__ADD(a__fib(mark(x0)), mark(y1))
A__SEL(s(0), cons(y1, y2)) → A__SEL(0, mark(y2))
A__SEL(s(add(y0, y1)), cons(y2, cons(x0, x1))) → A__SEL(a__add(mark(y0), mark(y1)), cons(mark(x0), x1))
MARK(sel(0, cons(x0, x1))) → A__SEL(0, cons(mark(x0), x1))
MARK(sel(add(y0, y1), cons(x0, x1))) → A__SEL(a__add(mark(y0), mark(y1)), cons(mark(x0), x1))
A__ADD(s(fib1(x0, x1)), y1) → A__ADD(a__fib1(mark(x0), mark(x1)), mark(y1))
A__SEL(s(sel(y0, y1)), cons(y2, fib(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__fib(mark(x0)))
MARK(fib(X)) → MARK(X)
A__SEL(s(add(y0, y1)), cons(y2, fib(x0))) → A__SEL(a__add(mark(y0), mark(y1)), a__fib(mark(x0)))
A__SEL(s(fib1(x0, x1)), cons(y1, y2)) → A__SEL(a__fib1(mark(x0), mark(x1)), mark(y2))
A__FIB1(X, Y) → MARK(X)
A__ADD(0, X) → MARK(X)
MARK(s(X)) → MARK(X)
MARK(add(X1, X2)) → MARK(X2)
MARK(fib(X)) → A__FIB(mark(X))
MARK(sel(s(y0), sel(x0, x1))) → A__SEL(s(mark(y0)), a__sel(mark(x0), mark(x1)))
MARK(fib1(X1, X2)) → MARK(X2)
MARK(sel(fib(y0), sel(x0, x1))) → A__SEL(a__fib(mark(y0)), a__sel(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), fib(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__fib(mark(x0)))
A__FIB(0) → A__SEL(0, a__fib1(s(0), s(0)))
A__SEL(0, cons(X, XS)) → MARK(X)
A__ADD(s(sel(x0, x1)), y1) → A__ADD(a__sel(mark(x0), mark(x1)), mark(y1))
MARK(add(X1, X2)) → MARK(X1)
MARK(add(fib1(x0, x1), y1)) → A__ADD(a__fib1(mark(x0), mark(x1)), mark(y1))
MARK(sel(0, fib1(x0, x1))) → A__SEL(0, a__fib1(mark(x0), mark(x1)))
A__FIB(fib1(x0, x1)) → A__SEL(a__fib1(mark(x0), mark(x1)), a__fib1(s(0), s(0)))
A__SEL(s(add(y0, y1)), cons(y2, fib1(x0, x1))) → A__SEL(a__add(mark(y0), mark(y1)), a__fib1(mark(x0), mark(x1)))
A__SEL(s(fib(x0)), cons(y1, y2)) → A__SEL(a__fib(mark(x0)), mark(y2))
A__SEL(s(sel(y0, y1)), cons(y2, fib1(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), a__fib1(mark(x0), mark(x1)))
A__SEL(s(N), cons(X, XS)) → MARK(N)
A__ADD(s(s(x0)), y1) → A__ADD(s(mark(x0)), mark(y1))
MARK(fib1(X1, X2)) → A__FIB1(mark(X1), mark(X2))
A__SEL(s(s(x0)), cons(y1, y2)) → A__SEL(s(mark(x0)), mark(y2))
MARK(sel(s(y0), fib1(x0, x1))) → A__SEL(s(mark(y0)), a__fib1(mark(x0), mark(x1)))
MARK(sel(fib(y0), fib(x0))) → A__SEL(a__fib(mark(y0)), a__fib(mark(x0)))
MARK(sel(add(y0, y1), add(x0, x1))) → A__SEL(a__add(mark(y0), mark(y1)), a__add(mark(x0), mark(x1)))
MARK(sel(fib1(y0, y1), sel(x0, x1))) → A__SEL(a__fib1(mark(y0), mark(y1)), a__sel(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), fib1(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), a__fib1(mark(x0), mark(x1)))
A__SEL(s(sel(y0, y1)), cons(y2, s(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), s(mark(x0)))
MARK(sel(s(y0), add(x0, x1))) → A__SEL(s(mark(y0)), a__add(mark(x0), mark(x1)))
MARK(sel(0, sel(x0, x1))) → A__SEL(0, a__sel(mark(x0), mark(x1)))
A__SEL(s(sel(y0, y1)), cons(y2, add(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), a__add(mark(x0), mark(x1)))
A__SEL(s(add(y0, y1)), cons(y2, sel(x0, x1))) → A__SEL(a__add(mark(y0), mark(y1)), a__sel(mark(x0), mark(x1)))
A__FIB(add(x0, x1)) → A__SEL(a__add(mark(x0), mark(x1)), a__fib1(s(0), s(0)))
MARK(sel(X1, X2)) → MARK(X2)

The TRS R consists of the following rules:

a__fib(N) → a__sel(mark(N), a__fib1(s(0), s(0)))
a__fib1(X, Y) → cons(mark(X), fib1(Y, add(X, Y)))
a__add(0, X) → mark(X)
a__add(s(X), Y) → s(a__add(mark(X), mark(Y)))
a__sel(0, cons(X, XS)) → mark(X)
a__sel(s(N), cons(X, XS)) → a__sel(mark(N), mark(XS))
mark(fib(X)) → a__fib(mark(X))
mark(sel(X1, X2)) → a__sel(mark(X1), mark(X2))
mark(fib1(X1, X2)) → a__fib1(mark(X1), mark(X2))
mark(add(X1, X2)) → a__add(mark(X1), mark(X2))
mark(s(X)) → s(mark(X))
mark(0) → 0
mark(cons(X1, X2)) → cons(mark(X1), X2)
a__fib(X) → fib(X)
a__sel(X1, X2) → sel(X1, X2)
a__fib1(X1, X2) → fib1(X1, X2)
a__add(X1, X2) → add(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 1 SCC with 2 less nodes.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ Narrowing

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ QDP

              ↳ Narrowing

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ QDP

                      ↳ Narrowing

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ QDP

                              ↳ Narrowing

                                ↳ QDP

                                  ↳ DependencyGraphProof

                                    ↳ QDP

                                      ↳ Narrowing

                                        ↳ QDP

                                          ↳ DependencyGraphProof

                                            ↳ QDP

                                              ↳ Narrowing

                                                ↳ QDP

                                                  ↳ DependencyGraphProof

                                                    ↳ QDP

                                                      ↳ Narrowing

                                                        ↳ QDP

                                                          ↳ DependencyGraphProof

                                                            ↳ QDP

                                                              ↳ Narrowing

                                                                ↳ QDP

                                                                  ↳ DependencyGraphProof

                                                                    ↳ QDP

                                                                      ↳ Narrowing

                                                                        ↳ QDP

                                                                          ↳ DependencyGraphProof

                                                                            ↳ QDP

                                                                              ↳ Narrowing

                                                                                ↳ QDP

                                                                                  ↳ DependencyGraphProof

                                                                                    ↳ QDP

                                                                                      ↳ Narrowing

                                                                                        ↳ QDP

                                                                                          ↳ DependencyGraphProof

                                                                                            ↳ QDP

                                                                                              ↳ Narrowing

                                                                                                ↳ QDP

                                                                                                  ↳ DependencyGraphProof

                                                                                                    ↳ QDP

                                                                                                      ↳ Narrowing

                                                                                                        ↳ QDP

                                                                                                          ↳ DependencyGraphProof

                                                                                                            ↳ QDP

                                                                                                              ↳ Narrowing

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

MARK(sel(sel(y0, y1), sel(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), a__sel(mark(x0), mark(x1)))
MARK(sel(fib1(y0, y1), fib1(x0, x1))) → A__SEL(a__fib1(mark(y0), mark(y1)), a__fib1(mark(x0), mark(x1)))
MARK(fib1(X1, X2)) → MARK(X1)
MARK(sel(s(y0), cons(x0, x1))) → A__SEL(s(mark(y0)), cons(mark(x0), x1))
MARK(cons(X1, X2)) → MARK(X1)
A__ADD(s(add(x0, x1)), y1) → A__ADD(a__add(mark(x0), mark(x1)), mark(y1))
MARK(sel(0, add(x0, x1))) → A__SEL(0, a__add(mark(x0), mark(x1)))
MARK(sel(0, fib(x0))) → A__SEL(0, a__fib(mark(x0)))
A__FIB(N) → MARK(N)
MARK(sel(fib1(y0, y1), add(x0, x1))) → A__SEL(a__fib1(mark(y0), mark(y1)), a__add(mark(x0), mark(x1)))
MARK(sel(add(y0, y1), fib1(x0, x1))) → A__SEL(a__add(mark(y0), mark(y1)), a__fib1(mark(x0), mark(x1)))
A__FIB(s(x0)) → A__SEL(s(mark(x0)), a__fib1(s(0), s(0)))
A__SEL(s(N), cons(X, XS)) → MARK(XS)
A__ADD(s(X), Y) → MARK(Y)
A__ADD(s(0), y1) → A__ADD(0, mark(y1))
MARK(add(0, y1)) → A__ADD(0, mark(y1))
MARK(sel(fib(y0), cons(x0, x1))) → A__SEL(a__fib(mark(y0)), cons(mark(x0), x1))
A__FIB(sel(x0, x1)) → A__SEL(a__sel(mark(x0), mark(x1)), a__fib1(s(0), s(0)))
MARK(add(s(x0), y1)) → A__ADD(s(mark(x0)), mark(y1))
MARK(sel(sel(y0, y1), add(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), a__add(mark(x0), mark(x1)))
MARK(sel(add(y0, y1), sel(x0, x1))) → A__SEL(a__add(mark(y0), mark(y1)), a__sel(mark(x0), mark(x1)))
MARK(sel(s(y0), fib(x0))) → A__SEL(s(mark(y0)), a__fib(mark(x0)))
MARK(sel(fib1(y0, y1), fib(x0))) → A__SEL(a__fib1(mark(y0), mark(y1)), a__fib(mark(x0)))
MARK(sel(fib(y0), fib1(x0, x1))) → A__SEL(a__fib(mark(y0)), a__fib1(mark(x0), mark(x1)))
MARK(add(sel(x0, x1), y1)) → A__ADD(a__sel(mark(x0), mark(x1)), mark(y1))
MARK(sel(fib1(y0, y1), cons(x0, x1))) → A__SEL(a__fib1(mark(y0), mark(y1)), cons(mark(x0), 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)))
MARK(sel(X1, X2)) → MARK(X1)
A__SEL(s(add(y0, y1)), cons(y2, add(x0, x1))) → A__SEL(a__add(mark(y0), mark(y1)), a__add(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), cons(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), cons(mark(x0), x1))
MARK(sel(add(y0, y1), fib(x0))) → A__SEL(a__add(mark(y0), mark(y1)), a__fib(mark(x0)))
MARK(sel(fib(y0), add(x0, x1))) → A__SEL(a__fib(mark(y0)), a__add(mark(x0), mark(x1)))
MARK(add(add(x0, x1), y1)) → A__ADD(a__add(mark(x0), mark(x1)), mark(y1))
A__ADD(s(X), Y) → MARK(X)
A__SEL(s(sel(y0, y1)), cons(y2, cons(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), cons(mark(x0), x1))
A__FIB(N) → A__FIB1(s(0), s(0))
MARK(add(fib(x0), y1)) → A__ADD(a__fib(mark(x0)), mark(y1))
A__FIB(fib(x0)) → A__SEL(a__fib(mark(x0)), a__fib1(s(0), s(0)))
A__ADD(s(fib(x0)), y1) → A__ADD(a__fib(mark(x0)), mark(y1))
A__SEL(s(0), cons(y1, y2)) → A__SEL(0, mark(y2))
A__SEL(s(add(y0, y1)), cons(y2, cons(x0, x1))) → A__SEL(a__add(mark(y0), mark(y1)), cons(mark(x0), x1))
MARK(sel(0, cons(x0, x1))) → A__SEL(0, cons(mark(x0), x1))
MARK(sel(add(y0, y1), cons(x0, x1))) → A__SEL(a__add(mark(y0), mark(y1)), cons(mark(x0), x1))
A__ADD(s(fib1(x0, x1)), y1) → A__ADD(a__fib1(mark(x0), mark(x1)), mark(y1))
MARK(fib(X)) → MARK(X)
A__SEL(s(sel(y0, y1)), cons(y2, fib(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__fib(mark(x0)))
A__SEL(s(fib1(x0, x1)), cons(y1, y2)) → A__SEL(a__fib1(mark(x0), mark(x1)), mark(y2))
A__SEL(s(add(y0, y1)), cons(y2, fib(x0))) → A__SEL(a__add(mark(y0), mark(y1)), a__fib(mark(x0)))
A__ADD(0, X) → MARK(X)
A__FIB1(X, Y) → MARK(X)
MARK(s(X)) → MARK(X)
MARK(add(X1, X2)) → MARK(X2)
MARK(sel(s(y0), sel(x0, x1))) → A__SEL(s(mark(y0)), a__sel(mark(x0), mark(x1)))
MARK(fib(X)) → A__FIB(mark(X))
MARK(fib1(X1, X2)) → MARK(X2)
A__FIB(0) → A__SEL(0, a__fib1(s(0), s(0)))
MARK(sel(sel(y0, y1), fib(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__fib(mark(x0)))
MARK(sel(fib(y0), sel(x0, x1))) → A__SEL(a__fib(mark(y0)), a__sel(mark(x0), mark(x1)))
A__ADD(s(sel(x0, x1)), y1) → A__ADD(a__sel(mark(x0), mark(x1)), mark(y1))
A__SEL(0, cons(X, XS)) → MARK(X)
MARK(add(X1, X2)) → MARK(X1)
MARK(add(fib1(x0, x1), y1)) → A__ADD(a__fib1(mark(x0), mark(x1)), mark(y1))
MARK(sel(0, fib1(x0, x1))) → A__SEL(0, a__fib1(mark(x0), mark(x1)))
A__FIB(fib1(x0, x1)) → A__SEL(a__fib1(mark(x0), mark(x1)), a__fib1(s(0), s(0)))
A__SEL(s(fib(x0)), cons(y1, y2)) → A__SEL(a__fib(mark(x0)), mark(y2))
A__SEL(s(add(y0, y1)), cons(y2, fib1(x0, x1))) → A__SEL(a__add(mark(y0), mark(y1)), a__fib1(mark(x0), mark(x1)))
A__SEL(s(sel(y0, y1)), cons(y2, fib1(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), a__fib1(mark(x0), mark(x1)))
A__SEL(s(N), cons(X, XS)) → MARK(N)
A__ADD(s(s(x0)), y1) → A__ADD(s(mark(x0)), mark(y1))
MARK(fib1(X1, X2)) → A__FIB1(mark(X1), mark(X2))
MARK(sel(s(y0), fib1(x0, x1))) → A__SEL(s(mark(y0)), a__fib1(mark(x0), mark(x1)))
A__SEL(s(s(x0)), cons(y1, y2)) → A__SEL(s(mark(x0)), mark(y2))
MARK(sel(fib(y0), fib(x0))) → A__SEL(a__fib(mark(y0)), a__fib(mark(x0)))
MARK(sel(sel(y0, y1), fib1(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), a__fib1(mark(x0), mark(x1)))
MARK(sel(fib1(y0, y1), sel(x0, x1))) → A__SEL(a__fib1(mark(y0), mark(y1)), a__sel(mark(x0), mark(x1)))
MARK(sel(add(y0, y1), add(x0, x1))) → A__SEL(a__add(mark(y0), mark(y1)), a__add(mark(x0), mark(x1)))
MARK(sel(s(y0), add(x0, x1))) → A__SEL(s(mark(y0)), a__add(mark(x0), mark(x1)))
MARK(sel(0, sel(x0, x1))) → A__SEL(0, a__sel(mark(x0), mark(x1)))
A__SEL(s(add(y0, y1)), cons(y2, sel(x0, x1))) → A__SEL(a__add(mark(y0), mark(y1)), a__sel(mark(x0), mark(x1)))
A__SEL(s(sel(y0, y1)), cons(y2, add(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), a__add(mark(x0), mark(x1)))
A__FIB(add(x0, x1)) → A__SEL(a__add(mark(x0), mark(x1)), a__fib1(s(0), s(0)))
MARK(sel(X1, X2)) → MARK(X2)

The TRS R consists of the following rules:

a__fib(N) → a__sel(mark(N), a__fib1(s(0), s(0)))
a__fib1(X, Y) → cons(mark(X), fib1(Y, add(X, Y)))
a__add(0, X) → mark(X)
a__add(s(X), Y) → s(a__add(mark(X), mark(Y)))
a__sel(0, cons(X, XS)) → mark(X)
a__sel(s(N), cons(X, XS)) → a__sel(mark(N), mark(XS))
mark(fib(X)) → a__fib(mark(X))
mark(sel(X1, X2)) → a__sel(mark(X1), mark(X2))
mark(fib1(X1, X2)) → a__fib1(mark(X1), mark(X2))
mark(add(X1, X2)) → a__add(mark(X1), mark(X2))
mark(s(X)) → s(mark(X))
mark(0) → 0
mark(cons(X1, X2)) → cons(mark(X1), X2)
a__fib(X) → fib(X)
a__sel(X1, X2) → sel(X1, X2)
a__fib1(X1, X2) → fib1(X1, X2)
a__add(X1, X2) → add(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule A__SEL(s(fib1(x0, x1)), cons(y1, y2)) → A__SEL(a__fib1(mark(x0), mark(x1)), mark(y2)) at position [1] we obtained the following new rules:

A__SEL(s(fib1(y0, y1)), cons(y2, 0)) → A__SEL(a__fib1(mark(y0), mark(y1)), 0)
A__SEL(s(fib1(y0, y1)), cons(y2, fib1(x0, x1))) → A__SEL(a__fib1(mark(y0), mark(y1)), a__fib1(mark(x0), mark(x1)))
A__SEL(s(fib1(y0, y1)), cons(y2, fib(x0))) → A__SEL(a__fib1(mark(y0), mark(y1)), a__fib(mark(x0)))
A__SEL(s(fib1(y0, y1)), cons(y2, add(x0, x1))) → A__SEL(a__fib1(mark(y0), mark(y1)), a__add(mark(x0), mark(x1)))
A__SEL(s(fib1(y0, y1)), cons(y2, cons(x0, x1))) → A__SEL(a__fib1(mark(y0), mark(y1)), cons(mark(x0), x1))
A__SEL(s(fib1(y0, y1)), cons(y2, s(x0))) → A__SEL(a__fib1(mark(y0), mark(y1)), s(mark(x0)))
A__SEL(s(fib1(y0, y1)), cons(y2, sel(x0, x1))) → A__SEL(a__fib1(mark(y0), mark(y1)), a__sel(mark(x0), mark(x1)))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ Narrowing

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ QDP

              ↳ Narrowing

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ QDP

                      ↳ Narrowing

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ QDP

                              ↳ Narrowing

                                ↳ QDP

                                  ↳ DependencyGraphProof

                                    ↳ QDP

                                      ↳ Narrowing

                                        ↳ QDP

                                          ↳ DependencyGraphProof

                                            ↳ QDP

                                              ↳ Narrowing

                                                ↳ QDP

                                                  ↳ DependencyGraphProof

                                                    ↳ QDP

                                                      ↳ Narrowing

                                                        ↳ QDP

                                                          ↳ DependencyGraphProof

                                                            ↳ QDP

                                                              ↳ Narrowing

                                                                ↳ QDP

                                                                  ↳ DependencyGraphProof

                                                                    ↳ QDP

                                                                      ↳ Narrowing

                                                                        ↳ QDP

                                                                          ↳ DependencyGraphProof

                                                                            ↳ QDP

                                                                              ↳ Narrowing

                                                                                ↳ QDP

                                                                                  ↳ DependencyGraphProof

                                                                                    ↳ QDP

                                                                                      ↳ Narrowing

                                                                                        ↳ QDP

                                                                                          ↳ DependencyGraphProof

                                                                                            ↳ QDP

                                                                                              ↳ Narrowing

                                                                                                ↳ QDP

                                                                                                  ↳ DependencyGraphProof

                                                                                                    ↳ QDP

                                                                                                      ↳ Narrowing

                                                                                                        ↳ QDP

                                                                                                          ↳ DependencyGraphProof

                                                                                                            ↳ QDP

                                                                                                              ↳ Narrowing

                                                                                                                ↳ QDP

                                                                                                                  ↳ DependencyGraphProof

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

MARK(sel(sel(y0, y1), sel(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), a__sel(mark(x0), mark(x1)))
MARK(sel(fib1(y0, y1), fib1(x0, x1))) → A__SEL(a__fib1(mark(y0), mark(y1)), a__fib1(mark(x0), mark(x1)))
MARK(fib1(X1, X2)) → MARK(X1)
A__SEL(s(fib1(y0, y1)), cons(y2, 0)) → A__SEL(a__fib1(mark(y0), mark(y1)), 0)
MARK(sel(s(y0), cons(x0, x1))) → A__SEL(s(mark(y0)), cons(mark(x0), x1))
MARK(cons(X1, X2)) → MARK(X1)
A__ADD(s(add(x0, x1)), y1) → A__ADD(a__add(mark(x0), mark(x1)), mark(y1))
A__SEL(s(fib1(y0, y1)), cons(y2, fib1(x0, x1))) → A__SEL(a__fib1(mark(y0), mark(y1)), a__fib1(mark(x0), mark(x1)))
MARK(sel(0, add(x0, x1))) → A__SEL(0, a__add(mark(x0), mark(x1)))
A__FIB(N) → MARK(N)
MARK(sel(0, fib(x0))) → A__SEL(0, a__fib(mark(x0)))
MARK(sel(add(y0, y1), fib1(x0, x1))) → A__SEL(a__add(mark(y0), mark(y1)), a__fib1(mark(x0), mark(x1)))
MARK(sel(fib1(y0, y1), add(x0, x1))) → A__SEL(a__fib1(mark(y0), mark(y1)), a__add(mark(x0), mark(x1)))
A__SEL(s(N), cons(X, XS)) → MARK(XS)
A__FIB(s(x0)) → A__SEL(s(mark(x0)), a__fib1(s(0), s(0)))
A__ADD(s(X), Y) → MARK(Y)
A__ADD(s(0), y1) → A__ADD(0, mark(y1))
MARK(add(0, y1)) → A__ADD(0, mark(y1))
MARK(sel(fib(y0), cons(x0, x1))) → A__SEL(a__fib(mark(y0)), cons(mark(x0), x1))
A__FIB(sel(x0, x1)) → A__SEL(a__sel(mark(x0), mark(x1)), a__fib1(s(0), s(0)))
MARK(add(s(x0), y1)) → A__ADD(s(mark(x0)), mark(y1))
MARK(sel(add(y0, y1), sel(x0, x1))) → A__SEL(a__add(mark(y0), mark(y1)), a__sel(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), add(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), a__add(mark(x0), mark(x1)))
MARK(add(sel(x0, x1), y1)) → A__ADD(a__sel(mark(x0), mark(x1)), mark(y1))
MARK(sel(fib(y0), fib1(x0, x1))) → A__SEL(a__fib(mark(y0)), a__fib1(mark(x0), mark(x1)))
MARK(sel(fib1(y0, y1), fib(x0))) → A__SEL(a__fib1(mark(y0), mark(y1)), a__fib(mark(x0)))
MARK(sel(s(y0), fib(x0))) → A__SEL(s(mark(y0)), a__fib(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)))
MARK(sel(fib1(y0, y1), cons(x0, x1))) → A__SEL(a__fib1(mark(y0), mark(y1)), cons(mark(x0), x1))
A__SEL(s(add(y0, y1)), cons(y2, add(x0, x1))) → A__SEL(a__add(mark(y0), mark(y1)), a__add(mark(x0), mark(x1)))
MARK(sel(X1, X2)) → MARK(X1)
MARK(sel(sel(y0, y1), cons(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), cons(mark(x0), x1))
MARK(sel(fib(y0), add(x0, x1))) → A__SEL(a__fib(mark(y0)), a__add(mark(x0), mark(x1)))
MARK(sel(add(y0, y1), fib(x0))) → A__SEL(a__add(mark(y0), mark(y1)), a__fib(mark(x0)))
MARK(add(add(x0, x1), y1)) → A__ADD(a__add(mark(x0), mark(x1)), mark(y1))
A__ADD(s(X), Y) → MARK(X)
A__SEL(s(sel(y0, y1)), cons(y2, cons(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), cons(mark(x0), x1))
A__FIB(N) → A__FIB1(s(0), s(0))
MARK(add(fib(x0), y1)) → A__ADD(a__fib(mark(x0)), mark(y1))
A__FIB(fib(x0)) → A__SEL(a__fib(mark(x0)), a__fib1(s(0), s(0)))
A__ADD(s(fib(x0)), y1) → A__ADD(a__fib(mark(x0)), mark(y1))
A__SEL(s(0), cons(y1, y2)) → A__SEL(0, mark(y2))
A__SEL(s(add(y0, y1)), cons(y2, cons(x0, x1))) → A__SEL(a__add(mark(y0), mark(y1)), cons(mark(x0), x1))
MARK(sel(0, cons(x0, x1))) → A__SEL(0, cons(mark(x0), x1))
MARK(sel(add(y0, y1), cons(x0, x1))) → A__SEL(a__add(mark(y0), mark(y1)), cons(mark(x0), x1))
A__ADD(s(fib1(x0, x1)), y1) → A__ADD(a__fib1(mark(x0), mark(x1)), mark(y1))
A__SEL(s(sel(y0, y1)), cons(y2, fib(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__fib(mark(x0)))
MARK(fib(X)) → MARK(X)
A__SEL(s(add(y0, y1)), cons(y2, fib(x0))) → A__SEL(a__add(mark(y0), mark(y1)), a__fib(mark(x0)))
A__FIB1(X, Y) → MARK(X)
A__ADD(0, X) → MARK(X)
MARK(s(X)) → MARK(X)
MARK(add(X1, X2)) → MARK(X2)
MARK(fib(X)) → A__FIB(mark(X))
MARK(sel(s(y0), sel(x0, x1))) → A__SEL(s(mark(y0)), a__sel(mark(x0), mark(x1)))
MARK(fib1(X1, X2)) → MARK(X2)
MARK(sel(fib(y0), sel(x0, x1))) → A__SEL(a__fib(mark(y0)), a__sel(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), fib(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__fib(mark(x0)))
A__FIB(0) → A__SEL(0, a__fib1(s(0), s(0)))
A__SEL(0, cons(X, XS)) → MARK(X)
A__ADD(s(sel(x0, x1)), y1) → A__ADD(a__sel(mark(x0), mark(x1)), mark(y1))
MARK(add(X1, X2)) → MARK(X1)
MARK(add(fib1(x0, x1), y1)) → A__ADD(a__fib1(mark(x0), mark(x1)), mark(y1))
MARK(sel(0, fib1(x0, x1))) → A__SEL(0, a__fib1(mark(x0), mark(x1)))
A__FIB(fib1(x0, x1)) → A__SEL(a__fib1(mark(x0), mark(x1)), a__fib1(s(0), s(0)))
A__SEL(s(fib1(y0, y1)), cons(y2, add(x0, x1))) → A__SEL(a__fib1(mark(y0), mark(y1)), a__add(mark(x0), mark(x1)))
A__SEL(s(add(y0, y1)), cons(y2, fib1(x0, x1))) → A__SEL(a__add(mark(y0), mark(y1)), a__fib1(mark(x0), mark(x1)))
A__SEL(s(fib(x0)), cons(y1, y2)) → A__SEL(a__fib(mark(x0)), mark(y2))
A__SEL(s(fib1(y0, y1)), cons(y2, cons(x0, x1))) → A__SEL(a__fib1(mark(y0), mark(y1)), cons(mark(x0), x1))
A__SEL(s(fib1(y0, y1)), cons(y2, sel(x0, x1))) → A__SEL(a__fib1(mark(y0), mark(y1)), a__sel(mark(x0), mark(x1)))
A__SEL(s(sel(y0, y1)), cons(y2, fib1(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), a__fib1(mark(x0), mark(x1)))
A__SEL(s(N), cons(X, XS)) → MARK(N)
A__ADD(s(s(x0)), y1) → A__ADD(s(mark(x0)), mark(y1))
MARK(fib1(X1, X2)) → A__FIB1(mark(X1), mark(X2))
A__SEL(s(s(x0)), cons(y1, y2)) → A__SEL(s(mark(x0)), mark(y2))
MARK(sel(s(y0), fib1(x0, x1))) → A__SEL(s(mark(y0)), a__fib1(mark(x0), mark(x1)))
MARK(sel(fib(y0), fib(x0))) → A__SEL(a__fib(mark(y0)), a__fib(mark(x0)))
MARK(sel(add(y0, y1), add(x0, x1))) → A__SEL(a__add(mark(y0), mark(y1)), a__add(mark(x0), mark(x1)))
MARK(sel(fib1(y0, y1), sel(x0, x1))) → A__SEL(a__fib1(mark(y0), mark(y1)), a__sel(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), fib1(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), a__fib1(mark(x0), mark(x1)))
MARK(sel(s(y0), add(x0, x1))) → A__SEL(s(mark(y0)), a__add(mark(x0), mark(x1)))
A__SEL(s(fib1(y0, y1)), cons(y2, fib(x0))) → A__SEL(a__fib1(mark(y0), mark(y1)), a__fib(mark(x0)))
MARK(sel(0, sel(x0, x1))) → A__SEL(0, a__sel(mark(x0), mark(x1)))
A__SEL(s(sel(y0, y1)), cons(y2, add(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), a__add(mark(x0), mark(x1)))
A__SEL(s(add(y0, y1)), cons(y2, sel(x0, x1))) → A__SEL(a__add(mark(y0), mark(y1)), a__sel(mark(x0), mark(x1)))
A__FIB(add(x0, x1)) → A__SEL(a__add(mark(x0), mark(x1)), a__fib1(s(0), s(0)))
A__SEL(s(fib1(y0, y1)), cons(y2, s(x0))) → A__SEL(a__fib1(mark(y0), mark(y1)), s(mark(x0)))
MARK(sel(X1, X2)) → MARK(X2)

The TRS R consists of the following rules:

a__fib(N) → a__sel(mark(N), a__fib1(s(0), s(0)))
a__fib1(X, Y) → cons(mark(X), fib1(Y, add(X, Y)))
a__add(0, X) → mark(X)
a__add(s(X), Y) → s(a__add(mark(X), mark(Y)))
a__sel(0, cons(X, XS)) → mark(X)
a__sel(s(N), cons(X, XS)) → a__sel(mark(N), mark(XS))
mark(fib(X)) → a__fib(mark(X))
mark(sel(X1, X2)) → a__sel(mark(X1), mark(X2))
mark(fib1(X1, X2)) → a__fib1(mark(X1), mark(X2))
mark(add(X1, X2)) → a__add(mark(X1), mark(X2))
mark(s(X)) → s(mark(X))
mark(0) → 0
mark(cons(X1, X2)) → cons(mark(X1), X2)
a__fib(X) → fib(X)
a__sel(X1, X2) → sel(X1, X2)
a__fib1(X1, X2) → fib1(X1, X2)
a__add(X1, X2) → add(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 1 SCC with 2 less nodes.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ Narrowing

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ QDP

              ↳ Narrowing

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ QDP

                      ↳ Narrowing

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ QDP

                              ↳ Narrowing

                                ↳ QDP

                                  ↳ DependencyGraphProof

                                    ↳ QDP

                                      ↳ Narrowing

                                        ↳ QDP

                                          ↳ DependencyGraphProof

                                            ↳ QDP

                                              ↳ Narrowing

                                                ↳ QDP

                                                  ↳ DependencyGraphProof

                                                    ↳ QDP

                                                      ↳ Narrowing

                                                        ↳ QDP

                                                          ↳ DependencyGraphProof

                                                            ↳ QDP

                                                              ↳ Narrowing

                                                                ↳ QDP

                                                                  ↳ DependencyGraphProof

                                                                    ↳ QDP

                                                                      ↳ Narrowing

                                                                        ↳ QDP

                                                                          ↳ DependencyGraphProof

                                                                            ↳ QDP

                                                                              ↳ Narrowing

                                                                                ↳ QDP

                                                                                  ↳ DependencyGraphProof

                                                                                    ↳ QDP

                                                                                      ↳ Narrowing

                                                                                        ↳ QDP

                                                                                          ↳ DependencyGraphProof

                                                                                            ↳ QDP

                                                                                              ↳ Narrowing

                                                                                                ↳ QDP

                                                                                                  ↳ DependencyGraphProof

                                                                                                    ↳ QDP

                                                                                                      ↳ Narrowing

                                                                                                        ↳ QDP

                                                                                                          ↳ DependencyGraphProof

                                                                                                            ↳ QDP

                                                                                                              ↳ Narrowing

                                                                                                                ↳ QDP

                                                                                                                  ↳ DependencyGraphProof

                                                                                                                    ↳ QDP

                                                                                                                      ↳ Narrowing

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

MARK(sel(sel(y0, y1), sel(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), a__sel(mark(x0), mark(x1)))
MARK(sel(fib1(y0, y1), fib1(x0, x1))) → A__SEL(a__fib1(mark(y0), mark(y1)), a__fib1(mark(x0), mark(x1)))
MARK(fib1(X1, X2)) → MARK(X1)
MARK(sel(s(y0), cons(x0, x1))) → A__SEL(s(mark(y0)), cons(mark(x0), x1))
MARK(cons(X1, X2)) → MARK(X1)
A__ADD(s(add(x0, x1)), y1) → A__ADD(a__add(mark(x0), mark(x1)), mark(y1))
MARK(sel(0, add(x0, x1))) → A__SEL(0, a__add(mark(x0), mark(x1)))
A__SEL(s(fib1(y0, y1)), cons(y2, fib1(x0, x1))) → A__SEL(a__fib1(mark(y0), mark(y1)), a__fib1(mark(x0), mark(x1)))
MARK(sel(0, fib(x0))) → A__SEL(0, a__fib(mark(x0)))
A__FIB(N) → MARK(N)
MARK(sel(fib1(y0, y1), add(x0, x1))) → A__SEL(a__fib1(mark(y0), mark(y1)), a__add(mark(x0), mark(x1)))
MARK(sel(add(y0, y1), fib1(x0, x1))) → A__SEL(a__add(mark(y0), mark(y1)), a__fib1(mark(x0), mark(x1)))
A__FIB(s(x0)) → A__SEL(s(mark(x0)), a__fib1(s(0), s(0)))
A__SEL(s(N), cons(X, XS)) → MARK(XS)
A__ADD(s(X), Y) → MARK(Y)
A__ADD(s(0), y1) → A__ADD(0, mark(y1))
MARK(add(0, y1)) → A__ADD(0, mark(y1))
MARK(sel(fib(y0), cons(x0, x1))) → A__SEL(a__fib(mark(y0)), cons(mark(x0), x1))
A__FIB(sel(x0, x1)) → A__SEL(a__sel(mark(x0), mark(x1)), a__fib1(s(0), s(0)))
MARK(add(s(x0), y1)) → A__ADD(s(mark(x0)), mark(y1))
MARK(sel(sel(y0, y1), add(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), a__add(mark(x0), mark(x1)))
MARK(sel(add(y0, y1), sel(x0, x1))) → A__SEL(a__add(mark(y0), mark(y1)), a__sel(mark(x0), mark(x1)))
MARK(sel(s(y0), fib(x0))) → A__SEL(s(mark(y0)), a__fib(mark(x0)))
MARK(sel(fib1(y0, y1), fib(x0))) → A__SEL(a__fib1(mark(y0), mark(y1)), a__fib(mark(x0)))
MARK(sel(fib(y0), fib1(x0, x1))) → A__SEL(a__fib(mark(y0)), a__fib1(mark(x0), mark(x1)))
MARK(add(sel(x0, x1), y1)) → A__ADD(a__sel(mark(x0), mark(x1)), mark(y1))
MARK(sel(fib1(y0, y1), cons(x0, x1))) → A__SEL(a__fib1(mark(y0), mark(y1)), cons(mark(x0), 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)))
MARK(sel(X1, X2)) → MARK(X1)
A__SEL(s(add(y0, y1)), cons(y2, add(x0, x1))) → A__SEL(a__add(mark(y0), mark(y1)), a__add(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), cons(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), cons(mark(x0), x1))
MARK(sel(add(y0, y1), fib(x0))) → A__SEL(a__add(mark(y0), mark(y1)), a__fib(mark(x0)))
MARK(sel(fib(y0), add(x0, x1))) → A__SEL(a__fib(mark(y0)), a__add(mark(x0), mark(x1)))
MARK(add(add(x0, x1), y1)) → A__ADD(a__add(mark(x0), mark(x1)), mark(y1))
A__ADD(s(X), Y) → MARK(X)
A__SEL(s(sel(y0, y1)), cons(y2, cons(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), cons(mark(x0), x1))
A__FIB(N) → A__FIB1(s(0), s(0))
MARK(add(fib(x0), y1)) → A__ADD(a__fib(mark(x0)), mark(y1))
A__FIB(fib(x0)) → A__SEL(a__fib(mark(x0)), a__fib1(s(0), s(0)))
A__ADD(s(fib(x0)), y1) → A__ADD(a__fib(mark(x0)), mark(y1))
A__SEL(s(0), cons(y1, y2)) → A__SEL(0, mark(y2))
A__SEL(s(add(y0, y1)), cons(y2, cons(x0, x1))) → A__SEL(a__add(mark(y0), mark(y1)), cons(mark(x0), x1))
MARK(sel(0, cons(x0, x1))) → A__SEL(0, cons(mark(x0), x1))
MARK(sel(add(y0, y1), cons(x0, x1))) → A__SEL(a__add(mark(y0), mark(y1)), cons(mark(x0), x1))
A__ADD(s(fib1(x0, x1)), y1) → A__ADD(a__fib1(mark(x0), mark(x1)), mark(y1))
MARK(fib(X)) → MARK(X)
A__SEL(s(sel(y0, y1)), cons(y2, fib(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__fib(mark(x0)))
A__SEL(s(add(y0, y1)), cons(y2, fib(x0))) → A__SEL(a__add(mark(y0), mark(y1)), a__fib(mark(x0)))
A__ADD(0, X) → MARK(X)
A__FIB1(X, Y) → MARK(X)
MARK(s(X)) → MARK(X)
MARK(add(X1, X2)) → MARK(X2)
MARK(sel(s(y0), sel(x0, x1))) → A__SEL(s(mark(y0)), a__sel(mark(x0), mark(x1)))
MARK(fib(X)) → A__FIB(mark(X))
MARK(fib1(X1, X2)) → MARK(X2)
A__FIB(0) → A__SEL(0, a__fib1(s(0), s(0)))
MARK(sel(sel(y0, y1), fib(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__fib(mark(x0)))
MARK(sel(fib(y0), sel(x0, x1))) → A__SEL(a__fib(mark(y0)), a__sel(mark(x0), mark(x1)))
A__ADD(s(sel(x0, x1)), y1) → A__ADD(a__sel(mark(x0), mark(x1)), mark(y1))
A__SEL(0, cons(X, XS)) → MARK(X)
MARK(add(X1, X2)) → MARK(X1)
MARK(add(fib1(x0, x1), y1)) → A__ADD(a__fib1(mark(x0), mark(x1)), mark(y1))
MARK(sel(0, fib1(x0, x1))) → A__SEL(0, a__fib1(mark(x0), mark(x1)))
A__FIB(fib1(x0, x1)) → A__SEL(a__fib1(mark(x0), mark(x1)), a__fib1(s(0), s(0)))
A__SEL(s(fib1(y0, y1)), cons(y2, add(x0, x1))) → A__SEL(a__fib1(mark(y0), mark(y1)), a__add(mark(x0), mark(x1)))
A__SEL(s(fib(x0)), cons(y1, y2)) → A__SEL(a__fib(mark(x0)), mark(y2))
A__SEL(s(add(y0, y1)), cons(y2, fib1(x0, x1))) → A__SEL(a__add(mark(y0), mark(y1)), a__fib1(mark(x0), mark(x1)))
A__SEL(s(fib1(y0, y1)), cons(y2, cons(x0, x1))) → A__SEL(a__fib1(mark(y0), mark(y1)), cons(mark(x0), x1))
A__SEL(s(fib1(y0, y1)), cons(y2, sel(x0, x1))) → A__SEL(a__fib1(mark(y0), mark(y1)), a__sel(mark(x0), mark(x1)))
A__SEL(s(sel(y0, y1)), cons(y2, fib1(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), a__fib1(mark(x0), mark(x1)))
A__SEL(s(N), cons(X, XS)) → MARK(N)
A__ADD(s(s(x0)), y1) → A__ADD(s(mark(x0)), mark(y1))
MARK(fib1(X1, X2)) → A__FIB1(mark(X1), mark(X2))
MARK(sel(s(y0), fib1(x0, x1))) → A__SEL(s(mark(y0)), a__fib1(mark(x0), mark(x1)))
A__SEL(s(s(x0)), cons(y1, y2)) → A__SEL(s(mark(x0)), mark(y2))
MARK(sel(fib(y0), fib(x0))) → A__SEL(a__fib(mark(y0)), a__fib(mark(x0)))
MARK(sel(sel(y0, y1), fib1(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), a__fib1(mark(x0), mark(x1)))
MARK(sel(fib1(y0, y1), sel(x0, x1))) → A__SEL(a__fib1(mark(y0), mark(y1)), a__sel(mark(x0), mark(x1)))
MARK(sel(add(y0, y1), add(x0, x1))) → A__SEL(a__add(mark(y0), mark(y1)), a__add(mark(x0), mark(x1)))
MARK(sel(s(y0), add(x0, x1))) → A__SEL(s(mark(y0)), a__add(mark(x0), mark(x1)))
A__SEL(s(fib1(y0, y1)), cons(y2, fib(x0))) → A__SEL(a__fib1(mark(y0), mark(y1)), a__fib(mark(x0)))
MARK(sel(0, sel(x0, x1))) → A__SEL(0, a__sel(mark(x0), mark(x1)))
A__SEL(s(add(y0, y1)), cons(y2, sel(x0, x1))) → A__SEL(a__add(mark(y0), mark(y1)), a__sel(mark(x0), mark(x1)))
A__SEL(s(sel(y0, y1)), cons(y2, add(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), a__add(mark(x0), mark(x1)))
A__FIB(add(x0, x1)) → A__SEL(a__add(mark(x0), mark(x1)), a__fib1(s(0), s(0)))
MARK(sel(X1, X2)) → MARK(X2)

The TRS R consists of the following rules:

a__fib(N) → a__sel(mark(N), a__fib1(s(0), s(0)))
a__fib1(X, Y) → cons(mark(X), fib1(Y, add(X, Y)))
a__add(0, X) → mark(X)
a__add(s(X), Y) → s(a__add(mark(X), mark(Y)))
a__sel(0, cons(X, XS)) → mark(X)
a__sel(s(N), cons(X, XS)) → a__sel(mark(N), mark(XS))
mark(fib(X)) → a__fib(mark(X))
mark(sel(X1, X2)) → a__sel(mark(X1), mark(X2))
mark(fib1(X1, X2)) → a__fib1(mark(X1), mark(X2))
mark(add(X1, X2)) → a__add(mark(X1), mark(X2))
mark(s(X)) → s(mark(X))
mark(0) → 0
mark(cons(X1, X2)) → cons(mark(X1), X2)
a__fib(X) → fib(X)
a__sel(X1, X2) → sel(X1, X2)
a__fib1(X1, X2) → fib1(X1, X2)
a__add(X1, X2) → add(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule A__SEL(s(0), cons(y1, y2)) → A__SEL(0, mark(y2)) at position [1] we obtained the following new rules:

A__SEL(s(0), cons(y0, fib(x0))) → A__SEL(0, a__fib(mark(x0)))
A__SEL(s(0), cons(y0, add(x0, x1))) → A__SEL(0, a__add(mark(x0), mark(x1)))
A__SEL(s(0), cons(y0, s(x0))) → A__SEL(0, s(mark(x0)))
A__SEL(s(0), cons(y0, cons(x0, x1))) → A__SEL(0, cons(mark(x0), x1))
A__SEL(s(0), cons(y0, sel(x0, x1))) → A__SEL(0, a__sel(mark(x0), mark(x1)))
A__SEL(s(0), cons(y0, fib1(x0, x1))) → A__SEL(0, a__fib1(mark(x0), mark(x1)))
A__SEL(s(0), cons(y0, 0)) → A__SEL(0, 0)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ Narrowing

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ QDP

              ↳ Narrowing

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ QDP

                      ↳ Narrowing

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ QDP

                              ↳ Narrowing

                                ↳ QDP

                                  ↳ DependencyGraphProof

                                    ↳ QDP

                                      ↳ Narrowing

                                        ↳ QDP

                                          ↳ DependencyGraphProof

                                            ↳ QDP

                                              ↳ Narrowing

                                                ↳ QDP

                                                  ↳ DependencyGraphProof

                                                    ↳ QDP

                                                      ↳ Narrowing

                                                        ↳ QDP

                                                          ↳ DependencyGraphProof

                                                            ↳ QDP

                                                              ↳ Narrowing

                                                                ↳ QDP

                                                                  ↳ DependencyGraphProof

                                                                    ↳ QDP

                                                                      ↳ Narrowing

                                                                        ↳ QDP

                                                                          ↳ DependencyGraphProof

                                                                            ↳ QDP

                                                                              ↳ Narrowing

                                                                                ↳ QDP

                                                                                  ↳ DependencyGraphProof

                                                                                    ↳ QDP

                                                                                      ↳ Narrowing

                                                                                        ↳ QDP

                                                                                          ↳ DependencyGraphProof

                                                                                            ↳ QDP

                                                                                              ↳ Narrowing

                                                                                                ↳ QDP

                                                                                                  ↳ DependencyGraphProof

                                                                                                    ↳ QDP

                                                                                                      ↳ Narrowing

                                                                                                        ↳ QDP

                                                                                                          ↳ DependencyGraphProof

                                                                                                            ↳ QDP

                                                                                                              ↳ Narrowing

                                                                                                                ↳ QDP

                                                                                                                  ↳ DependencyGraphProof

                                                                                                                    ↳ QDP

                                                                                                                      ↳ Narrowing

                                                                                                                        ↳ QDP

                                                                                                                          ↳ DependencyGraphProof

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

MARK(sel(sel(y0, y1), sel(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), a__sel(mark(x0), mark(x1)))
MARK(sel(fib1(y0, y1), fib1(x0, x1))) → A__SEL(a__fib1(mark(y0), mark(y1)), a__fib1(mark(x0), mark(x1)))
MARK(fib1(X1, X2)) → MARK(X1)
MARK(sel(s(y0), cons(x0, x1))) → A__SEL(s(mark(y0)), cons(mark(x0), x1))
A__SEL(s(0), cons(y0, s(x0))) → A__SEL(0, s(mark(x0)))
MARK(cons(X1, X2)) → MARK(X1)
A__ADD(s(add(x0, x1)), y1) → A__ADD(a__add(mark(x0), mark(x1)), mark(y1))
A__SEL(s(fib1(y0, y1)), cons(y2, fib1(x0, x1))) → A__SEL(a__fib1(mark(y0), mark(y1)), a__fib1(mark(x0), mark(x1)))
MARK(sel(0, add(x0, x1))) → A__SEL(0, a__add(mark(x0), mark(x1)))
A__FIB(N) → MARK(N)
MARK(sel(0, fib(x0))) → A__SEL(0, a__fib(mark(x0)))
MARK(sel(add(y0, y1), fib1(x0, x1))) → A__SEL(a__add(mark(y0), mark(y1)), a__fib1(mark(x0), mark(x1)))
MARK(sel(fib1(y0, y1), add(x0, x1))) → A__SEL(a__fib1(mark(y0), mark(y1)), a__add(mark(x0), mark(x1)))
A__SEL(s(N), cons(X, XS)) → MARK(XS)
A__FIB(s(x0)) → A__SEL(s(mark(x0)), a__fib1(s(0), s(0)))
A__ADD(s(X), Y) → MARK(Y)
A__ADD(s(0), y1) → A__ADD(0, mark(y1))
MARK(add(0, y1)) → A__ADD(0, mark(y1))
MARK(sel(fib(y0), cons(x0, x1))) → A__SEL(a__fib(mark(y0)), cons(mark(x0), x1))
A__FIB(sel(x0, x1)) → A__SEL(a__sel(mark(x0), mark(x1)), a__fib1(s(0), s(0)))
MARK(add(s(x0), y1)) → A__ADD(s(mark(x0)), mark(y1))
MARK(sel(add(y0, y1), sel(x0, x1))) → A__SEL(a__add(mark(y0), mark(y1)), a__sel(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), add(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), a__add(mark(x0), mark(x1)))
MARK(add(sel(x0, x1), y1)) → A__ADD(a__sel(mark(x0), mark(x1)), mark(y1))
MARK(sel(fib(y0), fib1(x0, x1))) → A__SEL(a__fib(mark(y0)), a__fib1(mark(x0), mark(x1)))
MARK(sel(fib1(y0, y1), fib(x0))) → A__SEL(a__fib1(mark(y0), mark(y1)), a__fib(mark(x0)))
MARK(sel(s(y0), fib(x0))) → A__SEL(s(mark(y0)), a__fib(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)))
MARK(sel(fib1(y0, y1), cons(x0, x1))) → A__SEL(a__fib1(mark(y0), mark(y1)), cons(mark(x0), x1))
A__SEL(s(add(y0, y1)), cons(y2, add(x0, x1))) → A__SEL(a__add(mark(y0), mark(y1)), a__add(mark(x0), mark(x1)))
MARK(sel(X1, X2)) → MARK(X1)
A__SEL(s(0), cons(y0, add(x0, x1))) → A__SEL(0, a__add(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), cons(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), cons(mark(x0), x1))
MARK(sel(fib(y0), add(x0, x1))) → A__SEL(a__fib(mark(y0)), a__add(mark(x0), mark(x1)))
MARK(sel(add(y0, y1), fib(x0))) → A__SEL(a__add(mark(y0), mark(y1)), a__fib(mark(x0)))
MARK(add(add(x0, x1), y1)) → A__ADD(a__add(mark(x0), mark(x1)), mark(y1))
A__ADD(s(X), Y) → MARK(X)
A__SEL(s(sel(y0, y1)), cons(y2, cons(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), cons(mark(x0), x1))
A__FIB(N) → A__FIB1(s(0), s(0))
MARK(add(fib(x0), y1)) → A__ADD(a__fib(mark(x0)), mark(y1))
A__FIB(fib(x0)) → A__SEL(a__fib(mark(x0)), a__fib1(s(0), s(0)))
A__ADD(s(fib(x0)), y1) → A__ADD(a__fib(mark(x0)), mark(y1))
A__SEL(s(add(y0, y1)), cons(y2, cons(x0, x1))) → A__SEL(a__add(mark(y0), mark(y1)), cons(mark(x0), x1))
MARK(sel(0, cons(x0, x1))) → A__SEL(0, cons(mark(x0), x1))
A__SEL(s(0), cons(y0, sel(x0, x1))) → A__SEL(0, a__sel(mark(x0), mark(x1)))
MARK(sel(add(y0, y1), cons(x0, x1))) → A__SEL(a__add(mark(y0), mark(y1)), cons(mark(x0), x1))
A__ADD(s(fib1(x0, x1)), y1) → A__ADD(a__fib1(mark(x0), mark(x1)), mark(y1))
A__SEL(s(sel(y0, y1)), cons(y2, fib(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__fib(mark(x0)))
MARK(fib(X)) → MARK(X)
A__SEL(s(add(y0, y1)), cons(y2, fib(x0))) → A__SEL(a__add(mark(y0), mark(y1)), a__fib(mark(x0)))
A__FIB1(X, Y) → MARK(X)
A__ADD(0, X) → MARK(X)
MARK(s(X)) → MARK(X)
MARK(add(X1, X2)) → MARK(X2)
MARK(fib(X)) → A__FIB(mark(X))
MARK(sel(s(y0), sel(x0, x1))) → A__SEL(s(mark(y0)), a__sel(mark(x0), mark(x1)))
MARK(fib1(X1, X2)) → MARK(X2)
MARK(sel(fib(y0), sel(x0, x1))) → A__SEL(a__fib(mark(y0)), a__sel(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), fib(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__fib(mark(x0)))
A__FIB(0) → A__SEL(0, a__fib1(s(0), s(0)))
A__SEL(0, cons(X, XS)) → MARK(X)
A__ADD(s(sel(x0, x1)), y1) → A__ADD(a__sel(mark(x0), mark(x1)), mark(y1))
MARK(add(X1, X2)) → MARK(X1)
MARK(add(fib1(x0, x1), y1)) → A__ADD(a__fib1(mark(x0), mark(x1)), mark(y1))
MARK(sel(0, fib1(x0, x1))) → A__SEL(0, a__fib1(mark(x0), mark(x1)))
A__FIB(fib1(x0, x1)) → A__SEL(a__fib1(mark(x0), mark(x1)), a__fib1(s(0), s(0)))
A__SEL(s(0), cons(y0, cons(x0, x1))) → A__SEL(0, cons(mark(x0), x1))
A__SEL(s(add(y0, y1)), cons(y2, fib1(x0, x1))) → A__SEL(a__add(mark(y0), mark(y1)), a__fib1(mark(x0), mark(x1)))
A__SEL(s(fib(x0)), cons(y1, y2)) → A__SEL(a__fib(mark(x0)), mark(y2))
A__SEL(s(fib1(y0, y1)), cons(y2, add(x0, x1))) → A__SEL(a__fib1(mark(y0), mark(y1)), a__add(mark(x0), mark(x1)))
A__SEL(s(fib1(y0, y1)), cons(y2, cons(x0, x1))) → A__SEL(a__fib1(mark(y0), mark(y1)), cons(mark(x0), x1))
A__SEL(s(sel(y0, y1)), cons(y2, fib1(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), a__fib1(mark(x0), mark(x1)))
A__SEL(s(fib1(y0, y1)), cons(y2, sel(x0, x1))) → A__SEL(a__fib1(mark(y0), mark(y1)), a__sel(mark(x0), mark(x1)))
A__SEL(s(N), cons(X, XS)) → MARK(N)
A__ADD(s(s(x0)), y1) → A__ADD(s(mark(x0)), mark(y1))
A__SEL(s(0), cons(y0, fib(x0))) → A__SEL(0, a__fib(mark(x0)))
MARK(fib1(X1, X2)) → A__FIB1(mark(X1), mark(X2))
A__SEL(s(s(x0)), cons(y1, y2)) → A__SEL(s(mark(x0)), mark(y2))
MARK(sel(s(y0), fib1(x0, x1))) → A__SEL(s(mark(y0)), a__fib1(mark(x0), mark(x1)))
MARK(sel(fib(y0), fib(x0))) → A__SEL(a__fib(mark(y0)), a__fib(mark(x0)))
A__SEL(s(0), cons(y0, 0)) → A__SEL(0, 0)
MARK(sel(add(y0, y1), add(x0, x1))) → A__SEL(a__add(mark(y0), mark(y1)), a__add(mark(x0), mark(x1)))
MARK(sel(fib1(y0, y1), sel(x0, x1))) → A__SEL(a__fib1(mark(y0), mark(y1)), a__sel(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), fib1(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), a__fib1(mark(x0), mark(x1)))
MARK(sel(s(y0), add(x0, x1))) → A__SEL(s(mark(y0)), a__add(mark(x0), mark(x1)))
A__SEL(s(fib1(y0, y1)), cons(y2, fib(x0))) → A__SEL(a__fib1(mark(y0), mark(y1)), a__fib(mark(x0)))
MARK(sel(0, sel(x0, x1))) → A__SEL(0, a__sel(mark(x0), mark(x1)))
A__SEL(s(sel(y0, y1)), cons(y2, add(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), a__add(mark(x0), mark(x1)))
A__SEL(s(add(y0, y1)), cons(y2, sel(x0, x1))) → A__SEL(a__add(mark(y0), mark(y1)), a__sel(mark(x0), mark(x1)))
A__FIB(add(x0, x1)) → A__SEL(a__add(mark(x0), mark(x1)), a__fib1(s(0), s(0)))
A__SEL(s(0), cons(y0, fib1(x0, x1))) → A__SEL(0, a__fib1(mark(x0), mark(x1)))
MARK(sel(X1, X2)) → MARK(X2)

The TRS R consists of the following rules:

a__fib(N) → a__sel(mark(N), a__fib1(s(0), s(0)))
a__fib1(X, Y) → cons(mark(X), fib1(Y, add(X, Y)))
a__add(0, X) → mark(X)
a__add(s(X), Y) → s(a__add(mark(X), mark(Y)))
a__sel(0, cons(X, XS)) → mark(X)
a__sel(s(N), cons(X, XS)) → a__sel(mark(N), mark(XS))
mark(fib(X)) → a__fib(mark(X))
mark(sel(X1, X2)) → a__sel(mark(X1), mark(X2))
mark(fib1(X1, X2)) → a__fib1(mark(X1), mark(X2))
mark(add(X1, X2)) → a__add(mark(X1), mark(X2))
mark(s(X)) → s(mark(X))
mark(0) → 0
mark(cons(X1, X2)) → cons(mark(X1), X2)
a__fib(X) → fib(X)
a__sel(X1, X2) → sel(X1, X2)
a__fib1(X1, X2) → fib1(X1, X2)
a__add(X1, X2) → add(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 1 SCC with 2 less nodes.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ Narrowing

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ QDP

              ↳ Narrowing

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ QDP

                      ↳ Narrowing

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ QDP

                              ↳ Narrowing

                                ↳ QDP

                                  ↳ DependencyGraphProof

                                    ↳ QDP

                                      ↳ Narrowing

                                        ↳ QDP

                                          ↳ DependencyGraphProof

                                            ↳ QDP

                                              ↳ Narrowing

                                                ↳ QDP

                                                  ↳ DependencyGraphProof

                                                    ↳ QDP

                                                      ↳ Narrowing

                                                        ↳ QDP

                                                          ↳ DependencyGraphProof

                                                            ↳ QDP

                                                              ↳ Narrowing

                                                                ↳ QDP

                                                                  ↳ DependencyGraphProof

                                                                    ↳ QDP

                                                                      ↳ Narrowing

                                                                        ↳ QDP

                                                                          ↳ DependencyGraphProof

                                                                            ↳ QDP

                                                                              ↳ Narrowing

                                                                                ↳ QDP

                                                                                  ↳ DependencyGraphProof

                                                                                    ↳ QDP

                                                                                      ↳ Narrowing

                                                                                        ↳ QDP

                                                                                          ↳ DependencyGraphProof

                                                                                            ↳ QDP

                                                                                              ↳ Narrowing

                                                                                                ↳ QDP

                                                                                                  ↳ DependencyGraphProof

                                                                                                    ↳ QDP

                                                                                                      ↳ Narrowing

                                                                                                        ↳ QDP

                                                                                                          ↳ DependencyGraphProof

                                                                                                            ↳ QDP

                                                                                                              ↳ Narrowing

                                                                                                                ↳ QDP

                                                                                                                  ↳ DependencyGraphProof

                                                                                                                    ↳ QDP

                                                                                                                      ↳ Narrowing

                                                                                                                        ↳ QDP

                                                                                                                          ↳ DependencyGraphProof

                                                                                                                            ↳ QDP

                                                                                                                              ↳ Narrowing

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

MARK(sel(sel(y0, y1), sel(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), a__sel(mark(x0), mark(x1)))
MARK(sel(fib1(y0, y1), fib1(x0, x1))) → A__SEL(a__fib1(mark(y0), mark(y1)), a__fib1(mark(x0), mark(x1)))
MARK(fib1(X1, X2)) → MARK(X1)
MARK(sel(s(y0), cons(x0, x1))) → A__SEL(s(mark(y0)), cons(mark(x0), x1))
MARK(cons(X1, X2)) → MARK(X1)
A__ADD(s(add(x0, x1)), y1) → A__ADD(a__add(mark(x0), mark(x1)), mark(y1))
MARK(sel(0, add(x0, x1))) → A__SEL(0, a__add(mark(x0), mark(x1)))
A__SEL(s(fib1(y0, y1)), cons(y2, fib1(x0, x1))) → A__SEL(a__fib1(mark(y0), mark(y1)), a__fib1(mark(x0), mark(x1)))
MARK(sel(0, fib(x0))) → A__SEL(0, a__fib(mark(x0)))
A__FIB(N) → MARK(N)
MARK(sel(fib1(y0, y1), add(x0, x1))) → A__SEL(a__fib1(mark(y0), mark(y1)), a__add(mark(x0), mark(x1)))
MARK(sel(add(y0, y1), fib1(x0, x1))) → A__SEL(a__add(mark(y0), mark(y1)), a__fib1(mark(x0), mark(x1)))
A__FIB(s(x0)) → A__SEL(s(mark(x0)), a__fib1(s(0), s(0)))
A__SEL(s(N), cons(X, XS)) → MARK(XS)
A__ADD(s(X), Y) → MARK(Y)
A__ADD(s(0), y1) → A__ADD(0, mark(y1))
MARK(add(0, y1)) → A__ADD(0, mark(y1))
MARK(sel(fib(y0), cons(x0, x1))) → A__SEL(a__fib(mark(y0)), cons(mark(x0), x1))
A__FIB(sel(x0, x1)) → A__SEL(a__sel(mark(x0), mark(x1)), a__fib1(s(0), s(0)))
MARK(add(s(x0), y1)) → A__ADD(s(mark(x0)), mark(y1))
MARK(sel(sel(y0, y1), add(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), a__add(mark(x0), mark(x1)))
MARK(sel(add(y0, y1), sel(x0, x1))) → A__SEL(a__add(mark(y0), mark(y1)), a__sel(mark(x0), mark(x1)))
MARK(sel(s(y0), fib(x0))) → A__SEL(s(mark(y0)), a__fib(mark(x0)))
MARK(sel(fib1(y0, y1), fib(x0))) → A__SEL(a__fib1(mark(y0), mark(y1)), a__fib(mark(x0)))
MARK(sel(fib(y0), fib1(x0, x1))) → A__SEL(a__fib(mark(y0)), a__fib1(mark(x0), mark(x1)))
MARK(add(sel(x0, x1), y1)) → A__ADD(a__sel(mark(x0), mark(x1)), mark(y1))
MARK(sel(fib1(y0, y1), cons(x0, x1))) → A__SEL(a__fib1(mark(y0), mark(y1)), cons(mark(x0), 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)))
MARK(sel(X1, X2)) → MARK(X1)
A__SEL(s(add(y0, y1)), cons(y2, add(x0, x1))) → A__SEL(a__add(mark(y0), mark(y1)), a__add(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), cons(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), cons(mark(x0), x1))
A__SEL(s(0), cons(y0, add(x0, x1))) → A__SEL(0, a__add(mark(x0), mark(x1)))
MARK(sel(add(y0, y1), fib(x0))) → A__SEL(a__add(mark(y0), mark(y1)), a__fib(mark(x0)))
MARK(sel(fib(y0), add(x0, x1))) → A__SEL(a__fib(mark(y0)), a__add(mark(x0), mark(x1)))
MARK(add(add(x0, x1), y1)) → A__ADD(a__add(mark(x0), mark(x1)), mark(y1))
A__ADD(s(X), Y) → MARK(X)
A__SEL(s(sel(y0, y1)), cons(y2, cons(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), cons(mark(x0), x1))
A__FIB(N) → A__FIB1(s(0), s(0))
MARK(add(fib(x0), y1)) → A__ADD(a__fib(mark(x0)), mark(y1))
A__FIB(fib(x0)) → A__SEL(a__fib(mark(x0)), a__fib1(s(0), s(0)))
A__ADD(s(fib(x0)), y1) → A__ADD(a__fib(mark(x0)), mark(y1))
A__SEL(s(add(y0, y1)), cons(y2, cons(x0, x1))) → A__SEL(a__add(mark(y0), mark(y1)), cons(mark(x0), x1))
MARK(sel(0, cons(x0, x1))) → A__SEL(0, cons(mark(x0), x1))
A__SEL(s(0), cons(y0, sel(x0, x1))) → A__SEL(0, a__sel(mark(x0), mark(x1)))
MARK(sel(add(y0, y1), cons(x0, x1))) → A__SEL(a__add(mark(y0), mark(y1)), cons(mark(x0), x1))
A__ADD(s(fib1(x0, x1)), y1) → A__ADD(a__fib1(mark(x0), mark(x1)), mark(y1))
MARK(fib(X)) → MARK(X)
A__SEL(s(sel(y0, y1)), cons(y2, fib(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__fib(mark(x0)))
A__SEL(s(add(y0, y1)), cons(y2, fib(x0))) → A__SEL(a__add(mark(y0), mark(y1)), a__fib(mark(x0)))
A__ADD(0, X) → MARK(X)
A__FIB1(X, Y) → MARK(X)
MARK(s(X)) → MARK(X)
MARK(add(X1, X2)) → MARK(X2)
MARK(sel(s(y0), sel(x0, x1))) → A__SEL(s(mark(y0)), a__sel(mark(x0), mark(x1)))
MARK(fib(X)) → A__FIB(mark(X))
MARK(fib1(X1, X2)) → MARK(X2)
A__FIB(0) → A__SEL(0, a__fib1(s(0), s(0)))
MARK(sel(sel(y0, y1), fib(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__fib(mark(x0)))
MARK(sel(fib(y0), sel(x0, x1))) → A__SEL(a__fib(mark(y0)), a__sel(mark(x0), mark(x1)))
A__ADD(s(sel(x0, x1)), y1) → A__ADD(a__sel(mark(x0), mark(x1)), mark(y1))
A__SEL(0, cons(X, XS)) → MARK(X)
MARK(add(X1, X2)) → MARK(X1)
MARK(add(fib1(x0, x1), y1)) → A__ADD(a__fib1(mark(x0), mark(x1)), mark(y1))
MARK(sel(0, fib1(x0, x1))) → A__SEL(0, a__fib1(mark(x0), mark(x1)))
A__FIB(fib1(x0, x1)) → A__SEL(a__fib1(mark(x0), mark(x1)), a__fib1(s(0), s(0)))
A__SEL(s(0), cons(y0, cons(x0, x1))) → A__SEL(0, cons(mark(x0), x1))
A__SEL(s(fib1(y0, y1)), cons(y2, add(x0, x1))) → A__SEL(a__fib1(mark(y0), mark(y1)), a__add(mark(x0), mark(x1)))
A__SEL(s(fib(x0)), cons(y1, y2)) → A__SEL(a__fib(mark(x0)), mark(y2))
A__SEL(s(add(y0, y1)), cons(y2, fib1(x0, x1))) → A__SEL(a__add(mark(y0), mark(y1)), a__fib1(mark(x0), mark(x1)))
A__SEL(s(fib1(y0, y1)), cons(y2, cons(x0, x1))) → A__SEL(a__fib1(mark(y0), mark(y1)), cons(mark(x0), x1))
A__SEL(s(fib1(y0, y1)), cons(y2, sel(x0, x1))) → A__SEL(a__fib1(mark(y0), mark(y1)), a__sel(mark(x0), mark(x1)))
A__SEL(s(sel(y0, y1)), cons(y2, fib1(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), a__fib1(mark(x0), mark(x1)))
A__SEL(s(N), cons(X, XS)) → MARK(N)
A__ADD(s(s(x0)), y1) → A__ADD(s(mark(x0)), mark(y1))
A__SEL(s(0), cons(y0, fib(x0))) → A__SEL(0, a__fib(mark(x0)))
MARK(fib1(X1, X2)) → A__FIB1(mark(X1), mark(X2))
MARK(sel(s(y0), fib1(x0, x1))) → A__SEL(s(mark(y0)), a__fib1(mark(x0), mark(x1)))
A__SEL(s(s(x0)), cons(y1, y2)) → A__SEL(s(mark(x0)), mark(y2))
MARK(sel(fib(y0), fib(x0))) → A__SEL(a__fib(mark(y0)), a__fib(mark(x0)))
MARK(sel(sel(y0, y1), fib1(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), a__fib1(mark(x0), mark(x1)))
MARK(sel(fib1(y0, y1), sel(x0, x1))) → A__SEL(a__fib1(mark(y0), mark(y1)), a__sel(mark(x0), mark(x1)))
MARK(sel(add(y0, y1), add(x0, x1))) → A__SEL(a__add(mark(y0), mark(y1)), a__add(mark(x0), mark(x1)))
MARK(sel(s(y0), add(x0, x1))) → A__SEL(s(mark(y0)), a__add(mark(x0), mark(x1)))
A__SEL(s(fib1(y0, y1)), cons(y2, fib(x0))) → A__SEL(a__fib1(mark(y0), mark(y1)), a__fib(mark(x0)))
MARK(sel(0, sel(x0, x1))) → A__SEL(0, a__sel(mark(x0), mark(x1)))
A__SEL(s(add(y0, y1)), cons(y2, sel(x0, x1))) → A__SEL(a__add(mark(y0), mark(y1)), a__sel(mark(x0), mark(x1)))
A__SEL(s(sel(y0, y1)), cons(y2, add(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), a__add(mark(x0), mark(x1)))
A__FIB(add(x0, x1)) → A__SEL(a__add(mark(x0), mark(x1)), a__fib1(s(0), s(0)))
A__SEL(s(0), cons(y0, fib1(x0, x1))) → A__SEL(0, a__fib1(mark(x0), mark(x1)))
MARK(sel(X1, X2)) → MARK(X2)

The TRS R consists of the following rules:

a__fib(N) → a__sel(mark(N), a__fib1(s(0), s(0)))
a__fib1(X, Y) → cons(mark(X), fib1(Y, add(X, Y)))
a__add(0, X) → mark(X)
a__add(s(X), Y) → s(a__add(mark(X), mark(Y)))
a__sel(0, cons(X, XS)) → mark(X)
a__sel(s(N), cons(X, XS)) → a__sel(mark(N), mark(XS))
mark(fib(X)) → a__fib(mark(X))
mark(sel(X1, X2)) → a__sel(mark(X1), mark(X2))
mark(fib1(X1, X2)) → a__fib1(mark(X1), mark(X2))
mark(add(X1, X2)) → a__add(mark(X1), mark(X2))
mark(s(X)) → s(mark(X))
mark(0) → 0
mark(cons(X1, X2)) → cons(mark(X1), X2)
a__fib(X) → fib(X)
a__sel(X1, X2) → sel(X1, X2)
a__fib1(X1, X2) → fib1(X1, X2)
a__add(X1, X2) → add(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, fib(x0))) → A__SEL(s(mark(y0)), a__fib(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, fib1(x0, x1))) → A__SEL(s(mark(y0)), a__fib1(mark(x0), mark(x1)))
A__SEL(s(s(y0)), cons(y1, add(x0, x1))) → A__SEL(s(mark(y0)), a__add(mark(x0), mark(x1)))
A__SEL(s(s(y0)), cons(y1, 0)) → A__SEL(s(mark(y0)), 0)
A__SEL(s(s(y0)), cons(y1, cons(x0, x1))) → A__SEL(s(mark(y0)), cons(mark(x0), x1))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ Narrowing

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ QDP

              ↳ Narrowing

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ QDP

                      ↳ Narrowing

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ QDP

                              ↳ Narrowing

                                ↳ QDP

                                  ↳ DependencyGraphProof

                                    ↳ QDP

                                      ↳ Narrowing

                                        ↳ QDP

                                          ↳ DependencyGraphProof

                                            ↳ QDP

                                              ↳ Narrowing

                                                ↳ QDP

                                                  ↳ DependencyGraphProof

                                                    ↳ QDP

                                                      ↳ Narrowing

                                                        ↳ QDP

                                                          ↳ DependencyGraphProof

                                                            ↳ QDP

                                                              ↳ Narrowing

                                                                ↳ QDP

                                                                  ↳ DependencyGraphProof

                                                                    ↳ QDP

                                                                      ↳ Narrowing

                                                                        ↳ QDP

                                                                          ↳ DependencyGraphProof

                                                                            ↳ QDP

                                                                              ↳ Narrowing

                                                                                ↳ QDP

                                                                                  ↳ DependencyGraphProof

                                                                                    ↳ QDP

                                                                                      ↳ Narrowing

                                                                                        ↳ QDP

                                                                                          ↳ DependencyGraphProof

                                                                                            ↳ QDP

                                                                                              ↳ Narrowing

                                                                                                ↳ QDP

                                                                                                  ↳ DependencyGraphProof

                                                                                                    ↳ QDP

                                                                                                      ↳ Narrowing

                                                                                                        ↳ QDP

                                                                                                          ↳ DependencyGraphProof

                                                                                                            ↳ QDP

                                                                                                              ↳ Narrowing

                                                                                                                ↳ QDP

                                                                                                                  ↳ DependencyGraphProof

                                                                                                                    ↳ QDP

                                                                                                                      ↳ Narrowing

                                                                                                                        ↳ QDP

                                                                                                                          ↳ DependencyGraphProof

                                                                                                                            ↳ QDP

                                                                                                                              ↳ Narrowing

                                                                                                                                ↳ QDP

                                                                                                                                  ↳ DependencyGraphProof

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

MARK(sel(fib1(y0, y1), fib1(x0, x1))) → A__SEL(a__fib1(mark(y0), mark(y1)), a__fib1(mark(x0), mark(x1)))
MARK(sel(s(y0), cons(x0, x1))) → A__SEL(s(mark(y0)), cons(mark(x0), x1))
A__ADD(s(add(x0, x1)), y1) → A__ADD(a__add(mark(x0), mark(x1)), mark(y1))
MARK(sel(0, add(x0, x1))) → A__SEL(0, a__add(mark(x0), mark(x1)))
MARK(sel(fib1(y0, y1), add(x0, x1))) → A__SEL(a__fib1(mark(y0), mark(y1)), a__add(mark(x0), mark(x1)))
MARK(sel(add(y0, y1), fib1(x0, x1))) → A__SEL(a__add(mark(y0), mark(y1)), a__fib1(mark(x0), mark(x1)))
A__ADD(s(0), y1) → A__ADD(0, mark(y1))
MARK(add(0, y1)) → A__ADD(0, mark(y1))
MARK(sel(fib(y0), cons(x0, x1))) → A__SEL(a__fib(mark(y0)), cons(mark(x0), x1))
A__FIB(sel(x0, x1)) → A__SEL(a__sel(mark(x0), mark(x1)), a__fib1(s(0), s(0)))
MARK(sel(sel(y0, y1), add(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), a__add(mark(x0), mark(x1)))
MARK(sel(add(y0, y1), sel(x0, x1))) → A__SEL(a__add(mark(y0), mark(y1)), a__sel(mark(x0), mark(x1)))
A__SEL(s(0), cons(y0, add(x0, x1))) → A__SEL(0, a__add(mark(x0), mark(x1)))
MARK(add(add(x0, x1), y1)) → A__ADD(a__add(mark(x0), mark(x1)), mark(y1))
A__ADD(s(X), Y) → MARK(X)
A__FIB(N) → A__FIB1(s(0), s(0))
MARK(add(fib(x0), y1)) → A__ADD(a__fib(mark(x0)), mark(y1))
A__ADD(s(fib(x0)), y1) → A__ADD(a__fib(mark(x0)), mark(y1))
A__SEL(s(add(y0, y1)), cons(y2, cons(x0, x1))) → A__SEL(a__add(mark(y0), mark(y1)), cons(mark(x0), x1))
MARK(sel(add(y0, y1), cons(x0, x1))) → A__SEL(a__add(mark(y0), mark(y1)), cons(mark(x0), x1))
A__ADD(0, X) → MARK(X)
A__FIB1(X, Y) → MARK(X)
MARK(s(X)) → MARK(X)
MARK(sel(s(y0), sel(x0, x1))) → A__SEL(s(mark(y0)), a__sel(mark(x0), mark(x1)))
A__FIB(0) → A__SEL(0, a__fib1(s(0), s(0)))
A__SEL(0, cons(X, XS)) → MARK(X)
MARK(add(X1, X2)) → MARK(X1)
A__FIB(fib1(x0, x1)) → A__SEL(a__fib1(mark(x0), mark(x1)), a__fib1(s(0), s(0)))
A__SEL(s(fib1(y0, y1)), cons(y2, cons(x0, x1))) → A__SEL(a__fib1(mark(y0), mark(y1)), cons(mark(x0), x1))
A__SEL(s(fib1(y0, y1)), cons(y2, sel(x0, x1))) → A__SEL(a__fib1(mark(y0), mark(y1)), a__sel(mark(x0), mark(x1)))
A__SEL(s(sel(y0, y1)), cons(y2, fib1(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), a__fib1(mark(x0), mark(x1)))
A__SEL(s(N), cons(X, XS)) → MARK(N)
A__ADD(s(s(x0)), y1) → A__ADD(s(mark(x0)), mark(y1))
MARK(fib1(X1, X2)) → A__FIB1(mark(X1), mark(X2))
MARK(sel(fib(y0), fib(x0))) → A__SEL(a__fib(mark(y0)), a__fib(mark(x0)))
MARK(sel(add(y0, y1), add(x0, x1))) → A__SEL(a__add(mark(y0), mark(y1)), a__add(mark(x0), mark(x1)))
MARK(sel(s(y0), add(x0, x1))) → A__SEL(s(mark(y0)), a__add(mark(x0), mark(x1)))
A__FIB(add(x0, x1)) → A__SEL(a__add(mark(x0), mark(x1)), a__fib1(s(0), s(0)))
MARK(sel(X1, X2)) → MARK(X2)
A__SEL(s(s(y0)), cons(y1, s(x0))) → A__SEL(s(mark(y0)), s(mark(x0)))
MARK(sel(sel(y0, y1), sel(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), a__sel(mark(x0), mark(x1)))
MARK(fib1(X1, X2)) → MARK(X1)
MARK(cons(X1, X2)) → MARK(X1)
A__SEL(s(fib1(y0, y1)), cons(y2, fib1(x0, x1))) → A__SEL(a__fib1(mark(y0), mark(y1)), a__fib1(mark(x0), mark(x1)))
MARK(sel(0, fib(x0))) → A__SEL(0, a__fib(mark(x0)))
A__FIB(N) → MARK(N)
A__FIB(s(x0)) → A__SEL(s(mark(x0)), a__fib1(s(0), s(0)))
A__SEL(s(N), cons(X, XS)) → MARK(XS)
A__ADD(s(X), Y) → MARK(Y)
MARK(add(s(x0), y1)) → A__ADD(s(mark(x0)), mark(y1))
MARK(sel(s(y0), fib(x0))) → A__SEL(s(mark(y0)), a__fib(mark(x0)))
MARK(sel(fib1(y0, y1), fib(x0))) → A__SEL(a__fib1(mark(y0), mark(y1)), a__fib(mark(x0)))
MARK(sel(fib(y0), fib1(x0, x1))) → A__SEL(a__fib(mark(y0)), a__fib1(mark(x0), mark(x1)))
MARK(add(sel(x0, x1), y1)) → A__ADD(a__sel(mark(x0), mark(x1)), mark(y1))
MARK(sel(fib1(y0, y1), cons(x0, x1))) → A__SEL(a__fib1(mark(y0), mark(y1)), cons(mark(x0), 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)))
MARK(sel(X1, X2)) → MARK(X1)
A__SEL(s(add(y0, y1)), cons(y2, add(x0, x1))) → A__SEL(a__add(mark(y0), mark(y1)), a__add(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), cons(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), cons(mark(x0), x1))
MARK(sel(add(y0, y1), fib(x0))) → A__SEL(a__add(mark(y0), mark(y1)), a__fib(mark(x0)))
MARK(sel(fib(y0), add(x0, x1))) → A__SEL(a__fib(mark(y0)), a__add(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__FIB(fib(x0)) → A__SEL(a__fib(mark(x0)), a__fib1(s(0), s(0)))
A__SEL(s(s(y0)), cons(y1, 0)) → A__SEL(s(mark(y0)), 0)
MARK(sel(0, cons(x0, x1))) → A__SEL(0, cons(mark(x0), x1))
A__SEL(s(0), cons(y0, sel(x0, x1))) → A__SEL(0, a__sel(mark(x0), mark(x1)))
A__ADD(s(fib1(x0, x1)), y1) → A__ADD(a__fib1(mark(x0), mark(x1)), mark(y1))
MARK(fib(X)) → MARK(X)
A__SEL(s(sel(y0, y1)), cons(y2, fib(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__fib(mark(x0)))
A__SEL(s(add(y0, y1)), cons(y2, fib(x0))) → A__SEL(a__add(mark(y0), mark(y1)), a__fib(mark(x0)))
MARK(add(X1, X2)) → MARK(X2)
MARK(fib(X)) → A__FIB(mark(X))
MARK(fib1(X1, X2)) → MARK(X2)
MARK(sel(sel(y0, y1), fib(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__fib(mark(x0)))
MARK(sel(fib(y0), sel(x0, x1))) → A__SEL(a__fib(mark(y0)), a__sel(mark(x0), mark(x1)))
A__ADD(s(sel(x0, x1)), y1) → A__ADD(a__sel(mark(x0), mark(x1)), mark(y1))
MARK(add(fib1(x0, x1), y1)) → A__ADD(a__fib1(mark(x0), mark(x1)), mark(y1))
A__SEL(s(s(y0)), cons(y1, fib(x0))) → A__SEL(s(mark(y0)), a__fib(mark(x0)))
MARK(sel(0, fib1(x0, x1))) → A__SEL(0, a__fib1(mark(x0), mark(x1)))
A__SEL(s(s(y0)), cons(y1, fib1(x0, x1))) → A__SEL(s(mark(y0)), a__fib1(mark(x0), mark(x1)))
A__SEL(s(0), cons(y0, cons(x0, x1))) → A__SEL(0, cons(mark(x0), x1))
A__SEL(s(fib1(y0, y1)), cons(y2, add(x0, x1))) → A__SEL(a__fib1(mark(y0), mark(y1)), a__add(mark(x0), mark(x1)))
A__SEL(s(fib(x0)), cons(y1, y2)) → A__SEL(a__fib(mark(x0)), mark(y2))
A__SEL(s(add(y0, y1)), cons(y2, fib1(x0, x1))) → A__SEL(a__add(mark(y0), mark(y1)), a__fib1(mark(x0), mark(x1)))
A__SEL(s(0), cons(y0, fib(x0))) → A__SEL(0, a__fib(mark(x0)))
MARK(sel(s(y0), fib1(x0, x1))) → A__SEL(s(mark(y0)), a__fib1(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), fib1(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), a__fib1(mark(x0), mark(x1)))
MARK(sel(fib1(y0, y1), sel(x0, x1))) → A__SEL(a__fib1(mark(y0), mark(y1)), a__sel(mark(x0), mark(x1)))
A__SEL(s(s(y0)), cons(y1, add(x0, x1))) → A__SEL(s(mark(y0)), a__add(mark(x0), mark(x1)))
A__SEL(s(fib1(y0, y1)), cons(y2, fib(x0))) → A__SEL(a__fib1(mark(y0), mark(y1)), a__fib(mark(x0)))
MARK(sel(0, sel(x0, x1))) → A__SEL(0, a__sel(mark(x0), mark(x1)))
A__SEL(s(add(y0, y1)), cons(y2, sel(x0, x1))) → A__SEL(a__add(mark(y0), mark(y1)), a__sel(mark(x0), mark(x1)))
A__SEL(s(sel(y0, y1)), cons(y2, add(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), a__add(mark(x0), mark(x1)))
A__SEL(s(0), cons(y0, fib1(x0, x1))) → A__SEL(0, a__fib1(mark(x0), mark(x1)))

The TRS R consists of the following rules:

a__fib(N) → a__sel(mark(N), a__fib1(s(0), s(0)))
a__fib1(X, Y) → cons(mark(X), fib1(Y, add(X, Y)))
a__add(0, X) → mark(X)
a__add(s(X), Y) → s(a__add(mark(X), mark(Y)))
a__sel(0, cons(X, XS)) → mark(X)
a__sel(s(N), cons(X, XS)) → a__sel(mark(N), mark(XS))
mark(fib(X)) → a__fib(mark(X))
mark(sel(X1, X2)) → a__sel(mark(X1), mark(X2))
mark(fib1(X1, X2)) → a__fib1(mark(X1), mark(X2))
mark(add(X1, X2)) → a__add(mark(X1), mark(X2))
mark(s(X)) → s(mark(X))
mark(0) → 0
mark(cons(X1, X2)) → cons(mark(X1), X2)
a__fib(X) → fib(X)
a__sel(X1, X2) → sel(X1, X2)
a__fib1(X1, X2) → fib1(X1, X2)
a__add(X1, X2) → add(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 1 SCC with 2 less nodes.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ Narrowing

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ QDP

              ↳ Narrowing

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ QDP

                      ↳ Narrowing

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ QDP

                              ↳ Narrowing

                                ↳ QDP

                                  ↳ DependencyGraphProof

                                    ↳ QDP

                                      ↳ Narrowing

                                        ↳ QDP

                                          ↳ DependencyGraphProof

                                            ↳ QDP

                                              ↳ Narrowing

                                                ↳ QDP

                                                  ↳ DependencyGraphProof

                                                    ↳ QDP

                                                      ↳ Narrowing

                                                        ↳ QDP

                                                          ↳ DependencyGraphProof

                                                            ↳ QDP

                                                              ↳ Narrowing

                                                                ↳ QDP

                                                                  ↳ DependencyGraphProof

                                                                    ↳ QDP

                                                                      ↳ Narrowing

                                                                        ↳ QDP

                                                                          ↳ DependencyGraphProof

                                                                            ↳ QDP

                                                                              ↳ Narrowing

                                                                                ↳ QDP

                                                                                  ↳ DependencyGraphProof

                                                                                    ↳ QDP

                                                                                      ↳ Narrowing

                                                                                        ↳ QDP

                                                                                          ↳ DependencyGraphProof

                                                                                            ↳ QDP

                                                                                              ↳ Narrowing

                                                                                                ↳ QDP

                                                                                                  ↳ DependencyGraphProof

                                                                                                    ↳ QDP

                                                                                                      ↳ Narrowing

                                                                                                        ↳ QDP

                                                                                                          ↳ DependencyGraphProof

                                                                                                            ↳ QDP

                                                                                                              ↳ Narrowing

                                                                                                                ↳ QDP

                                                                                                                  ↳ DependencyGraphProof

                                                                                                                    ↳ QDP

                                                                                                                      ↳ Narrowing

                                                                                                                        ↳ QDP

                                                                                                                          ↳ DependencyGraphProof

                                                                                                                            ↳ QDP

                                                                                                                              ↳ Narrowing

                                                                                                                                ↳ QDP

                                                                                                                                  ↳ DependencyGraphProof

                                                                                                                                    ↳ QDP

                                                                                                                                      ↳ Narrowing

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

MARK(sel(sel(y0, y1), sel(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), a__sel(mark(x0), mark(x1)))
MARK(sel(fib1(y0, y1), fib1(x0, x1))) → A__SEL(a__fib1(mark(y0), mark(y1)), a__fib1(mark(x0), mark(x1)))
MARK(fib1(X1, X2)) → MARK(X1)
MARK(sel(s(y0), cons(x0, x1))) → A__SEL(s(mark(y0)), cons(mark(x0), x1))
MARK(cons(X1, X2)) → MARK(X1)
A__ADD(s(add(x0, x1)), y1) → A__ADD(a__add(mark(x0), mark(x1)), mark(y1))
MARK(sel(0, add(x0, x1))) → A__SEL(0, a__add(mark(x0), mark(x1)))
A__SEL(s(fib1(y0, y1)), cons(y2, fib1(x0, x1))) → A__SEL(a__fib1(mark(y0), mark(y1)), a__fib1(mark(x0), mark(x1)))
MARK(sel(0, fib(x0))) → A__SEL(0, a__fib(mark(x0)))
A__FIB(N) → MARK(N)
MARK(sel(fib1(y0, y1), add(x0, x1))) → A__SEL(a__fib1(mark(y0), mark(y1)), a__add(mark(x0), mark(x1)))
MARK(sel(add(y0, y1), fib1(x0, x1))) → A__SEL(a__add(mark(y0), mark(y1)), a__fib1(mark(x0), mark(x1)))
A__FIB(s(x0)) → A__SEL(s(mark(x0)), a__fib1(s(0), s(0)))
A__SEL(s(N), cons(X, XS)) → MARK(XS)
A__ADD(s(X), Y) → MARK(Y)
A__ADD(s(0), y1) → A__ADD(0, mark(y1))
MARK(add(0, y1)) → A__ADD(0, mark(y1))
MARK(sel(fib(y0), cons(x0, x1))) → A__SEL(a__fib(mark(y0)), cons(mark(x0), x1))
A__FIB(sel(x0, x1)) → A__SEL(a__sel(mark(x0), mark(x1)), a__fib1(s(0), s(0)))
MARK(add(s(x0), y1)) → A__ADD(s(mark(x0)), mark(y1))
MARK(sel(sel(y0, y1), add(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), a__add(mark(x0), mark(x1)))
MARK(sel(add(y0, y1), sel(x0, x1))) → A__SEL(a__add(mark(y0), mark(y1)), a__sel(mark(x0), mark(x1)))
MARK(sel(s(y0), fib(x0))) → A__SEL(s(mark(y0)), a__fib(mark(x0)))
MARK(sel(fib1(y0, y1), fib(x0))) → A__SEL(a__fib1(mark(y0), mark(y1)), a__fib(mark(x0)))
MARK(sel(fib(y0), fib1(x0, x1))) → A__SEL(a__fib(mark(y0)), a__fib1(mark(x0), mark(x1)))
MARK(add(sel(x0, x1), y1)) → A__ADD(a__sel(mark(x0), mark(x1)), mark(y1))
MARK(sel(fib1(y0, y1), cons(x0, x1))) → A__SEL(a__fib1(mark(y0), mark(y1)), cons(mark(x0), 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)))
MARK(sel(X1, X2)) → MARK(X1)
A__SEL(s(add(y0, y1)), cons(y2, add(x0, x1))) → A__SEL(a__add(mark(y0), mark(y1)), a__add(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), cons(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), cons(mark(x0), x1))
A__SEL(s(0), cons(y0, add(x0, x1))) → A__SEL(0, a__add(mark(x0), mark(x1)))
MARK(sel(add(y0, y1), fib(x0))) → A__SEL(a__add(mark(y0), mark(y1)), a__fib(mark(x0)))
MARK(sel(fib(y0), add(x0, x1))) → A__SEL(a__fib(mark(y0)), a__add(mark(x0), mark(x1)))
MARK(add(add(x0, x1), y1)) → A__ADD(a__add(mark(x0), mark(x1)), mark(y1))
A__ADD(s(X), Y) → MARK(X)
A__SEL(s(sel(y0, y1)), cons(y2, cons(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), cons(mark(x0), x1))
A__FIB(N) → A__FIB1(s(0), s(0))
A__SEL(s(s(y0)), cons(y1, cons(x0, x1))) → A__SEL(s(mark(y0)), cons(mark(x0), x1))
MARK(add(fib(x0), y1)) → A__ADD(a__fib(mark(x0)), mark(y1))
A__FIB(fib(x0)) → A__SEL(a__fib(mark(x0)), a__fib1(s(0), s(0)))
A__ADD(s(fib(x0)), y1) → A__ADD(a__fib(mark(x0)), mark(y1))
A__SEL(s(add(y0, y1)), cons(y2, cons(x0, x1))) → A__SEL(a__add(mark(y0), mark(y1)), cons(mark(x0), x1))
MARK(sel(0, cons(x0, x1))) → A__SEL(0, cons(mark(x0), x1))
A__SEL(s(0), cons(y0, sel(x0, x1))) → A__SEL(0, a__sel(mark(x0), mark(x1)))
MARK(sel(add(y0, y1), cons(x0, x1))) → A__SEL(a__add(mark(y0), mark(y1)), cons(mark(x0), x1))
A__ADD(s(fib1(x0, x1)), y1) → A__ADD(a__fib1(mark(x0), mark(x1)), mark(y1))
MARK(fib(X)) → MARK(X)
A__SEL(s(sel(y0, y1)), cons(y2, fib(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__fib(mark(x0)))
A__SEL(s(add(y0, y1)), cons(y2, fib(x0))) → A__SEL(a__add(mark(y0), mark(y1)), a__fib(mark(x0)))
A__ADD(0, X) → MARK(X)
A__FIB1(X, Y) → MARK(X)
MARK(s(X)) → MARK(X)
MARK(add(X1, X2)) → MARK(X2)
MARK(sel(s(y0), sel(x0, x1))) → A__SEL(s(mark(y0)), a__sel(mark(x0), mark(x1)))
MARK(fib(X)) → A__FIB(mark(X))
MARK(fib1(X1, X2)) → MARK(X2)
A__FIB(0) → A__SEL(0, a__fib1(s(0), s(0)))
MARK(sel(sel(y0, y1), fib(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__fib(mark(x0)))
MARK(sel(fib(y0), sel(x0, x1))) → A__SEL(a__fib(mark(y0)), a__sel(mark(x0), mark(x1)))
A__ADD(s(sel(x0, x1)), y1) → A__ADD(a__sel(mark(x0), mark(x1)), mark(y1))
A__SEL(0, cons(X, XS)) → MARK(X)
MARK(add(X1, X2)) → MARK(X1)
MARK(add(fib1(x0, x1), y1)) → A__ADD(a__fib1(mark(x0), mark(x1)), mark(y1))
A__SEL(s(s(y0)), cons(y1, fib(x0))) → A__SEL(s(mark(y0)), a__fib(mark(x0)))
MARK(sel(0, fib1(x0, x1))) → A__SEL(0, a__fib1(mark(x0), mark(x1)))
A__SEL(s(s(y0)), cons(y1, fib1(x0, x1))) → A__SEL(s(mark(y0)), a__fib1(mark(x0), mark(x1)))
A__FIB(fib1(x0, x1)) → A__SEL(a__fib1(mark(x0), mark(x1)), a__fib1(s(0), s(0)))
A__SEL(s(0), cons(y0, cons(x0, x1))) → A__SEL(0, cons(mark(x0), x1))
A__SEL(s(fib1(y0, y1)), cons(y2, add(x0, x1))) → A__SEL(a__fib1(mark(y0), mark(y1)), a__add(mark(x0), mark(x1)))
A__SEL(s(fib(x0)), cons(y1, y2)) → A__SEL(a__fib(mark(x0)), mark(y2))
A__SEL(s(add(y0, y1)), cons(y2, fib1(x0, x1))) → A__SEL(a__add(mark(y0), mark(y1)), a__fib1(mark(x0), mark(x1)))
A__SEL(s(fib1(y0, y1)), cons(y2, cons(x0, x1))) → A__SEL(a__fib1(mark(y0), mark(y1)), cons(mark(x0), x1))
A__SEL(s(fib1(y0, y1)), cons(y2, sel(x0, x1))) → A__SEL(a__fib1(mark(y0), mark(y1)), a__sel(mark(x0), mark(x1)))
A__SEL(s(sel(y0, y1)), cons(y2, fib1(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), a__fib1(mark(x0), mark(x1)))
A__SEL(s(N), cons(X, XS)) → MARK(N)
A__ADD(s(s(x0)), y1) → A__ADD(s(mark(x0)), mark(y1))
A__SEL(s(0), cons(y0, fib(x0))) → A__SEL(0, a__fib(mark(x0)))
MARK(fib1(X1, X2)) → A__FIB1(mark(X1), mark(X2))
MARK(sel(s(y0), fib1(x0, x1))) → A__SEL(s(mark(y0)), a__fib1(mark(x0), mark(x1)))
MARK(sel(fib(y0), fib(x0))) → A__SEL(a__fib(mark(y0)), a__fib(mark(x0)))
MARK(sel(sel(y0, y1), fib1(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), a__fib1(mark(x0), mark(x1)))
MARK(sel(fib1(y0, y1), sel(x0, x1))) → A__SEL(a__fib1(mark(y0), mark(y1)), a__sel(mark(x0), mark(x1)))
MARK(sel(add(y0, y1), add(x0, x1))) → A__SEL(a__add(mark(y0), mark(y1)), a__add(mark(x0), mark(x1)))
MARK(sel(s(y0), add(x0, x1))) → A__SEL(s(mark(y0)), a__add(mark(x0), mark(x1)))
A__SEL(s(s(y0)), cons(y1, add(x0, x1))) → A__SEL(s(mark(y0)), a__add(mark(x0), mark(x1)))
A__SEL(s(fib1(y0, y1)), cons(y2, fib(x0))) → A__SEL(a__fib1(mark(y0), mark(y1)), a__fib(mark(x0)))
MARK(sel(0, sel(x0, x1))) → A__SEL(0, a__sel(mark(x0), mark(x1)))
A__SEL(s(add(y0, y1)), cons(y2, sel(x0, x1))) → A__SEL(a__add(mark(y0), mark(y1)), a__sel(mark(x0), mark(x1)))
A__SEL(s(sel(y0, y1)), cons(y2, add(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), a__add(mark(x0), mark(x1)))
A__FIB(add(x0, x1)) → A__SEL(a__add(mark(x0), mark(x1)), a__fib1(s(0), s(0)))
A__SEL(s(0), cons(y0, fib1(x0, x1))) → A__SEL(0, a__fib1(mark(x0), mark(x1)))
MARK(sel(X1, X2)) → MARK(X2)

The TRS R consists of the following rules:

a__fib(N) → a__sel(mark(N), a__fib1(s(0), s(0)))
a__fib1(X, Y) → cons(mark(X), fib1(Y, add(X, Y)))
a__add(0, X) → mark(X)
a__add(s(X), Y) → s(a__add(mark(X), mark(Y)))
a__sel(0, cons(X, XS)) → mark(X)
a__sel(s(N), cons(X, XS)) → a__sel(mark(N), mark(XS))
mark(fib(X)) → a__fib(mark(X))
mark(sel(X1, X2)) → a__sel(mark(X1), mark(X2))
mark(fib1(X1, X2)) → a__fib1(mark(X1), mark(X2))
mark(add(X1, X2)) → a__add(mark(X1), mark(X2))
mark(s(X)) → s(mark(X))
mark(0) → 0
mark(cons(X1, X2)) → cons(mark(X1), X2)
a__fib(X) → fib(X)
a__sel(X1, X2) → sel(X1, X2)
a__fib1(X1, X2) → fib1(X1, X2)
a__add(X1, X2) → add(X1, X2)

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

A__SEL(s(fib(y0)), cons(y1, s(x0))) → A__SEL(a__fib(mark(y0)), s(mark(x0)))
A__SEL(s(fib(y0)), cons(y1, 0)) → A__SEL(a__fib(mark(y0)), 0)
A__SEL(s(fib(y0)), cons(y1, fib(x0))) → A__SEL(a__fib(mark(y0)), a__fib(mark(x0)))
A__SEL(s(fib(y0)), cons(y1, cons(x0, x1))) → A__SEL(a__fib(mark(y0)), cons(mark(x0), x1))
A__SEL(s(fib(y0)), cons(y1, add(x0, x1))) → A__SEL(a__fib(mark(y0)), a__add(mark(x0), mark(x1)))
A__SEL(s(fib(y0)), cons(y1, sel(x0, x1))) → A__SEL(a__fib(mark(y0)), a__sel(mark(x0), mark(x1)))
A__SEL(s(fib(y0)), cons(y1, fib1(x0, x1))) → A__SEL(a__fib(mark(y0)), a__fib1(mark(x0), mark(x1)))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ Narrowing

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ QDP

              ↳ Narrowing

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ QDP

                      ↳ Narrowing

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ QDP

                              ↳ Narrowing

                                ↳ QDP

                                  ↳ DependencyGraphProof

                                    ↳ QDP

                                      ↳ Narrowing

                                        ↳ QDP

                                          ↳ DependencyGraphProof

                                            ↳ QDP

                                              ↳ Narrowing

                                                ↳ QDP

                                                  ↳ DependencyGraphProof

                                                    ↳ QDP

                                                      ↳ Narrowing

                                                        ↳ QDP

                                                          ↳ DependencyGraphProof

                                                            ↳ QDP

                                                              ↳ Narrowing

                                                                ↳ QDP

                                                                  ↳ DependencyGraphProof

                                                                    ↳ QDP

                                                                      ↳ Narrowing

                                                                        ↳ QDP

                                                                          ↳ DependencyGraphProof

                                                                            ↳ QDP

                                                                              ↳ Narrowing

                                                                                ↳ QDP

                                                                                  ↳ DependencyGraphProof

                                                                                    ↳ QDP

                                                                                      ↳ Narrowing

                                                                                        ↳ QDP

                                                                                          ↳ DependencyGraphProof

                                                                                            ↳ QDP

                                                                                              ↳ Narrowing

                                                                                                ↳ QDP

                                                                                                  ↳ DependencyGraphProof

                                                                                                    ↳ QDP

                                                                                                      ↳ Narrowing

                                                                                                        ↳ QDP

                                                                                                          ↳ DependencyGraphProof

                                                                                                            ↳ QDP

                                                                                                              ↳ Narrowing

                                                                                                                ↳ QDP

                                                                                                                  ↳ DependencyGraphProof

                                                                                                                    ↳ QDP

                                                                                                                      ↳ Narrowing

                                                                                                                        ↳ QDP

                                                                                                                          ↳ DependencyGraphProof

                                                                                                                            ↳ QDP

                                                                                                                              ↳ Narrowing

                                                                                                                                ↳ QDP

                                                                                                                                  ↳ DependencyGraphProof

                                                                                                                                    ↳ QDP

                                                                                                                                      ↳ Narrowing

                                                                                                                                        ↳ QDP

                                                                                                                                          ↳ DependencyGraphProof

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

MARK(sel(fib1(y0, y1), fib1(x0, x1))) → A__SEL(a__fib1(mark(y0), mark(y1)), a__fib1(mark(x0), mark(x1)))
MARK(sel(s(y0), cons(x0, x1))) → A__SEL(s(mark(y0)), cons(mark(x0), x1))
A__SEL(s(fib(y0)), cons(y1, add(x0, x1))) → A__SEL(a__fib(mark(y0)), a__add(mark(x0), mark(x1)))
A__ADD(s(add(x0, x1)), y1) → A__ADD(a__add(mark(x0), mark(x1)), mark(y1))
MARK(sel(0, add(x0, x1))) → A__SEL(0, a__add(mark(x0), mark(x1)))
MARK(sel(fib1(y0, y1), add(x0, x1))) → A__SEL(a__fib1(mark(y0), mark(y1)), a__add(mark(x0), mark(x1)))
MARK(sel(add(y0, y1), fib1(x0, x1))) → A__SEL(a__add(mark(y0), mark(y1)), a__fib1(mark(x0), mark(x1)))
A__ADD(s(0), y1) → A__ADD(0, mark(y1))
MARK(add(0, y1)) → A__ADD(0, mark(y1))
MARK(sel(fib(y0), cons(x0, x1))) → A__SEL(a__fib(mark(y0)), cons(mark(x0), x1))
A__FIB(sel(x0, x1)) → A__SEL(a__sel(mark(x0), mark(x1)), a__fib1(s(0), s(0)))
MARK(sel(sel(y0, y1), add(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), a__add(mark(x0), mark(x1)))
MARK(sel(add(y0, y1), sel(x0, x1))) → A__SEL(a__add(mark(y0), mark(y1)), a__sel(mark(x0), mark(x1)))
A__SEL(s(0), cons(y0, add(x0, x1))) → A__SEL(0, a__add(mark(x0), mark(x1)))
MARK(add(add(x0, x1), y1)) → A__ADD(a__add(mark(x0), mark(x1)), mark(y1))
A__ADD(s(X), Y) → MARK(X)
A__FIB(N) → A__FIB1(s(0), s(0))
MARK(add(fib(x0), y1)) → A__ADD(a__fib(mark(x0)), mark(y1))
A__ADD(s(fib(x0)), y1) → A__ADD(a__fib(mark(x0)), mark(y1))
A__SEL(s(add(y0, y1)), cons(y2, cons(x0, x1))) → A__SEL(a__add(mark(y0), mark(y1)), cons(mark(x0), x1))
A__SEL(s(fib(y0)), cons(y1, fib(x0))) → A__SEL(a__fib(mark(y0)), a__fib(mark(x0)))
MARK(sel(add(y0, y1), cons(x0, x1))) → A__SEL(a__add(mark(y0), mark(y1)), cons(mark(x0), x1))
A__ADD(0, X) → MARK(X)
A__FIB1(X, Y) → MARK(X)
MARK(s(X)) → MARK(X)
MARK(sel(s(y0), sel(x0, x1))) → A__SEL(s(mark(y0)), a__sel(mark(x0), mark(x1)))
A__FIB(0) → A__SEL(0, a__fib1(s(0), s(0)))
A__SEL(0, cons(X, XS)) → MARK(X)
MARK(add(X1, X2)) → MARK(X1)
A__FIB(fib1(x0, x1)) → A__SEL(a__fib1(mark(x0), mark(x1)), a__fib1(s(0), s(0)))
A__SEL(s(fib1(y0, y1)), cons(y2, cons(x0, x1))) → A__SEL(a__fib1(mark(y0), mark(y1)), cons(mark(x0), x1))
A__SEL(s(fib1(y0, y1)), cons(y2, sel(x0, x1))) → A__SEL(a__fib1(mark(y0), mark(y1)), a__sel(mark(x0), mark(x1)))
A__SEL(s(sel(y0, y1)), cons(y2, fib1(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), a__fib1(mark(x0), mark(x1)))
A__SEL(s(N), cons(X, XS)) → MARK(N)
A__ADD(s(s(x0)), y1) → A__ADD(s(mark(x0)), mark(y1))
MARK(fib1(X1, X2)) → A__FIB1(mark(X1), mark(X2))
A__SEL(s(fib(y0)), cons(y1, 0)) → A__SEL(a__fib(mark(y0)), 0)
MARK(sel(fib(y0), fib(x0))) → A__SEL(a__fib(mark(y0)), a__fib(mark(x0)))
MARK(sel(add(y0, y1), add(x0, x1))) → A__SEL(a__add(mark(y0), mark(y1)), a__add(mark(x0), mark(x1)))
MARK(sel(s(y0), add(x0, x1))) → A__SEL(s(mark(y0)), a__add(mark(x0), mark(x1)))
A__FIB(add(x0, x1)) → A__SEL(a__add(mark(x0), mark(x1)), a__fib1(s(0), s(0)))
MARK(sel(X1, X2)) → MARK(X2)
MARK(sel(sel(y0, y1), sel(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), a__sel(mark(x0), mark(x1)))
MARK(fib1(X1, X2)) → MARK(X1)
MARK(cons(X1, X2)) → MARK(X1)
A__SEL(s(fib(y0)), cons(y1, cons(x0, x1))) → A__SEL(a__fib(mark(y0)), cons(mark(x0), x1))
A__SEL(s(fib1(y0, y1)), cons(y2, fib1(x0, x1))) → A__SEL(a__fib1(mark(y0), mark(y1)), a__fib1(mark(x0), mark(x1)))
MARK(sel(0, fib(x0))) → A__SEL(0, a__fib(mark(x0)))
A__FIB(N) → MARK(N)
A__FIB(s(x0)) → A__SEL(s(mark(x0)), a__fib1(s(0), s(0)))
A__SEL(s(N), cons(X, XS)) → MARK(XS)
A__ADD(s(X), Y) → MARK(Y)
MARK(add(s(x0), y1)) → A__ADD(s(mark(x0)), mark(y1))
MARK(sel(s(y0), fib(x0))) → A__SEL(s(mark(y0)), a__fib(mark(x0)))
MARK(sel(fib1(y0, y1), fib(x0))) → A__SEL(a__fib1(mark(y0), mark(y1)), a__fib(mark(x0)))
MARK(sel(fib(y0), fib1(x0, x1))) → A__SEL(a__fib(mark(y0)), a__fib1(mark(x0), mark(x1)))
MARK(add(sel(x0, x1), y1)) → A__ADD(a__sel(mark(x0), mark(x1)), mark(y1))
MARK(sel(fib1(y0, y1), cons(x0, x1))) → A__SEL(a__fib1(mark(y0), mark(y1)), cons(mark(x0), 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)))
MARK(sel(X1, X2)) → MARK(X1)
A__SEL(s(add(y0, y1)), cons(y2, add(x0, x1))) → A__SEL(a__add(mark(y0), mark(y1)), a__add(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), cons(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), cons(mark(x0), x1))
MARK(sel(add(y0, y1), fib(x0))) → A__SEL(a__add(mark(y0), mark(y1)), a__fib(mark(x0)))
MARK(sel(fib(y0), add(x0, x1))) → A__SEL(a__fib(mark(y0)), a__add(mark(x0), mark(x1)))
A__SEL(s(fib(y0)), cons(y1, sel(x0, x1))) → A__SEL(a__fib(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__FIB(fib(x0)) → A__SEL(a__fib(mark(x0)), a__fib1(s(0), s(0)))
MARK(sel(0, cons(x0, x1))) → A__SEL(0, cons(mark(x0), x1))
A__SEL(s(0), cons(y0, sel(x0, x1))) → A__SEL(0, a__sel(mark(x0), mark(x1)))
A__ADD(s(fib1(x0, x1)), y1) → A__ADD(a__fib1(mark(x0), mark(x1)), mark(y1))
MARK(fib(X)) → MARK(X)
A__SEL(s(sel(y0, y1)), cons(y2, fib(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__fib(mark(x0)))
A__SEL(s(add(y0, y1)), cons(y2, fib(x0))) → A__SEL(a__add(mark(y0), mark(y1)), a__fib(mark(x0)))
MARK(add(X1, X2)) → MARK(X2)
MARK(fib(X)) → A__FIB(mark(X))
MARK(fib1(X1, X2)) → MARK(X2)
MARK(sel(sel(y0, y1), fib(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__fib(mark(x0)))
MARK(sel(fib(y0), sel(x0, x1))) → A__SEL(a__fib(mark(y0)), a__sel(mark(x0), mark(x1)))
A__ADD(s(sel(x0, x1)), y1) → A__ADD(a__sel(mark(x0), mark(x1)), mark(y1))
MARK(add(fib1(x0, x1), y1)) → A__ADD(a__fib1(mark(x0), mark(x1)), mark(y1))
A__SEL(s(s(y0)), cons(y1, fib(x0))) → A__SEL(s(mark(y0)), a__fib(mark(x0)))
A__SEL(s(fib(y0)), cons(y1, s(x0))) → A__SEL(a__fib(mark(y0)), s(mark(x0)))
MARK(sel(0, fib1(x0, x1))) → A__SEL(0, a__fib1(mark(x0), mark(x1)))
A__SEL(s(s(y0)), cons(y1, fib1(x0, x1))) → A__SEL(s(mark(y0)), a__fib1(mark(x0), mark(x1)))
A__SEL(s(0), cons(y0, cons(x0, x1))) → A__SEL(0, cons(mark(x0), x1))
A__SEL(s(fib1(y0, y1)), cons(y2, add(x0, x1))) → A__SEL(a__fib1(mark(y0), mark(y1)), a__add(mark(x0), mark(x1)))
A__SEL(s(add(y0, y1)), cons(y2, fib1(x0, x1))) → A__SEL(a__add(mark(y0), mark(y1)), a__fib1(mark(x0), mark(x1)))
A__SEL(s(fib(y0)), cons(y1, fib1(x0, x1))) → A__SEL(a__fib(mark(y0)), a__fib1(mark(x0), mark(x1)))
A__SEL(s(0), cons(y0, fib(x0))) → A__SEL(0, a__fib(mark(x0)))
MARK(sel(s(y0), fib1(x0, x1))) → A__SEL(s(mark(y0)), a__fib1(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), fib1(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), a__fib1(mark(x0), mark(x1)))
MARK(sel(fib1(y0, y1), sel(x0, x1))) → A__SEL(a__fib1(mark(y0), mark(y1)), a__sel(mark(x0), mark(x1)))
A__SEL(s(s(y0)), cons(y1, add(x0, x1))) → A__SEL(s(mark(y0)), a__add(mark(x0), mark(x1)))
A__SEL(s(fib1(y0, y1)), cons(y2, fib(x0))) → A__SEL(a__fib1(mark(y0), mark(y1)), a__fib(mark(x0)))
MARK(sel(0, sel(x0, x1))) → A__SEL(0, a__sel(mark(x0), mark(x1)))
A__SEL(s(add(y0, y1)), cons(y2, sel(x0, x1))) → A__SEL(a__add(mark(y0), mark(y1)), a__sel(mark(x0), mark(x1)))
A__SEL(s(sel(y0, y1)), cons(y2, add(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), a__add(mark(x0), mark(x1)))
A__SEL(s(0), cons(y0, fib1(x0, x1))) → A__SEL(0, a__fib1(mark(x0), mark(x1)))

The TRS R consists of the following rules:

a__fib(N) → a__sel(mark(N), a__fib1(s(0), s(0)))
a__fib1(X, Y) → cons(mark(X), fib1(Y, add(X, Y)))
a__add(0, X) → mark(X)
a__add(s(X), Y) → s(a__add(mark(X), mark(Y)))
a__sel(0, cons(X, XS)) → mark(X)
a__sel(s(N), cons(X, XS)) → a__sel(mark(N), mark(XS))
mark(fib(X)) → a__fib(mark(X))
mark(sel(X1, X2)) → a__sel(mark(X1), mark(X2))
mark(fib1(X1, X2)) → a__fib1(mark(X1), mark(X2))
mark(add(X1, X2)) → a__add(mark(X1), mark(X2))
mark(s(X)) → s(mark(X))
mark(0) → 0
mark(cons(X1, X2)) → cons(mark(X1), X2)
a__fib(X) → fib(X)
a__sel(X1, X2) → sel(X1, X2)
a__fib1(X1, X2) → fib1(X1, X2)
a__add(X1, X2) → add(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 1 SCC with 2 less nodes.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ Narrowing

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ QDP

              ↳ Narrowing

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ QDP

                      ↳ Narrowing

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ QDP

                              ↳ Narrowing

                                ↳ QDP

                                  ↳ DependencyGraphProof

                                    ↳ QDP

                                      ↳ Narrowing

                                        ↳ QDP

                                          ↳ DependencyGraphProof

                                            ↳ QDP

                                              ↳ Narrowing

                                                ↳ QDP

                                                  ↳ DependencyGraphProof

                                                    ↳ QDP

                                                      ↳ Narrowing

                                                        ↳ QDP

                                                          ↳ DependencyGraphProof

                                                            ↳ QDP

                                                              ↳ Narrowing

                                                                ↳ QDP

                                                                  ↳ DependencyGraphProof

                                                                    ↳ QDP

                                                                      ↳ Narrowing

                                                                        ↳ QDP

                                                                          ↳ DependencyGraphProof

                                                                            ↳ QDP

                                                                              ↳ Narrowing

                                                                                ↳ QDP

                                                                                  ↳ DependencyGraphProof

                                                                                    ↳ QDP

                                                                                      ↳ Narrowing

                                                                                        ↳ QDP

                                                                                          ↳ DependencyGraphProof

                                                                                            ↳ QDP

                                                                                              ↳ Narrowing

                                                                                                ↳ QDP

                                                                                                  ↳ DependencyGraphProof

                                                                                                    ↳ QDP

                                                                                                      ↳ Narrowing

                                                                                                        ↳ QDP

                                                                                                          ↳ DependencyGraphProof

                                                                                                            ↳ QDP

                                                                                                              ↳ Narrowing

                                                                                                                ↳ QDP

                                                                                                                  ↳ DependencyGraphProof

                                                                                                                    ↳ QDP

                                                                                                                      ↳ Narrowing

                                                                                                                        ↳ QDP

                                                                                                                          ↳ DependencyGraphProof

                                                                                                                            ↳ QDP

                                                                                                                              ↳ Narrowing

                                                                                                                                ↳ QDP

                                                                                                                                  ↳ DependencyGraphProof

                                                                                                                                    ↳ QDP

                                                                                                                                      ↳ Narrowing

                                                                                                                                        ↳ QDP

                                                                                                                                          ↳ DependencyGraphProof

                                                                                                                                            ↳ QDP

                                                                                                                                              ↳ QDPOrderProof

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

MARK(sel(fib1(y0, y1), fib1(x0, x1))) → A__SEL(a__fib1(mark(y0), mark(y1)), a__fib1(mark(x0), mark(x1)))
MARK(sel(s(y0), cons(x0, x1))) → A__SEL(s(mark(y0)), cons(mark(x0), x1))
A__SEL(s(fib(y0)), cons(y1, add(x0, x1))) → A__SEL(a__fib(mark(y0)), a__add(mark(x0), mark(x1)))
A__ADD(s(add(x0, x1)), y1) → A__ADD(a__add(mark(x0), mark(x1)), mark(y1))
MARK(sel(0, add(x0, x1))) → A__SEL(0, a__add(mark(x0), mark(x1)))
MARK(sel(add(y0, y1), fib1(x0, x1))) → A__SEL(a__add(mark(y0), mark(y1)), a__fib1(mark(x0), mark(x1)))
MARK(sel(fib1(y0, y1), add(x0, x1))) → A__SEL(a__fib1(mark(y0), mark(y1)), a__add(mark(x0), mark(x1)))
A__ADD(s(0), y1) → A__ADD(0, mark(y1))
MARK(add(0, y1)) → A__ADD(0, mark(y1))
MARK(sel(fib(y0), cons(x0, x1))) → A__SEL(a__fib(mark(y0)), cons(mark(x0), x1))
A__FIB(sel(x0, x1)) → A__SEL(a__sel(mark(x0), mark(x1)), a__fib1(s(0), s(0)))
MARK(sel(add(y0, y1), sel(x0, x1))) → A__SEL(a__add(mark(y0), mark(y1)), a__sel(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), add(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), a__add(mark(x0), mark(x1)))
A__SEL(s(0), cons(y0, add(x0, x1))) → A__SEL(0, a__add(mark(x0), mark(x1)))
MARK(add(add(x0, x1), y1)) → A__ADD(a__add(mark(x0), mark(x1)), mark(y1))
A__ADD(s(X), Y) → MARK(X)
A__FIB(N) → A__FIB1(s(0), s(0))
MARK(add(fib(x0), y1)) → A__ADD(a__fib(mark(x0)), mark(y1))
A__ADD(s(fib(x0)), y1) → A__ADD(a__fib(mark(x0)), mark(y1))
A__SEL(s(add(y0, y1)), cons(y2, cons(x0, x1))) → A__SEL(a__add(mark(y0), mark(y1)), cons(mark(x0), x1))
A__SEL(s(fib(y0)), cons(y1, fib(x0))) → A__SEL(a__fib(mark(y0)), a__fib(mark(x0)))
MARK(sel(add(y0, y1), cons(x0, x1))) → A__SEL(a__add(mark(y0), mark(y1)), cons(mark(x0), x1))
A__FIB1(X, Y) → MARK(X)
A__ADD(0, X) → MARK(X)
MARK(s(X)) → MARK(X)
MARK(sel(s(y0), sel(x0, x1))) → A__SEL(s(mark(y0)), a__sel(mark(x0), mark(x1)))
A__FIB(0) → A__SEL(0, a__fib1(s(0), s(0)))
A__SEL(0, cons(X, XS)) → MARK(X)
MARK(add(X1, X2)) → MARK(X1)
A__FIB(fib1(x0, x1)) → A__SEL(a__fib1(mark(x0), mark(x1)), a__fib1(s(0), s(0)))
A__SEL(s(fib1(y0, y1)), cons(y2, cons(x0, x1))) → A__SEL(a__fib1(mark(y0), mark(y1)), cons(mark(x0), x1))
A__SEL(s(sel(y0, y1)), cons(y2, fib1(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), a__fib1(mark(x0), mark(x1)))
A__SEL(s(fib1(y0, y1)), cons(y2, sel(x0, x1))) → A__SEL(a__fib1(mark(y0), mark(y1)), a__sel(mark(x0), mark(x1)))
A__SEL(s(N), cons(X, XS)) → MARK(N)
A__ADD(s(s(x0)), y1) → A__ADD(s(mark(x0)), mark(y1))
MARK(fib1(X1, X2)) → A__FIB1(mark(X1), mark(X2))
MARK(sel(fib(y0), fib(x0))) → A__SEL(a__fib(mark(y0)), a__fib(mark(x0)))
MARK(sel(add(y0, y1), add(x0, x1))) → A__SEL(a__add(mark(y0), mark(y1)), a__add(mark(x0), mark(x1)))
MARK(sel(s(y0), add(x0, x1))) → A__SEL(s(mark(y0)), a__add(mark(x0), mark(x1)))
A__FIB(add(x0, x1)) → A__SEL(a__add(mark(x0), mark(x1)), a__fib1(s(0), s(0)))
MARK(sel(X1, X2)) → MARK(X2)
MARK(sel(sel(y0, y1), sel(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), a__sel(mark(x0), mark(x1)))
MARK(fib1(X1, X2)) → MARK(X1)
MARK(cons(X1, X2)) → MARK(X1)
A__SEL(s(fib(y0)), cons(y1, cons(x0, x1))) → A__SEL(a__fib(mark(y0)), cons(mark(x0), x1))
A__SEL(s(fib1(y0, y1)), cons(y2, fib1(x0, x1))) → A__SEL(a__fib1(mark(y0), mark(y1)), a__fib1(mark(x0), mark(x1)))
A__FIB(N) → MARK(N)
MARK(sel(0, fib(x0))) → A__SEL(0, a__fib(mark(x0)))
A__SEL(s(N), cons(X, XS)) → MARK(XS)
A__FIB(s(x0)) → A__SEL(s(mark(x0)), a__fib1(s(0), s(0)))
A__ADD(s(X), Y) → MARK(Y)
MARK(add(s(x0), y1)) → A__ADD(s(mark(x0)), mark(y1))
MARK(add(sel(x0, x1), y1)) → A__ADD(a__sel(mark(x0), mark(x1)), mark(y1))
MARK(sel(fib(y0), fib1(x0, x1))) → A__SEL(a__fib(mark(y0)), a__fib1(mark(x0), mark(x1)))
MARK(sel(fib1(y0, y1), fib(x0))) → A__SEL(a__fib1(mark(y0), mark(y1)), a__fib(mark(x0)))
MARK(sel(s(y0), fib(x0))) → A__SEL(s(mark(y0)), a__fib(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)))
MARK(sel(fib1(y0, y1), cons(x0, x1))) → A__SEL(a__fib1(mark(y0), mark(y1)), cons(mark(x0), 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(add(y0, y1)), cons(y2, add(x0, x1))) → A__SEL(a__add(mark(y0), mark(y1)), a__add(mark(x0), mark(x1)))
MARK(sel(X1, X2)) → MARK(X1)
MARK(sel(sel(y0, y1), cons(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), cons(mark(x0), x1))
MARK(sel(fib(y0), add(x0, x1))) → A__SEL(a__fib(mark(y0)), a__add(mark(x0), mark(x1)))
MARK(sel(add(y0, y1), fib(x0))) → A__SEL(a__add(mark(y0), mark(y1)), a__fib(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(fib(y0)), cons(y1, sel(x0, x1))) → A__SEL(a__fib(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__FIB(fib(x0)) → A__SEL(a__fib(mark(x0)), a__fib1(s(0), s(0)))
MARK(sel(0, cons(x0, x1))) → A__SEL(0, cons(mark(x0), x1))
A__SEL(s(0), cons(y0, sel(x0, x1))) → A__SEL(0, a__sel(mark(x0), mark(x1)))
A__ADD(s(fib1(x0, x1)), y1) → A__ADD(a__fib1(mark(x0), mark(x1)), mark(y1))
A__SEL(s(sel(y0, y1)), cons(y2, fib(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__fib(mark(x0)))
MARK(fib(X)) → MARK(X)
A__SEL(s(add(y0, y1)), cons(y2, fib(x0))) → A__SEL(a__add(mark(y0), mark(y1)), a__fib(mark(x0)))
MARK(add(X1, X2)) → MARK(X2)
MARK(fib(X)) → A__FIB(mark(X))
MARK(fib1(X1, X2)) → MARK(X2)
MARK(sel(fib(y0), sel(x0, x1))) → A__SEL(a__fib(mark(y0)), a__sel(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), fib(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__fib(mark(x0)))
A__ADD(s(sel(x0, x1)), y1) → A__ADD(a__sel(mark(x0), mark(x1)), mark(y1))
MARK(add(fib1(x0, x1), y1)) → A__ADD(a__fib1(mark(x0), mark(x1)), mark(y1))
A__SEL(s(s(y0)), cons(y1, fib(x0))) → A__SEL(s(mark(y0)), a__fib(mark(x0)))
A__SEL(s(s(y0)), cons(y1, fib1(x0, x1))) → A__SEL(s(mark(y0)), a__fib1(mark(x0), mark(x1)))
MARK(sel(0, fib1(x0, x1))) → A__SEL(0, a__fib1(mark(x0), mark(x1)))
A__SEL(s(fib1(y0, y1)), cons(y2, add(x0, x1))) → A__SEL(a__fib1(mark(y0), mark(y1)), a__add(mark(x0), mark(x1)))
A__SEL(s(add(y0, y1)), cons(y2, fib1(x0, x1))) → A__SEL(a__add(mark(y0), mark(y1)), a__fib1(mark(x0), mark(x1)))
A__SEL(s(0), cons(y0, cons(x0, x1))) → A__SEL(0, cons(mark(x0), x1))
A__SEL(s(fib(y0)), cons(y1, fib1(x0, x1))) → A__SEL(a__fib(mark(y0)), a__fib1(mark(x0), mark(x1)))
A__SEL(s(0), cons(y0, fib(x0))) → A__SEL(0, a__fib(mark(x0)))
MARK(sel(s(y0), fib1(x0, x1))) → A__SEL(s(mark(y0)), a__fib1(mark(x0), mark(x1)))
MARK(sel(fib1(y0, y1), sel(x0, x1))) → A__SEL(a__fib1(mark(y0), mark(y1)), a__sel(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), fib1(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), a__fib1(mark(x0), mark(x1)))
A__SEL(s(s(y0)), cons(y1, add(x0, x1))) → A__SEL(s(mark(y0)), a__add(mark(x0), mark(x1)))
A__SEL(s(fib1(y0, y1)), cons(y2, fib(x0))) → A__SEL(a__fib1(mark(y0), mark(y1)), a__fib(mark(x0)))
MARK(sel(0, sel(x0, x1))) → A__SEL(0, a__sel(mark(x0), mark(x1)))
A__SEL(s(sel(y0, y1)), cons(y2, add(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), a__add(mark(x0), mark(x1)))
A__SEL(s(add(y0, y1)), cons(y2, sel(x0, x1))) → A__SEL(a__add(mark(y0), mark(y1)), a__sel(mark(x0), mark(x1)))
A__SEL(s(0), cons(y0, fib1(x0, x1))) → A__SEL(0, a__fib1(mark(x0), mark(x1)))

The TRS R consists of the following rules:

a__fib(N) → a__sel(mark(N), a__fib1(s(0), s(0)))
a__fib1(X, Y) → cons(mark(X), fib1(Y, add(X, Y)))
a__add(0, X) → mark(X)
a__add(s(X), Y) → s(a__add(mark(X), mark(Y)))
a__sel(0, cons(X, XS)) → mark(X)
a__sel(s(N), cons(X, XS)) → a__sel(mark(N), mark(XS))
mark(fib(X)) → a__fib(mark(X))
mark(sel(X1, X2)) → a__sel(mark(X1), mark(X2))
mark(fib1(X1, X2)) → a__fib1(mark(X1), mark(X2))
mark(add(X1, X2)) → a__add(mark(X1), mark(X2))
mark(s(X)) → s(mark(X))
mark(0) → 0
mark(cons(X1, X2)) → cons(mark(X1), X2)
a__fib(X) → fib(X)
a__sel(X1, X2) → sel(X1, X2)
a__fib1(X1, X2) → fib1(X1, X2)
a__add(X1, X2) → add(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(fib1(y0, y1), fib1(x0, x1))) → A__SEL(a__fib1(mark(y0), mark(y1)), a__fib1(mark(x0), mark(x1)))
MARK(sel(fib1(y0, y1), add(x0, x1))) → A__SEL(a__fib1(mark(y0), mark(y1)), a__add(mark(x0), mark(x1)))
A__FIB(fib1(x0, x1)) → A__SEL(a__fib1(mark(x0), mark(x1)), a__fib1(s(0), s(0)))
A__SEL(s(fib1(y0, y1)), cons(y2, cons(x0, x1))) → A__SEL(a__fib1(mark(y0), mark(y1)), cons(mark(x0), x1))
A__SEL(s(fib1(y0, y1)), cons(y2, sel(x0, x1))) → A__SEL(a__fib1(mark(y0), mark(y1)), a__sel(mark(x0), mark(x1)))
A__SEL(s(fib1(y0, y1)), cons(y2, fib1(x0, x1))) → A__SEL(a__fib1(mark(y0), mark(y1)), a__fib1(mark(x0), mark(x1)))
MARK(sel(fib1(y0, y1), fib(x0))) → A__SEL(a__fib1(mark(y0), mark(y1)), a__fib(mark(x0)))
MARK(sel(fib1(y0, y1), cons(x0, x1))) → A__SEL(a__fib1(mark(y0), mark(y1)), cons(mark(x0), x1))
A__SEL(s(fib1(y0, y1)), cons(y2, add(x0, x1))) → A__SEL(a__fib1(mark(y0), mark(y1)), a__add(mark(x0), mark(x1)))
MARK(sel(fib1(y0, y1), sel(x0, x1))) → A__SEL(a__fib1(mark(y0), mark(y1)), a__sel(mark(x0), mark(x1)))
A__SEL(s(fib1(y0, y1)), cons(y2, fib(x0))) → A__SEL(a__fib1(mark(y0), mark(y1)), a__fib(mark(x0)))

The remaining pairs can at least be oriented weakly.

MARK(sel(s(y0), cons(x0, x1))) → A__SEL(s(mark(y0)), cons(mark(x0), x1))
A__SEL(s(fib(y0)), cons(y1, add(x0, x1))) → A__SEL(a__fib(mark(y0)), a__add(mark(x0), mark(x1)))
A__ADD(s(add(x0, x1)), y1) → A__ADD(a__add(mark(x0), mark(x1)), mark(y1))
MARK(sel(0, add(x0, x1))) → A__SEL(0, a__add(mark(x0), mark(x1)))
MARK(sel(add(y0, y1), fib1(x0, x1))) → A__SEL(a__add(mark(y0), mark(y1)), a__fib1(mark(x0), mark(x1)))
A__ADD(s(0), y1) → A__ADD(0, mark(y1))
MARK(add(0, y1)) → A__ADD(0, mark(y1))
MARK(sel(fib(y0), cons(x0, x1))) → A__SEL(a__fib(mark(y0)), cons(mark(x0), x1))
A__FIB(sel(x0, x1)) → A__SEL(a__sel(mark(x0), mark(x1)), a__fib1(s(0), s(0)))
MARK(sel(add(y0, y1), sel(x0, x1))) → A__SEL(a__add(mark(y0), mark(y1)), a__sel(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), add(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), a__add(mark(x0), mark(x1)))
A__SEL(s(0), cons(y0, add(x0, x1))) → A__SEL(0, a__add(mark(x0), mark(x1)))
MARK(add(add(x0, x1), y1)) → A__ADD(a__add(mark(x0), mark(x1)), mark(y1))
A__ADD(s(X), Y) → MARK(X)
A__FIB(N) → A__FIB1(s(0), s(0))
MARK(add(fib(x0), y1)) → A__ADD(a__fib(mark(x0)), mark(y1))
A__ADD(s(fib(x0)), y1) → A__ADD(a__fib(mark(x0)), mark(y1))
A__SEL(s(add(y0, y1)), cons(y2, cons(x0, x1))) → A__SEL(a__add(mark(y0), mark(y1)), cons(mark(x0), x1))
A__SEL(s(fib(y0)), cons(y1, fib(x0))) → A__SEL(a__fib(mark(y0)), a__fib(mark(x0)))
MARK(sel(add(y0, y1), cons(x0, x1))) → A__SEL(a__add(mark(y0), mark(y1)), cons(mark(x0), x1))
A__FIB1(X, Y) → MARK(X)
A__ADD(0, X) → MARK(X)
MARK(s(X)) → MARK(X)
MARK(sel(s(y0), sel(x0, x1))) → A__SEL(s(mark(y0)), a__sel(mark(x0), mark(x1)))
A__FIB(0) → A__SEL(0, a__fib1(s(0), s(0)))
A__SEL(0, cons(X, XS)) → MARK(X)
MARK(add(X1, X2)) → MARK(X1)
A__SEL(s(sel(y0, y1)), cons(y2, fib1(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), a__fib1(mark(x0), mark(x1)))
A__SEL(s(N), cons(X, XS)) → MARK(N)
A__ADD(s(s(x0)), y1) → A__ADD(s(mark(x0)), mark(y1))
MARK(fib1(X1, X2)) → A__FIB1(mark(X1), mark(X2))
MARK(sel(fib(y0), fib(x0))) → A__SEL(a__fib(mark(y0)), a__fib(mark(x0)))
MARK(sel(add(y0, y1), add(x0, x1))) → A__SEL(a__add(mark(y0), mark(y1)), a__add(mark(x0), mark(x1)))
MARK(sel(s(y0), add(x0, x1))) → A__SEL(s(mark(y0)), a__add(mark(x0), mark(x1)))
A__FIB(add(x0, x1)) → A__SEL(a__add(mark(x0), mark(x1)), a__fib1(s(0), s(0)))
MARK(sel(X1, X2)) → MARK(X2)
MARK(sel(sel(y0, y1), sel(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), a__sel(mark(x0), mark(x1)))
MARK(fib1(X1, X2)) → MARK(X1)
MARK(cons(X1, X2)) → MARK(X1)
A__SEL(s(fib(y0)), cons(y1, cons(x0, x1))) → A__SEL(a__fib(mark(y0)), cons(mark(x0), x1))
A__FIB(N) → MARK(N)
MARK(sel(0, fib(x0))) → A__SEL(0, a__fib(mark(x0)))
A__SEL(s(N), cons(X, XS)) → MARK(XS)
A__FIB(s(x0)) → A__SEL(s(mark(x0)), a__fib1(s(0), s(0)))
A__ADD(s(X), Y) → MARK(Y)
MARK(add(s(x0), y1)) → A__ADD(s(mark(x0)), mark(y1))
MARK(add(sel(x0, x1), y1)) → A__ADD(a__sel(mark(x0), mark(x1)), mark(y1))
MARK(sel(fib(y0), fib1(x0, x1))) → A__SEL(a__fib(mark(y0)), a__fib1(mark(x0), mark(x1)))
MARK(sel(s(y0), fib(x0))) → A__SEL(s(mark(y0)), a__fib(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(add(y0, y1)), cons(y2, add(x0, x1))) → A__SEL(a__add(mark(y0), mark(y1)), a__add(mark(x0), mark(x1)))
MARK(sel(X1, X2)) → MARK(X1)
MARK(sel(sel(y0, y1), cons(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), cons(mark(x0), x1))
MARK(sel(fib(y0), add(x0, x1))) → A__SEL(a__fib(mark(y0)), a__add(mark(x0), mark(x1)))
MARK(sel(add(y0, y1), fib(x0))) → A__SEL(a__add(mark(y0), mark(y1)), a__fib(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(fib(y0)), cons(y1, sel(x0, x1))) → A__SEL(a__fib(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__FIB(fib(x0)) → A__SEL(a__fib(mark(x0)), a__fib1(s(0), s(0)))
MARK(sel(0, cons(x0, x1))) → A__SEL(0, cons(mark(x0), x1))
A__SEL(s(0), cons(y0, sel(x0, x1))) → A__SEL(0, a__sel(mark(x0), mark(x1)))
A__ADD(s(fib1(x0, x1)), y1) → A__ADD(a__fib1(mark(x0), mark(x1)), mark(y1))
A__SEL(s(sel(y0, y1)), cons(y2, fib(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__fib(mark(x0)))
MARK(fib(X)) → MARK(X)
A__SEL(s(add(y0, y1)), cons(y2, fib(x0))) → A__SEL(a__add(mark(y0), mark(y1)), a__fib(mark(x0)))
MARK(add(X1, X2)) → MARK(X2)
MARK(fib(X)) → A__FIB(mark(X))
MARK(fib1(X1, X2)) → MARK(X2)
MARK(sel(fib(y0), sel(x0, x1))) → A__SEL(a__fib(mark(y0)), a__sel(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), fib(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__fib(mark(x0)))
A__ADD(s(sel(x0, x1)), y1) → A__ADD(a__sel(mark(x0), mark(x1)), mark(y1))
MARK(add(fib1(x0, x1), y1)) → A__ADD(a__fib1(mark(x0), mark(x1)), mark(y1))
A__SEL(s(s(y0)), cons(y1, fib(x0))) → A__SEL(s(mark(y0)), a__fib(mark(x0)))
A__SEL(s(s(y0)), cons(y1, fib1(x0, x1))) → A__SEL(s(mark(y0)), a__fib1(mark(x0), mark(x1)))
MARK(sel(0, fib1(x0, x1))) → A__SEL(0, a__fib1(mark(x0), mark(x1)))
A__SEL(s(add(y0, y1)), cons(y2, fib1(x0, x1))) → A__SEL(a__add(mark(y0), mark(y1)), a__fib1(mark(x0), mark(x1)))
A__SEL(s(0), cons(y0, cons(x0, x1))) → A__SEL(0, cons(mark(x0), x1))
A__SEL(s(fib(y0)), cons(y1, fib1(x0, x1))) → A__SEL(a__fib(mark(y0)), a__fib1(mark(x0), mark(x1)))
A__SEL(s(0), cons(y0, fib(x0))) → A__SEL(0, a__fib(mark(x0)))
MARK(sel(s(y0), fib1(x0, x1))) → A__SEL(s(mark(y0)), a__fib1(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), fib1(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), a__fib1(mark(x0), mark(x1)))
A__SEL(s(s(y0)), cons(y1, add(x0, x1))) → A__SEL(s(mark(y0)), a__add(mark(x0), mark(x1)))
MARK(sel(0, sel(x0, x1))) → A__SEL(0, a__sel(mark(x0), mark(x1)))
A__SEL(s(sel(y0, y1)), cons(y2, add(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), a__add(mark(x0), mark(x1)))
A__SEL(s(add(y0, y1)), cons(y2, sel(x0, x1))) → A__SEL(a__add(mark(y0), mark(y1)), a__sel(mark(x0), mark(x1)))
A__SEL(s(0), cons(y0, fib1(x0, x1))) → A__SEL(0, a__fib1(mark(x0), mark(x1)))

Used ordering: Polynomial interpretation [25]:

POL(0) = 1
POL(A__ADD(x₁, x₂)) = 1
POL(A__FIB(x₁)) = 1
POL(A__FIB1(x₁, x₂)) = 1
POL(A__SEL(x₁, x₂)) = x₁
POL(MARK(x₁)) = 1
POL(a__add(x₁, x₂)) = 1
POL(a__fib(x₁)) = 1
POL(a__fib1(x₁, x₂)) = 0
POL(a__sel(x₁, x₂)) = 1
POL(add(x₁, x₂)) = 0
POL(cons(x₁, x₂)) = 0
POL(fib(x₁)) = 0
POL(fib1(x₁, x₂)) = 0
POL(mark(x₁)) = 1
POL(s(x₁)) = 1
POL(sel(x₁, x₂)) = 0

The following usable rules [17] were oriented:

a__sel(s(N), cons(X, XS)) → a__sel(mark(N), mark(XS))
a__fib(N) → a__sel(mark(N), a__fib1(s(0), s(0)))
a__add(0, X) → mark(X)
a__sel(0, cons(X, XS)) → mark(X)
mark(fib(X)) → a__fib(mark(X))
mark(add(X1, X2)) → a__add(mark(X1), mark(X2))
mark(sel(X1, X2)) → a__sel(mark(X1), mark(X2))
a__add(s(X), Y) → s(a__add(mark(X), mark(Y)))
a__fib1(X, Y) → cons(mark(X), fib1(Y, add(X, Y)))
mark(fib1(X1, X2)) → a__fib1(mark(X1), mark(X2))
a__fib(X) → fib(X)
a__sel(X1, X2) → sel(X1, X2)
a__fib1(X1, X2) → fib1(X1, X2)
a__add(X1, X2) → add(X1, X2)
mark(s(X)) → s(mark(X))
mark(0) → 0
mark(cons(X1, X2)) → cons(mark(X1), X2)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ Narrowing

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ QDP

              ↳ Narrowing

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ QDP

                      ↳ Narrowing

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ QDP

                              ↳ Narrowing

                                ↳ QDP

                                  ↳ DependencyGraphProof

                                    ↳ QDP

                                      ↳ Narrowing

                                        ↳ QDP

                                          ↳ DependencyGraphProof

                                            ↳ QDP

                                              ↳ Narrowing

                                                ↳ QDP

                                                  ↳ DependencyGraphProof

                                                    ↳ QDP

                                                      ↳ Narrowing

                                                        ↳ QDP

                                                          ↳ DependencyGraphProof

                                                            ↳ QDP

                                                              ↳ Narrowing

                                                                ↳ QDP

                                                                  ↳ DependencyGraphProof

                                                                    ↳ QDP

                                                                      ↳ Narrowing

                                                                        ↳ QDP

                                                                          ↳ DependencyGraphProof

                                                                            ↳ QDP

                                                                              ↳ Narrowing

                                                                                ↳ QDP

                                                                                  ↳ DependencyGraphProof

                                                                                    ↳ QDP

                                                                                      ↳ Narrowing

                                                                                        ↳ QDP

                                                                                          ↳ DependencyGraphProof

                                                                                            ↳ QDP

                                                                                              ↳ Narrowing

                                                                                                ↳ QDP

                                                                                                  ↳ DependencyGraphProof

                                                                                                    ↳ QDP

                                                                                                      ↳ Narrowing

                                                                                                        ↳ QDP

                                                                                                          ↳ DependencyGraphProof

                                                                                                            ↳ QDP

                                                                                                              ↳ Narrowing

                                                                                                                ↳ QDP

                                                                                                                  ↳ DependencyGraphProof

                                                                                                                    ↳ QDP

                                                                                                                      ↳ Narrowing

                                                                                                                        ↳ QDP

                                                                                                                          ↳ DependencyGraphProof

                                                                                                                            ↳ QDP

                                                                                                                              ↳ Narrowing

                                                                                                                                ↳ QDP

                                                                                                                                  ↳ DependencyGraphProof

                                                                                                                                    ↳ QDP

                                                                                                                                      ↳ Narrowing

                                                                                                                                        ↳ QDP

                                                                                                                                          ↳ DependencyGraphProof

                                                                                                                                            ↳ QDP

                                                                                                                                              ↳ QDPOrderProof

                                                                                                                                                ↳ QDP

                                                                                                                                                  ↳ QDPOrderProof

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

MARK(sel(sel(y0, y1), sel(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), a__sel(mark(x0), mark(x1)))
MARK(fib1(X1, X2)) → MARK(X1)
MARK(sel(s(y0), cons(x0, x1))) → A__SEL(s(mark(y0)), cons(mark(x0), x1))
MARK(cons(X1, X2)) → MARK(X1)
A__SEL(s(fib(y0)), cons(y1, add(x0, x1))) → A__SEL(a__fib(mark(y0)), a__add(mark(x0), mark(x1)))
A__SEL(s(fib(y0)), cons(y1, cons(x0, x1))) → A__SEL(a__fib(mark(y0)), cons(mark(x0), x1))
A__ADD(s(add(x0, x1)), y1) → A__ADD(a__add(mark(x0), mark(x1)), mark(y1))
MARK(sel(0, add(x0, x1))) → A__SEL(0, a__add(mark(x0), mark(x1)))
A__FIB(N) → MARK(N)
MARK(sel(0, fib(x0))) → A__SEL(0, a__fib(mark(x0)))
MARK(sel(add(y0, y1), fib1(x0, x1))) → A__SEL(a__add(mark(y0), mark(y1)), a__fib1(mark(x0), mark(x1)))
A__SEL(s(N), cons(X, XS)) → MARK(XS)
A__FIB(s(x0)) → A__SEL(s(mark(x0)), a__fib1(s(0), s(0)))
A__ADD(s(X), Y) → MARK(Y)
A__ADD(s(0), y1) → A__ADD(0, mark(y1))
MARK(add(0, y1)) → A__ADD(0, mark(y1))
MARK(sel(fib(y0), cons(x0, x1))) → A__SEL(a__fib(mark(y0)), cons(mark(x0), x1))
A__FIB(sel(x0, x1)) → A__SEL(a__sel(mark(x0), mark(x1)), a__fib1(s(0), s(0)))
MARK(add(s(x0), y1)) → A__ADD(s(mark(x0)), mark(y1))
MARK(sel(add(y0, y1), sel(x0, x1))) → A__SEL(a__add(mark(y0), mark(y1)), a__sel(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), add(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), a__add(mark(x0), mark(x1)))
MARK(add(sel(x0, x1), y1)) → A__ADD(a__sel(mark(x0), mark(x1)), mark(y1))
MARK(sel(fib(y0), fib1(x0, x1))) → A__SEL(a__fib(mark(y0)), a__fib1(mark(x0), mark(x1)))
MARK(sel(s(y0), fib(x0))) → A__SEL(s(mark(y0)), a__fib(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)))
MARK(sel(X1, X2)) → MARK(X1)
A__SEL(s(add(y0, y1)), cons(y2, add(x0, x1))) → A__SEL(a__add(mark(y0), mark(y1)), a__add(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), cons(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), cons(mark(x0), x1))
A__SEL(s(0), cons(y0, add(x0, x1))) → A__SEL(0, a__add(mark(x0), mark(x1)))
MARK(sel(add(y0, y1), fib(x0))) → A__SEL(a__add(mark(y0), mark(y1)), a__fib(mark(x0)))
MARK(sel(fib(y0), add(x0, x1))) → A__SEL(a__fib(mark(y0)), a__add(mark(x0), mark(x1)))
MARK(add(add(x0, x1), y1)) → A__ADD(a__add(mark(x0), mark(x1)), mark(y1))
A__ADD(s(X), Y) → MARK(X)
A__SEL(s(fib(y0)), cons(y1, sel(x0, x1))) → A__SEL(a__fib(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__FIB(N) → A__FIB1(s(0), s(0))
A__SEL(s(s(y0)), cons(y1, cons(x0, x1))) → A__SEL(s(mark(y0)), cons(mark(x0), x1))
MARK(add(fib(x0), y1)) → A__ADD(a__fib(mark(x0)), mark(y1))
A__FIB(fib(x0)) → A__SEL(a__fib(mark(x0)), a__fib1(s(0), s(0)))
A__ADD(s(fib(x0)), y1) → A__ADD(a__fib(mark(x0)), mark(y1))
A__SEL(s(add(y0, y1)), cons(y2, cons(x0, x1))) → A__SEL(a__add(mark(y0), mark(y1)), cons(mark(x0), x1))
MARK(sel(0, cons(x0, x1))) → A__SEL(0, cons(mark(x0), x1))
A__SEL(s(fib(y0)), cons(y1, fib(x0))) → A__SEL(a__fib(mark(y0)), a__fib(mark(x0)))
A__SEL(s(0), cons(y0, sel(x0, x1))) → A__SEL(0, a__sel(mark(x0), mark(x1)))
MARK(sel(add(y0, y1), cons(x0, x1))) → A__SEL(a__add(mark(y0), mark(y1)), cons(mark(x0), x1))
A__ADD(s(fib1(x0, x1)), y1) → A__ADD(a__fib1(mark(x0), mark(x1)), mark(y1))
MARK(fib(X)) → MARK(X)
A__SEL(s(sel(y0, y1)), cons(y2, fib(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__fib(mark(x0)))
A__SEL(s(add(y0, y1)), cons(y2, fib(x0))) → A__SEL(a__add(mark(y0), mark(y1)), a__fib(mark(x0)))
A__ADD(0, X) → MARK(X)
A__FIB1(X, Y) → MARK(X)
MARK(s(X)) → MARK(X)
MARK(add(X1, X2)) → MARK(X2)
MARK(fib(X)) → A__FIB(mark(X))
MARK(sel(s(y0), sel(x0, x1))) → A__SEL(s(mark(y0)), a__sel(mark(x0), mark(x1)))
MARK(fib1(X1, X2)) → MARK(X2)
MARK(sel(sel(y0, y1), fib(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__fib(mark(x0)))
MARK(sel(fib(y0), sel(x0, x1))) → A__SEL(a__fib(mark(y0)), a__sel(mark(x0), mark(x1)))
A__FIB(0) → A__SEL(0, a__fib1(s(0), s(0)))
A__ADD(s(sel(x0, x1)), y1) → A__ADD(a__sel(mark(x0), mark(x1)), mark(y1))
A__SEL(0, cons(X, XS)) → MARK(X)
MARK(add(X1, X2)) → MARK(X1)
MARK(add(fib1(x0, x1), y1)) → A__ADD(a__fib1(mark(x0), mark(x1)), mark(y1))
A__SEL(s(s(y0)), cons(y1, fib(x0))) → A__SEL(s(mark(y0)), a__fib(mark(x0)))
MARK(sel(0, fib1(x0, x1))) → A__SEL(0, a__fib1(mark(x0), mark(x1)))
A__SEL(s(s(y0)), cons(y1, fib1(x0, x1))) → A__SEL(s(mark(y0)), a__fib1(mark(x0), mark(x1)))
A__SEL(s(0), cons(y0, cons(x0, x1))) → A__SEL(0, cons(mark(x0), x1))
A__SEL(s(add(y0, y1)), cons(y2, fib1(x0, x1))) → A__SEL(a__add(mark(y0), mark(y1)), a__fib1(mark(x0), mark(x1)))
A__SEL(s(fib(y0)), cons(y1, fib1(x0, x1))) → A__SEL(a__fib(mark(y0)), a__fib1(mark(x0), mark(x1)))
A__SEL(s(sel(y0, y1)), cons(y2, fib1(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), a__fib1(mark(x0), mark(x1)))
A__SEL(s(N), cons(X, XS)) → MARK(N)
A__ADD(s(s(x0)), y1) → A__ADD(s(mark(x0)), mark(y1))
A__SEL(s(0), cons(y0, fib(x0))) → A__SEL(0, a__fib(mark(x0)))
MARK(fib1(X1, X2)) → A__FIB1(mark(X1), mark(X2))
MARK(sel(s(y0), fib1(x0, x1))) → A__SEL(s(mark(y0)), a__fib1(mark(x0), mark(x1)))
MARK(sel(fib(y0), fib(x0))) → A__SEL(a__fib(mark(y0)), a__fib(mark(x0)))
MARK(sel(sel(y0, y1), fib1(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), a__fib1(mark(x0), mark(x1)))
MARK(sel(add(y0, y1), add(x0, x1))) → A__SEL(a__add(mark(y0), mark(y1)), a__add(mark(x0), mark(x1)))
MARK(sel(s(y0), add(x0, x1))) → A__SEL(s(mark(y0)), a__add(mark(x0), mark(x1)))
A__SEL(s(s(y0)), cons(y1, add(x0, x1))) → A__SEL(s(mark(y0)), a__add(mark(x0), mark(x1)))
MARK(sel(0, sel(x0, x1))) → A__SEL(0, a__sel(mark(x0), mark(x1)))
A__SEL(s(add(y0, y1)), cons(y2, sel(x0, x1))) → A__SEL(a__add(mark(y0), mark(y1)), a__sel(mark(x0), mark(x1)))
A__SEL(s(sel(y0, y1)), cons(y2, add(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), a__add(mark(x0), mark(x1)))
A__FIB(add(x0, x1)) → A__SEL(a__add(mark(x0), mark(x1)), a__fib1(s(0), s(0)))
A__SEL(s(0), cons(y0, fib1(x0, x1))) → A__SEL(0, a__fib1(mark(x0), mark(x1)))
MARK(sel(X1, X2)) → MARK(X2)

The TRS R consists of the following rules:

a__fib(N) → a__sel(mark(N), a__fib1(s(0), s(0)))
a__fib1(X, Y) → cons(mark(X), fib1(Y, add(X, Y)))
a__add(0, X) → mark(X)
a__add(s(X), Y) → s(a__add(mark(X), mark(Y)))
a__sel(0, cons(X, XS)) → mark(X)
a__sel(s(N), cons(X, XS)) → a__sel(mark(N), mark(XS))
mark(fib(X)) → a__fib(mark(X))
mark(sel(X1, X2)) → a__sel(mark(X1), mark(X2))
mark(fib1(X1, X2)) → a__fib1(mark(X1), mark(X2))
mark(add(X1, X2)) → a__add(mark(X1), mark(X2))
mark(s(X)) → s(mark(X))
mark(0) → 0
mark(cons(X1, X2)) → cons(mark(X1), X2)
a__fib(X) → fib(X)
a__sel(X1, X2) → sel(X1, X2)
a__fib1(X1, X2) → fib1(X1, X2)
a__add(X1, X2) → add(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__ADD(s(fib1(x0, x1)), y1) → A__ADD(a__fib1(mark(x0), mark(x1)), mark(y1))
MARK(add(fib1(x0, x1), y1)) → A__ADD(a__fib1(mark(x0), mark(x1)), mark(y1))

The remaining pairs can at least be oriented weakly.

MARK(sel(sel(y0, y1), sel(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), a__sel(mark(x0), mark(x1)))
MARK(fib1(X1, X2)) → MARK(X1)
MARK(sel(s(y0), cons(x0, x1))) → A__SEL(s(mark(y0)), cons(mark(x0), x1))
MARK(cons(X1, X2)) → MARK(X1)
A__SEL(s(fib(y0)), cons(y1, add(x0, x1))) → A__SEL(a__fib(mark(y0)), a__add(mark(x0), mark(x1)))
A__SEL(s(fib(y0)), cons(y1, cons(x0, x1))) → A__SEL(a__fib(mark(y0)), cons(mark(x0), x1))
A__ADD(s(add(x0, x1)), y1) → A__ADD(a__add(mark(x0), mark(x1)), mark(y1))
MARK(sel(0, add(x0, x1))) → A__SEL(0, a__add(mark(x0), mark(x1)))
A__FIB(N) → MARK(N)
MARK(sel(0, fib(x0))) → A__SEL(0, a__fib(mark(x0)))
MARK(sel(add(y0, y1), fib1(x0, x1))) → A__SEL(a__add(mark(y0), mark(y1)), a__fib1(mark(x0), mark(x1)))
A__SEL(s(N), cons(X, XS)) → MARK(XS)
A__FIB(s(x0)) → A__SEL(s(mark(x0)), a__fib1(s(0), s(0)))
A__ADD(s(X), Y) → MARK(Y)
A__ADD(s(0), y1) → A__ADD(0, mark(y1))
MARK(add(0, y1)) → A__ADD(0, mark(y1))
MARK(sel(fib(y0), cons(x0, x1))) → A__SEL(a__fib(mark(y0)), cons(mark(x0), x1))
A__FIB(sel(x0, x1)) → A__SEL(a__sel(mark(x0), mark(x1)), a__fib1(s(0), s(0)))
MARK(add(s(x0), y1)) → A__ADD(s(mark(x0)), mark(y1))
MARK(sel(add(y0, y1), sel(x0, x1))) → A__SEL(a__add(mark(y0), mark(y1)), a__sel(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), add(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), a__add(mark(x0), mark(x1)))
MARK(add(sel(x0, x1), y1)) → A__ADD(a__sel(mark(x0), mark(x1)), mark(y1))
MARK(sel(fib(y0), fib1(x0, x1))) → A__SEL(a__fib(mark(y0)), a__fib1(mark(x0), mark(x1)))
MARK(sel(s(y0), fib(x0))) → A__SEL(s(mark(y0)), a__fib(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)))
MARK(sel(X1, X2)) → MARK(X1)
A__SEL(s(add(y0, y1)), cons(y2, add(x0, x1))) → A__SEL(a__add(mark(y0), mark(y1)), a__add(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), cons(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), cons(mark(x0), x1))
A__SEL(s(0), cons(y0, add(x0, x1))) → A__SEL(0, a__add(mark(x0), mark(x1)))
MARK(sel(add(y0, y1), fib(x0))) → A__SEL(a__add(mark(y0), mark(y1)), a__fib(mark(x0)))
MARK(sel(fib(y0), add(x0, x1))) → A__SEL(a__fib(mark(y0)), a__add(mark(x0), mark(x1)))
MARK(add(add(x0, x1), y1)) → A__ADD(a__add(mark(x0), mark(x1)), mark(y1))
A__ADD(s(X), Y) → MARK(X)
A__SEL(s(fib(y0)), cons(y1, sel(x0, x1))) → A__SEL(a__fib(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__FIB(N) → A__FIB1(s(0), s(0))
A__SEL(s(s(y0)), cons(y1, cons(x0, x1))) → A__SEL(s(mark(y0)), cons(mark(x0), x1))
MARK(add(fib(x0), y1)) → A__ADD(a__fib(mark(x0)), mark(y1))
A__FIB(fib(x0)) → A__SEL(a__fib(mark(x0)), a__fib1(s(0), s(0)))
A__ADD(s(fib(x0)), y1) → A__ADD(a__fib(mark(x0)), mark(y1))
A__SEL(s(add(y0, y1)), cons(y2, cons(x0, x1))) → A__SEL(a__add(mark(y0), mark(y1)), cons(mark(x0), x1))
MARK(sel(0, cons(x0, x1))) → A__SEL(0, cons(mark(x0), x1))
A__SEL(s(fib(y0)), cons(y1, fib(x0))) → A__SEL(a__fib(mark(y0)), a__fib(mark(x0)))
A__SEL(s(0), cons(y0, sel(x0, x1))) → A__SEL(0, a__sel(mark(x0), mark(x1)))
MARK(sel(add(y0, y1), cons(x0, x1))) → A__SEL(a__add(mark(y0), mark(y1)), cons(mark(x0), x1))
MARK(fib(X)) → MARK(X)
A__SEL(s(sel(y0, y1)), cons(y2, fib(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__fib(mark(x0)))
A__SEL(s(add(y0, y1)), cons(y2, fib(x0))) → A__SEL(a__add(mark(y0), mark(y1)), a__fib(mark(x0)))
A__ADD(0, X) → MARK(X)
A__FIB1(X, Y) → MARK(X)
MARK(s(X)) → MARK(X)
MARK(add(X1, X2)) → MARK(X2)
MARK(fib(X)) → A__FIB(mark(X))
MARK(sel(s(y0), sel(x0, x1))) → A__SEL(s(mark(y0)), a__sel(mark(x0), mark(x1)))
MARK(fib1(X1, X2)) → MARK(X2)
MARK(sel(sel(y0, y1), fib(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__fib(mark(x0)))
MARK(sel(fib(y0), sel(x0, x1))) → A__SEL(a__fib(mark(y0)), a__sel(mark(x0), mark(x1)))
A__FIB(0) → A__SEL(0, a__fib1(s(0), s(0)))
A__ADD(s(sel(x0, x1)), y1) → A__ADD(a__sel(mark(x0), mark(x1)), mark(y1))
A__SEL(0, cons(X, XS)) → MARK(X)
MARK(add(X1, X2)) → MARK(X1)
A__SEL(s(s(y0)), cons(y1, fib(x0))) → A__SEL(s(mark(y0)), a__fib(mark(x0)))
MARK(sel(0, fib1(x0, x1))) → A__SEL(0, a__fib1(mark(x0), mark(x1)))
A__SEL(s(s(y0)), cons(y1, fib1(x0, x1))) → A__SEL(s(mark(y0)), a__fib1(mark(x0), mark(x1)))
A__SEL(s(0), cons(y0, cons(x0, x1))) → A__SEL(0, cons(mark(x0), x1))
A__SEL(s(add(y0, y1)), cons(y2, fib1(x0, x1))) → A__SEL(a__add(mark(y0), mark(y1)), a__fib1(mark(x0), mark(x1)))
A__SEL(s(fib(y0)), cons(y1, fib1(x0, x1))) → A__SEL(a__fib(mark(y0)), a__fib1(mark(x0), mark(x1)))
A__SEL(s(sel(y0, y1)), cons(y2, fib1(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), a__fib1(mark(x0), mark(x1)))
A__SEL(s(N), cons(X, XS)) → MARK(N)
A__ADD(s(s(x0)), y1) → A__ADD(s(mark(x0)), mark(y1))
A__SEL(s(0), cons(y0, fib(x0))) → A__SEL(0, a__fib(mark(x0)))
MARK(fib1(X1, X2)) → A__FIB1(mark(X1), mark(X2))
MARK(sel(s(y0), fib1(x0, x1))) → A__SEL(s(mark(y0)), a__fib1(mark(x0), mark(x1)))
MARK(sel(fib(y0), fib(x0))) → A__SEL(a__fib(mark(y0)), a__fib(mark(x0)))
MARK(sel(sel(y0, y1), fib1(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), a__fib1(mark(x0), mark(x1)))
MARK(sel(add(y0, y1), add(x0, x1))) → A__SEL(a__add(mark(y0), mark(y1)), a__add(mark(x0), mark(x1)))
MARK(sel(s(y0), add(x0, x1))) → A__SEL(s(mark(y0)), a__add(mark(x0), mark(x1)))
A__SEL(s(s(y0)), cons(y1, add(x0, x1))) → A__SEL(s(mark(y0)), a__add(mark(x0), mark(x1)))
MARK(sel(0, sel(x0, x1))) → A__SEL(0, a__sel(mark(x0), mark(x1)))
A__SEL(s(add(y0, y1)), cons(y2, sel(x0, x1))) → A__SEL(a__add(mark(y0), mark(y1)), a__sel(mark(x0), mark(x1)))
A__SEL(s(sel(y0, y1)), cons(y2, add(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), a__add(mark(x0), mark(x1)))
A__FIB(add(x0, x1)) → A__SEL(a__add(mark(x0), mark(x1)), a__fib1(s(0), s(0)))
A__SEL(s(0), cons(y0, fib1(x0, x1))) → A__SEL(0, a__fib1(mark(x0), mark(x1)))
MARK(sel(X1, X2)) → MARK(X2)

Used ordering: Polynomial interpretation [25]:

POL(0) = 1
POL(A__ADD(x₁, x₂)) = x₁
POL(A__FIB(x₁)) = 1
POL(A__FIB1(x₁, x₂)) = 1
POL(A__SEL(x₁, x₂)) = 1
POL(MARK(x₁)) = 1
POL(a__add(x₁, x₂)) = x₁
POL(a__fib(x₁)) = 1
POL(a__fib1(x₁, x₂)) = 0
POL(a__sel(x₁, x₂)) = 1
POL(add(x₁, x₂)) = 0
POL(cons(x₁, x₂)) = 0
POL(fib(x₁)) = 0
POL(fib1(x₁, x₂)) = 0
POL(mark(x₁)) = 1
POL(s(x₁)) = 1
POL(sel(x₁, x₂)) = 0

The following usable rules [17] were oriented:

a__sel(s(N), cons(X, XS)) → a__sel(mark(N), mark(XS))
a__fib(N) → a__sel(mark(N), a__fib1(s(0), s(0)))
a__add(0, X) → mark(X)
a__sel(0, cons(X, XS)) → mark(X)
mark(fib(X)) → a__fib(mark(X))
mark(add(X1, X2)) → a__add(mark(X1), mark(X2))
mark(sel(X1, X2)) → a__sel(mark(X1), mark(X2))
a__add(s(X), Y) → s(a__add(mark(X), mark(Y)))
a__fib1(X, Y) → cons(mark(X), fib1(Y, add(X, Y)))
mark(fib1(X1, X2)) → a__fib1(mark(X1), mark(X2))
a__fib(X) → fib(X)
a__sel(X1, X2) → sel(X1, X2)
a__fib1(X1, X2) → fib1(X1, X2)
a__add(X1, X2) → add(X1, X2)
mark(s(X)) → s(mark(X))
mark(0) → 0
mark(cons(X1, X2)) → cons(mark(X1), X2)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ Narrowing

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ QDP

              ↳ Narrowing

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ QDP

                      ↳ Narrowing

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ QDP

                              ↳ Narrowing

                                ↳ QDP

                                  ↳ DependencyGraphProof

                                    ↳ QDP

                                      ↳ Narrowing

                                        ↳ QDP

                                          ↳ DependencyGraphProof

                                            ↳ QDP

                                              ↳ Narrowing

                                                ↳ QDP

                                                  ↳ DependencyGraphProof

                                                    ↳ QDP

                                                      ↳ Narrowing

                                                        ↳ QDP

                                                          ↳ DependencyGraphProof

                                                            ↳ QDP

                                                              ↳ Narrowing

                                                                ↳ QDP

                                                                  ↳ DependencyGraphProof

                                                                    ↳ QDP

                                                                      ↳ Narrowing

                                                                        ↳ QDP

                                                                          ↳ DependencyGraphProof

                                                                            ↳ QDP

                                                                              ↳ Narrowing

                                                                                ↳ QDP

                                                                                  ↳ DependencyGraphProof

                                                                                    ↳ QDP

                                                                                      ↳ Narrowing

                                                                                        ↳ QDP

                                                                                          ↳ DependencyGraphProof

                                                                                            ↳ QDP

                                                                                              ↳ Narrowing

                                                                                                ↳ QDP

                                                                                                  ↳ DependencyGraphProof

                                                                                                    ↳ QDP

                                                                                                      ↳ Narrowing

                                                                                                        ↳ QDP

                                                                                                          ↳ DependencyGraphProof

                                                                                                            ↳ QDP

                                                                                                              ↳ Narrowing

                                                                                                                ↳ QDP

                                                                                                                  ↳ DependencyGraphProof

                                                                                                                    ↳ QDP

                                                                                                                      ↳ Narrowing

                                                                                                                        ↳ QDP

                                                                                                                          ↳ DependencyGraphProof

                                                                                                                            ↳ QDP

                                                                                                                              ↳ Narrowing

                                                                                                                                ↳ QDP

                                                                                                                                  ↳ DependencyGraphProof

                                                                                                                                    ↳ QDP

                                                                                                                                      ↳ Narrowing

                                                                                                                                        ↳ QDP

                                                                                                                                          ↳ DependencyGraphProof

                                                                                                                                            ↳ QDP

                                                                                                                                              ↳ QDPOrderProof

                                                                                                                                                ↳ QDP

                                                                                                                                                  ↳ QDPOrderProof

                                                                                                                                                    ↳ QDP

                                                                                                                                                      ↳ Narrowing

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

MARK(sel(sel(y0, y1), sel(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), a__sel(mark(x0), mark(x1)))
MARK(fib1(X1, X2)) → MARK(X1)
MARK(sel(s(y0), cons(x0, x1))) → A__SEL(s(mark(y0)), cons(mark(x0), x1))
MARK(cons(X1, X2)) → MARK(X1)
A__SEL(s(fib(y0)), cons(y1, add(x0, x1))) → A__SEL(a__fib(mark(y0)), a__add(mark(x0), mark(x1)))
A__SEL(s(fib(y0)), cons(y1, cons(x0, x1))) → A__SEL(a__fib(mark(y0)), cons(mark(x0), x1))
A__ADD(s(add(x0, x1)), y1) → A__ADD(a__add(mark(x0), mark(x1)), mark(y1))
MARK(sel(0, add(x0, x1))) → A__SEL(0, a__add(mark(x0), mark(x1)))
A__FIB(N) → MARK(N)
MARK(sel(0, fib(x0))) → A__SEL(0, a__fib(mark(x0)))
MARK(sel(add(y0, y1), fib1(x0, x1))) → A__SEL(a__add(mark(y0), mark(y1)), a__fib1(mark(x0), mark(x1)))
A__SEL(s(N), cons(X, XS)) → MARK(XS)
A__FIB(s(x0)) → A__SEL(s(mark(x0)), a__fib1(s(0), s(0)))
A__ADD(s(X), Y) → MARK(Y)
A__ADD(s(0), y1) → A__ADD(0, mark(y1))
MARK(add(0, y1)) → A__ADD(0, mark(y1))
MARK(sel(fib(y0), cons(x0, x1))) → A__SEL(a__fib(mark(y0)), cons(mark(x0), x1))
A__FIB(sel(x0, x1)) → A__SEL(a__sel(mark(x0), mark(x1)), a__fib1(s(0), s(0)))
MARK(add(s(x0), y1)) → A__ADD(s(mark(x0)), mark(y1))
MARK(sel(sel(y0, y1), add(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), a__add(mark(x0), mark(x1)))
MARK(sel(add(y0, y1), sel(x0, x1))) → A__SEL(a__add(mark(y0), mark(y1)), a__sel(mark(x0), mark(x1)))
MARK(sel(s(y0), fib(x0))) → A__SEL(s(mark(y0)), a__fib(mark(x0)))
MARK(sel(fib(y0), fib1(x0, x1))) → A__SEL(a__fib(mark(y0)), a__fib1(mark(x0), mark(x1)))
MARK(add(sel(x0, x1), y1)) → A__ADD(a__sel(mark(x0), mark(x1)), mark(y1))
A__SEL(s(sel(y0, y1)), cons(y2, sel(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), a__sel(mark(x0), mark(x1)))
A__SEL(s(s(y0)), cons(y1, sel(x0, x1))) → A__SEL(s(mark(y0)), a__sel(mark(x0), mark(x1)))
MARK(sel(X1, X2)) → MARK(X1)
A__SEL(s(add(y0, y1)), cons(y2, add(x0, x1))) → A__SEL(a__add(mark(y0), mark(y1)), a__add(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), cons(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), cons(mark(x0), x1))
A__SEL(s(0), cons(y0, add(x0, x1))) → A__SEL(0, a__add(mark(x0), mark(x1)))
MARK(sel(add(y0, y1), fib(x0))) → A__SEL(a__add(mark(y0), mark(y1)), a__fib(mark(x0)))
MARK(sel(fib(y0), add(x0, x1))) → A__SEL(a__fib(mark(y0)), a__add(mark(x0), mark(x1)))
MARK(add(add(x0, x1), y1)) → A__ADD(a__add(mark(x0), mark(x1)), mark(y1))
A__ADD(s(X), Y) → MARK(X)
A__SEL(s(fib(y0)), cons(y1, sel(x0, x1))) → A__SEL(a__fib(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__FIB(N) → A__FIB1(s(0), s(0))
A__SEL(s(s(y0)), cons(y1, cons(x0, x1))) → A__SEL(s(mark(y0)), cons(mark(x0), x1))
MARK(add(fib(x0), y1)) → A__ADD(a__fib(mark(x0)), mark(y1))
A__FIB(fib(x0)) → A__SEL(a__fib(mark(x0)), a__fib1(s(0), s(0)))
A__ADD(s(fib(x0)), y1) → A__ADD(a__fib(mark(x0)), mark(y1))
A__SEL(s(add(y0, y1)), cons(y2, cons(x0, x1))) → A__SEL(a__add(mark(y0), mark(y1)), cons(mark(x0), x1))
MARK(sel(0, cons(x0, x1))) → A__SEL(0, cons(mark(x0), x1))
A__SEL(s(fib(y0)), cons(y1, fib(x0))) → A__SEL(a__fib(mark(y0)), a__fib(mark(x0)))
A__SEL(s(0), cons(y0, sel(x0, x1))) → A__SEL(0, a__sel(mark(x0), mark(x1)))
MARK(sel(add(y0, y1), cons(x0, x1))) → A__SEL(a__add(mark(y0), mark(y1)), cons(mark(x0), x1))
MARK(fib(X)) → MARK(X)
A__SEL(s(sel(y0, y1)), cons(y2, fib(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__fib(mark(x0)))
A__SEL(s(add(y0, y1)), cons(y2, fib(x0))) → A__SEL(a__add(mark(y0), mark(y1)), a__fib(mark(x0)))
A__FIB1(X, Y) → MARK(X)
A__ADD(0, X) → MARK(X)
MARK(s(X)) → MARK(X)
MARK(add(X1, X2)) → MARK(X2)
MARK(sel(s(y0), sel(x0, x1))) → A__SEL(s(mark(y0)), a__sel(mark(x0), mark(x1)))
MARK(fib(X)) → A__FIB(mark(X))
MARK(fib1(X1, X2)) → MARK(X2)
A__FIB(0) → A__SEL(0, a__fib1(s(0), s(0)))
MARK(sel(fib(y0), sel(x0, x1))) → A__SEL(a__fib(mark(y0)), a__sel(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), fib(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__fib(mark(x0)))
A__SEL(0, cons(X, XS)) → MARK(X)
A__ADD(s(sel(x0, x1)), y1) → A__ADD(a__sel(mark(x0), mark(x1)), mark(y1))
MARK(add(X1, X2)) → MARK(X1)
A__SEL(s(s(y0)), cons(y1, fib(x0))) → A__SEL(s(mark(y0)), a__fib(mark(x0)))
A__SEL(s(s(y0)), cons(y1, fib1(x0, x1))) → A__SEL(s(mark(y0)), a__fib1(mark(x0), mark(x1)))
MARK(sel(0, fib1(x0, x1))) → A__SEL(0, a__fib1(mark(x0), mark(x1)))
A__SEL(s(add(y0, y1)), cons(y2, fib1(x0, x1))) → A__SEL(a__add(mark(y0), mark(y1)), a__fib1(mark(x0), mark(x1)))
A__SEL(s(0), cons(y0, cons(x0, x1))) → A__SEL(0, cons(mark(x0), x1))
A__SEL(s(fib(y0)), cons(y1, fib1(x0, x1))) → A__SEL(a__fib(mark(y0)), a__fib1(mark(x0), mark(x1)))
A__SEL(s(sel(y0, y1)), cons(y2, fib1(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), a__fib1(mark(x0), mark(x1)))
A__SEL(s(N), cons(X, XS)) → MARK(N)
A__ADD(s(s(x0)), y1) → A__ADD(s(mark(x0)), mark(y1))
MARK(fib1(X1, X2)) → A__FIB1(mark(X1), mark(X2))
A__SEL(s(0), cons(y0, fib(x0))) → A__SEL(0, a__fib(mark(x0)))
MARK(sel(s(y0), fib1(x0, x1))) → A__SEL(s(mark(y0)), a__fib1(mark(x0), mark(x1)))
MARK(sel(fib(y0), fib(x0))) → A__SEL(a__fib(mark(y0)), a__fib(mark(x0)))
MARK(sel(add(y0, y1), add(x0, x1))) → A__SEL(a__add(mark(y0), mark(y1)), a__add(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), fib1(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), a__fib1(mark(x0), mark(x1)))
MARK(sel(s(y0), add(x0, x1))) → A__SEL(s(mark(y0)), a__add(mark(x0), mark(x1)))
A__SEL(s(s(y0)), cons(y1, add(x0, x1))) → A__SEL(s(mark(y0)), a__add(mark(x0), mark(x1)))
MARK(sel(0, sel(x0, x1))) → A__SEL(0, a__sel(mark(x0), mark(x1)))
A__SEL(s(sel(y0, y1)), cons(y2, add(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), a__add(mark(x0), mark(x1)))
A__SEL(s(add(y0, y1)), cons(y2, sel(x0, x1))) → A__SEL(a__add(mark(y0), mark(y1)), a__sel(mark(x0), mark(x1)))
A__FIB(add(x0, x1)) → A__SEL(a__add(mark(x0), mark(x1)), a__fib1(s(0), s(0)))
A__SEL(s(0), cons(y0, fib1(x0, x1))) → A__SEL(0, a__fib1(mark(x0), mark(x1)))
MARK(sel(X1, X2)) → MARK(X2)

The TRS R consists of the following rules:

a__fib(N) → a__sel(mark(N), a__fib1(s(0), s(0)))
a__fib1(X, Y) → cons(mark(X), fib1(Y, add(X, Y)))
a__add(0, X) → mark(X)
a__add(s(X), Y) → s(a__add(mark(X), mark(Y)))
a__sel(0, cons(X, XS)) → mark(X)
a__sel(s(N), cons(X, XS)) → a__sel(mark(N), mark(XS))
mark(fib(X)) → a__fib(mark(X))
mark(sel(X1, X2)) → a__sel(mark(X1), mark(X2))
mark(fib1(X1, X2)) → a__fib1(mark(X1), mark(X2))
mark(add(X1, X2)) → a__add(mark(X1), mark(X2))
mark(s(X)) → s(mark(X))
mark(0) → 0
mark(cons(X1, X2)) → cons(mark(X1), X2)
a__fib(X) → fib(X)
a__sel(X1, X2) → sel(X1, X2)
a__fib1(X1, X2) → fib1(X1, X2)
a__add(X1, X2) → add(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule A__FIB(s(x0)) → A__SEL(s(mark(x0)), a__fib1(s(0), s(0))) at position [1] we obtained the following new rules:

A__FIB(s(y0)) → A__SEL(s(mark(y0)), fib1(s(0), s(0)))
A__FIB(s(y0)) → A__SEL(s(mark(y0)), cons(mark(s(0)), fib1(s(0), add(s(0), s(0)))))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ Narrowing

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ QDP

              ↳ Narrowing

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ QDP

                      ↳ Narrowing

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ QDP

                              ↳ Narrowing

                                ↳ QDP

                                  ↳ DependencyGraphProof

                                    ↳ QDP

                                      ↳ Narrowing

                                        ↳ QDP

                                          ↳ DependencyGraphProof

                                            ↳ QDP

                                              ↳ Narrowing

                                                ↳ QDP

                                                  ↳ DependencyGraphProof

                                                    ↳ QDP

                                                      ↳ Narrowing

                                                        ↳ QDP

                                                          ↳ DependencyGraphProof

                                                            ↳ QDP

                                                              ↳ Narrowing

                                                                ↳ QDP

                                                                  ↳ DependencyGraphProof

                                                                    ↳ QDP

                                                                      ↳ Narrowing

                                                                        ↳ QDP

                                                                          ↳ DependencyGraphProof

                                                                            ↳ QDP

                                                                              ↳ Narrowing

                                                                                ↳ QDP

                                                                                  ↳ DependencyGraphProof

                                                                                    ↳ QDP

                                                                                      ↳ Narrowing

                                                                                        ↳ QDP

                                                                                          ↳ DependencyGraphProof

                                                                                            ↳ QDP

                                                                                              ↳ Narrowing

                                                                                                ↳ QDP

                                                                                                  ↳ DependencyGraphProof

                                                                                                    ↳ QDP

                                                                                                      ↳ Narrowing

                                                                                                        ↳ QDP

                                                                                                          ↳ DependencyGraphProof

                                                                                                            ↳ QDP

                                                                                                              ↳ Narrowing

                                                                                                                ↳ QDP

                                                                                                                  ↳ DependencyGraphProof

                                                                                                                    ↳ QDP

                                                                                                                      ↳ Narrowing

                                                                                                                        ↳ QDP

                                                                                                                          ↳ DependencyGraphProof

                                                                                                                            ↳ QDP

                                                                                                                              ↳ Narrowing

                                                                                                                                ↳ QDP

                                                                                                                                  ↳ DependencyGraphProof

                                                                                                                                    ↳ QDP

                                                                                                                                      ↳ Narrowing

                                                                                                                                        ↳ QDP

                                                                                                                                          ↳ DependencyGraphProof

                                                                                                                                            ↳ QDP

                                                                                                                                              ↳ QDPOrderProof

                                                                                                                                                ↳ QDP

                                                                                                                                                  ↳ QDPOrderProof

                                                                                                                                                    ↳ QDP

                                                                                                                                                      ↳ Narrowing

                                                                                                                                                        ↳ QDP

                                                                                                                                                          ↳ DependencyGraphProof

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

MARK(sel(sel(y0, y1), sel(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), a__sel(mark(x0), mark(x1)))
MARK(fib1(X1, X2)) → MARK(X1)
MARK(sel(s(y0), cons(x0, x1))) → A__SEL(s(mark(y0)), cons(mark(x0), x1))
MARK(cons(X1, X2)) → MARK(X1)
A__SEL(s(fib(y0)), cons(y1, cons(x0, x1))) → A__SEL(a__fib(mark(y0)), cons(mark(x0), x1))
A__SEL(s(fib(y0)), cons(y1, add(x0, x1))) → A__SEL(a__fib(mark(y0)), a__add(mark(x0), mark(x1)))
A__ADD(s(add(x0, x1)), y1) → A__ADD(a__add(mark(x0), mark(x1)), mark(y1))
MARK(sel(0, add(x0, x1))) → A__SEL(0, a__add(mark(x0), mark(x1)))
MARK(sel(0, fib(x0))) → A__SEL(0, a__fib(mark(x0)))
A__FIB(N) → MARK(N)
MARK(sel(add(y0, y1), fib1(x0, x1))) → A__SEL(a__add(mark(y0), mark(y1)), a__fib1(mark(x0), mark(x1)))
A__SEL(s(N), cons(X, XS)) → MARK(XS)
A__ADD(s(X), Y) → MARK(Y)
A__ADD(s(0), y1) → A__ADD(0, mark(y1))
MARK(add(0, y1)) → A__ADD(0, mark(y1))
MARK(sel(fib(y0), cons(x0, x1))) → A__SEL(a__fib(mark(y0)), cons(mark(x0), x1))
A__FIB(sel(x0, x1)) → A__SEL(a__sel(mark(x0), mark(x1)), a__fib1(s(0), s(0)))
MARK(add(s(x0), y1)) → A__ADD(s(mark(x0)), mark(y1))
MARK(sel(add(y0, y1), sel(x0, x1))) → A__SEL(a__add(mark(y0), mark(y1)), a__sel(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), add(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), a__add(mark(x0), mark(x1)))
MARK(add(sel(x0, x1), y1)) → A__ADD(a__sel(mark(x0), mark(x1)), mark(y1))
MARK(sel(fib(y0), fib1(x0, x1))) → A__SEL(a__fib(mark(y0)), a__fib1(mark(x0), mark(x1)))
MARK(sel(s(y0), fib(x0))) → A__SEL(s(mark(y0)), a__fib(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(add(y0, y1)), cons(y2, add(x0, x1))) → A__SEL(a__add(mark(y0), mark(y1)), a__add(mark(x0), mark(x1)))
MARK(sel(X1, X2)) → MARK(X1)
A__SEL(s(0), cons(y0, add(x0, x1))) → A__SEL(0, a__add(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), cons(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), cons(mark(x0), x1))
MARK(sel(fib(y0), add(x0, x1))) → A__SEL(a__fib(mark(y0)), a__add(mark(x0), mark(x1)))
MARK(sel(add(y0, y1), fib(x0))) → A__SEL(a__add(mark(y0), mark(y1)), a__fib(mark(x0)))
MARK(add(add(x0, x1), y1)) → A__ADD(a__add(mark(x0), mark(x1)), mark(y1))
A__ADD(s(X), Y) → MARK(X)
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(fib(y0)), cons(y1, sel(x0, x1))) → A__SEL(a__fib(mark(y0)), a__sel(mark(x0), mark(x1)))
A__FIB(N) → A__FIB1(s(0), s(0))
MARK(add(fib(x0), y1)) → A__ADD(a__fib(mark(x0)), mark(y1))
A__SEL(s(s(y0)), cons(y1, cons(x0, x1))) → A__SEL(s(mark(y0)), cons(mark(x0), x1))
A__FIB(fib(x0)) → A__SEL(a__fib(mark(x0)), a__fib1(s(0), s(0)))
A__ADD(s(fib(x0)), y1) → A__ADD(a__fib(mark(x0)), mark(y1))
A__SEL(s(add(y0, y1)), cons(y2, cons(x0, x1))) → A__SEL(a__add(mark(y0), mark(y1)), cons(mark(x0), x1))
MARK(sel(0, cons(x0, x1))) → A__SEL(0, cons(mark(x0), x1))
A__SEL(s(fib(y0)), cons(y1, fib(x0))) → A__SEL(a__fib(mark(y0)), a__fib(mark(x0)))
A__FIB(s(y0)) → A__SEL(s(mark(y0)), cons(mark(s(0)), fib1(s(0), add(s(0), s(0)))))
A__SEL(s(0), cons(y0, sel(x0, x1))) → A__SEL(0, a__sel(mark(x0), mark(x1)))
MARK(sel(add(y0, y1), cons(x0, x1))) → A__SEL(a__add(mark(y0), mark(y1)), cons(mark(x0), x1))
A__SEL(s(sel(y0, y1)), cons(y2, fib(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__fib(mark(x0)))
MARK(fib(X)) → MARK(X)
A__SEL(s(add(y0, y1)), cons(y2, fib(x0))) → A__SEL(a__add(mark(y0), mark(y1)), a__fib(mark(x0)))
A__ADD(0, X) → MARK(X)
A__FIB1(X, Y) → MARK(X)
MARK(s(X)) → MARK(X)
MARK(add(X1, X2)) → MARK(X2)
MARK(fib(X)) → A__FIB(mark(X))
MARK(sel(s(y0), sel(x0, x1))) → A__SEL(s(mark(y0)), a__sel(mark(x0), mark(x1)))
MARK(fib1(X1, X2)) → MARK(X2)
MARK(sel(sel(y0, y1), fib(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__fib(mark(x0)))
MARK(sel(fib(y0), sel(x0, x1))) → A__SEL(a__fib(mark(y0)), a__sel(mark(x0), mark(x1)))
A__FIB(0) → A__SEL(0, a__fib1(s(0), s(0)))
A__ADD(s(sel(x0, x1)), y1) → A__ADD(a__sel(mark(x0), mark(x1)), mark(y1))
A__SEL(0, cons(X, XS)) → MARK(X)
MARK(add(X1, X2)) → MARK(X1)
A__SEL(s(s(y0)), cons(y1, fib(x0))) → A__SEL(s(mark(y0)), a__fib(mark(x0)))
MARK(sel(0, fib1(x0, x1))) → A__SEL(0, a__fib1(mark(x0), mark(x1)))
A__SEL(s(s(y0)), cons(y1, fib1(x0, x1))) → A__SEL(s(mark(y0)), a__fib1(mark(x0), mark(x1)))
A__SEL(s(0), cons(y0, cons(x0, x1))) → A__SEL(0, cons(mark(x0), x1))
A__SEL(s(add(y0, y1)), cons(y2, fib1(x0, x1))) → A__SEL(a__add(mark(y0), mark(y1)), a__fib1(mark(x0), mark(x1)))
A__SEL(s(fib(y0)), cons(y1, fib1(x0, x1))) → A__SEL(a__fib(mark(y0)), a__fib1(mark(x0), mark(x1)))
A__SEL(s(sel(y0, y1)), cons(y2, fib1(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), a__fib1(mark(x0), mark(x1)))
A__SEL(s(N), cons(X, XS)) → MARK(N)
A__FIB(s(y0)) → A__SEL(s(mark(y0)), fib1(s(0), s(0)))
A__ADD(s(s(x0)), y1) → A__ADD(s(mark(x0)), mark(y1))
A__SEL(s(0), cons(y0, fib(x0))) → A__SEL(0, a__fib(mark(x0)))
MARK(fib1(X1, X2)) → A__FIB1(mark(X1), mark(X2))
MARK(sel(s(y0), fib1(x0, x1))) → A__SEL(s(mark(y0)), a__fib1(mark(x0), mark(x1)))
MARK(sel(fib(y0), fib(x0))) → A__SEL(a__fib(mark(y0)), a__fib(mark(x0)))
MARK(sel(sel(y0, y1), fib1(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), a__fib1(mark(x0), mark(x1)))
MARK(sel(add(y0, y1), add(x0, x1))) → A__SEL(a__add(mark(y0), mark(y1)), a__add(mark(x0), mark(x1)))
MARK(sel(s(y0), add(x0, x1))) → A__SEL(s(mark(y0)), a__add(mark(x0), mark(x1)))
A__SEL(s(s(y0)), cons(y1, add(x0, x1))) → A__SEL(s(mark(y0)), a__add(mark(x0), mark(x1)))
MARK(sel(0, sel(x0, x1))) → A__SEL(0, a__sel(mark(x0), mark(x1)))
A__SEL(s(add(y0, y1)), cons(y2, sel(x0, x1))) → A__SEL(a__add(mark(y0), mark(y1)), a__sel(mark(x0), mark(x1)))
A__SEL(s(sel(y0, y1)), cons(y2, add(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), a__add(mark(x0), mark(x1)))
A__FIB(add(x0, x1)) → A__SEL(a__add(mark(x0), mark(x1)), a__fib1(s(0), s(0)))
A__SEL(s(0), cons(y0, fib1(x0, x1))) → A__SEL(0, a__fib1(mark(x0), mark(x1)))
MARK(sel(X1, X2)) → MARK(X2)

The TRS R consists of the following rules:

a__fib(N) → a__sel(mark(N), a__fib1(s(0), s(0)))
a__fib1(X, Y) → cons(mark(X), fib1(Y, add(X, Y)))
a__add(0, X) → mark(X)
a__add(s(X), Y) → s(a__add(mark(X), mark(Y)))
a__sel(0, cons(X, XS)) → mark(X)
a__sel(s(N), cons(X, XS)) → a__sel(mark(N), mark(XS))
mark(fib(X)) → a__fib(mark(X))
mark(sel(X1, X2)) → a__sel(mark(X1), mark(X2))
mark(fib1(X1, X2)) → a__fib1(mark(X1), mark(X2))
mark(add(X1, X2)) → a__add(mark(X1), mark(X2))
mark(s(X)) → s(mark(X))
mark(0) → 0
mark(cons(X1, X2)) → cons(mark(X1), X2)
a__fib(X) → fib(X)
a__sel(X1, X2) → sel(X1, X2)
a__fib1(X1, X2) → fib1(X1, X2)
a__add(X1, X2) → add(X1, X2)

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ Narrowing

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ QDP

              ↳ Narrowing

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ QDP

                      ↳ Narrowing

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ QDP

                              ↳ Narrowing

                                ↳ QDP

                                  ↳ DependencyGraphProof

                                    ↳ QDP

                                      ↳ Narrowing

                                        ↳ QDP

                                          ↳ DependencyGraphProof

                                            ↳ QDP

                                              ↳ Narrowing

                                                ↳ QDP

                                                  ↳ DependencyGraphProof

                                                    ↳ QDP

                                                      ↳ Narrowing

                                                        ↳ QDP

                                                          ↳ DependencyGraphProof

                                                            ↳ QDP

                                                              ↳ Narrowing

                                                                ↳ QDP

                                                                  ↳ DependencyGraphProof

                                                                    ↳ QDP

                                                                      ↳ Narrowing

                                                                        ↳ QDP

                                                                          ↳ DependencyGraphProof

                                                                            ↳ QDP

                                                                              ↳ Narrowing

                                                                                ↳ QDP

                                                                                  ↳ DependencyGraphProof

                                                                                    ↳ QDP

                                                                                      ↳ Narrowing

                                                                                        ↳ QDP

                                                                                          ↳ DependencyGraphProof

                                                                                            ↳ QDP

                                                                                              ↳ Narrowing

                                                                                                ↳ QDP

                                                                                                  ↳ DependencyGraphProof

                                                                                                    ↳ QDP

                                                                                                      ↳ Narrowing

                                                                                                        ↳ QDP

                                                                                                          ↳ DependencyGraphProof

                                                                                                            ↳ QDP

                                                                                                              ↳ Narrowing

                                                                                                                ↳ QDP

                                                                                                                  ↳ DependencyGraphProof

                                                                                                                    ↳ QDP

                                                                                                                      ↳ Narrowing

                                                                                                                        ↳ QDP

                                                                                                                          ↳ DependencyGraphProof

                                                                                                                            ↳ QDP

                                                                                                                              ↳ Narrowing

                                                                                                                                ↳ QDP

                                                                                                                                  ↳ DependencyGraphProof

                                                                                                                                    ↳ QDP

                                                                                                                                      ↳ Narrowing

                                                                                                                                        ↳ QDP

                                                                                                                                          ↳ DependencyGraphProof

                                                                                                                                            ↳ QDP

                                                                                                                                              ↳ QDPOrderProof

                                                                                                                                                ↳ QDP

                                                                                                                                                  ↳ QDPOrderProof

                                                                                                                                                    ↳ QDP

                                                                                                                                                      ↳ Narrowing

                                                                                                                                                        ↳ QDP

                                                                                                                                                          ↳ DependencyGraphProof

                                                                                                                                                            ↳ QDP

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

MARK(sel(sel(y0, y1), sel(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), a__sel(mark(x0), mark(x1)))
MARK(fib1(X1, X2)) → MARK(X1)
MARK(sel(s(y0), cons(x0, x1))) → A__SEL(s(mark(y0)), cons(mark(x0), x1))
MARK(cons(X1, X2)) → MARK(X1)
A__SEL(s(fib(y0)), cons(y1, add(x0, x1))) → A__SEL(a__fib(mark(y0)), a__add(mark(x0), mark(x1)))
A__SEL(s(fib(y0)), cons(y1, cons(x0, x1))) → A__SEL(a__fib(mark(y0)), cons(mark(x0), x1))
A__ADD(s(add(x0, x1)), y1) → A__ADD(a__add(mark(x0), mark(x1)), mark(y1))
MARK(sel(0, add(x0, x1))) → A__SEL(0, a__add(mark(x0), mark(x1)))
MARK(sel(0, fib(x0))) → A__SEL(0, a__fib(mark(x0)))
A__FIB(N) → MARK(N)
MARK(sel(add(y0, y1), fib1(x0, x1))) → A__SEL(a__add(mark(y0), mark(y1)), a__fib1(mark(x0), mark(x1)))
A__SEL(s(N), cons(X, XS)) → MARK(XS)
A__ADD(s(X), Y) → MARK(Y)
A__ADD(s(0), y1) → A__ADD(0, mark(y1))
MARK(add(0, y1)) → A__ADD(0, mark(y1))
MARK(sel(fib(y0), cons(x0, x1))) → A__SEL(a__fib(mark(y0)), cons(mark(x0), x1))
A__FIB(sel(x0, x1)) → A__SEL(a__sel(mark(x0), mark(x1)), a__fib1(s(0), s(0)))
MARK(add(s(x0), y1)) → A__ADD(s(mark(x0)), mark(y1))
MARK(sel(sel(y0, y1), add(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), a__add(mark(x0), mark(x1)))
MARK(sel(add(y0, y1), sel(x0, x1))) → A__SEL(a__add(mark(y0), mark(y1)), a__sel(mark(x0), mark(x1)))
MARK(sel(s(y0), fib(x0))) → A__SEL(s(mark(y0)), a__fib(mark(x0)))
MARK(sel(fib(y0), fib1(x0, x1))) → A__SEL(a__fib(mark(y0)), a__fib1(mark(x0), mark(x1)))
MARK(add(sel(x0, x1), y1)) → A__ADD(a__sel(mark(x0), mark(x1)), mark(y1))
A__SEL(s(sel(y0, y1)), cons(y2, sel(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), a__sel(mark(x0), mark(x1)))
A__SEL(s(s(y0)), cons(y1, sel(x0, x1))) → A__SEL(s(mark(y0)), a__sel(mark(x0), mark(x1)))
MARK(sel(X1, X2)) → MARK(X1)
A__SEL(s(add(y0, y1)), cons(y2, add(x0, x1))) → A__SEL(a__add(mark(y0), mark(y1)), a__add(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), cons(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), cons(mark(x0), x1))
A__SEL(s(0), cons(y0, add(x0, x1))) → A__SEL(0, a__add(mark(x0), mark(x1)))
MARK(sel(add(y0, y1), fib(x0))) → A__SEL(a__add(mark(y0), mark(y1)), a__fib(mark(x0)))
MARK(sel(fib(y0), add(x0, x1))) → A__SEL(a__fib(mark(y0)), a__add(mark(x0), mark(x1)))
MARK(add(add(x0, x1), y1)) → A__ADD(a__add(mark(x0), mark(x1)), mark(y1))
A__ADD(s(X), Y) → MARK(X)
A__SEL(s(fib(y0)), cons(y1, sel(x0, x1))) → A__SEL(a__fib(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__FIB(N) → A__FIB1(s(0), s(0))
A__SEL(s(s(y0)), cons(y1, cons(x0, x1))) → A__SEL(s(mark(y0)), cons(mark(x0), x1))
MARK(add(fib(x0), y1)) → A__ADD(a__fib(mark(x0)), mark(y1))
A__FIB(fib(x0)) → A__SEL(a__fib(mark(x0)), a__fib1(s(0), s(0)))
A__ADD(s(fib(x0)), y1) → A__ADD(a__fib(mark(x0)), mark(y1))
A__SEL(s(add(y0, y1)), cons(y2, cons(x0, x1))) → A__SEL(a__add(mark(y0), mark(y1)), cons(mark(x0), x1))
MARK(sel(0, cons(x0, x1))) → A__SEL(0, cons(mark(x0), x1))
A__SEL(s(fib(y0)), cons(y1, fib(x0))) → A__SEL(a__fib(mark(y0)), a__fib(mark(x0)))
A__FIB(s(y0)) → A__SEL(s(mark(y0)), cons(mark(s(0)), fib1(s(0), add(s(0), s(0)))))
A__SEL(s(0), cons(y0, sel(x0, x1))) → A__SEL(0, a__sel(mark(x0), mark(x1)))
MARK(sel(add(y0, y1), cons(x0, x1))) → A__SEL(a__add(mark(y0), mark(y1)), cons(mark(x0), x1))
MARK(fib(X)) → MARK(X)
A__SEL(s(sel(y0, y1)), cons(y2, fib(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__fib(mark(x0)))
A__SEL(s(add(y0, y1)), cons(y2, fib(x0))) → A__SEL(a__add(mark(y0), mark(y1)), a__fib(mark(x0)))
A__ADD(0, X) → MARK(X)
A__FIB1(X, Y) → MARK(X)
MARK(s(X)) → MARK(X)
MARK(add(X1, X2)) → MARK(X2)
MARK(sel(s(y0), sel(x0, x1))) → A__SEL(s(mark(y0)), a__sel(mark(x0), mark(x1)))
MARK(fib(X)) → A__FIB(mark(X))
MARK(fib1(X1, X2)) → MARK(X2)
A__FIB(0) → A__SEL(0, a__fib1(s(0), s(0)))
MARK(sel(sel(y0, y1), fib(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__fib(mark(x0)))
MARK(sel(fib(y0), sel(x0, x1))) → A__SEL(a__fib(mark(y0)), a__sel(mark(x0), mark(x1)))
A__SEL(0, cons(X, XS)) → MARK(X)
A__ADD(s(sel(x0, x1)), y1) → A__ADD(a__sel(mark(x0), mark(x1)), mark(y1))
MARK(add(X1, X2)) → MARK(X1)
A__SEL(s(s(y0)), cons(y1, fib(x0))) → A__SEL(s(mark(y0)), a__fib(mark(x0)))
MARK(sel(0, fib1(x0, x1))) → A__SEL(0, a__fib1(mark(x0), mark(x1)))
A__SEL(s(s(y0)), cons(y1, fib1(x0, x1))) → A__SEL(s(mark(y0)), a__fib1(mark(x0), mark(x1)))
A__SEL(s(add(y0, y1)), cons(y2, fib1(x0, x1))) → A__SEL(a__add(mark(y0), mark(y1)), a__fib1(mark(x0), mark(x1)))
A__SEL(s(0), cons(y0, cons(x0, x1))) → A__SEL(0, cons(mark(x0), x1))
A__SEL(s(fib(y0)), cons(y1, fib1(x0, x1))) → A__SEL(a__fib(mark(y0)), a__fib1(mark(x0), mark(x1)))
A__SEL(s(sel(y0, y1)), cons(y2, fib1(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), a__fib1(mark(x0), mark(x1)))
A__SEL(s(N), cons(X, XS)) → MARK(N)
A__ADD(s(s(x0)), y1) → A__ADD(s(mark(x0)), mark(y1))
MARK(fib1(X1, X2)) → A__FIB1(mark(X1), mark(X2))
A__SEL(s(0), cons(y0, fib(x0))) → A__SEL(0, a__fib(mark(x0)))
MARK(sel(s(y0), fib1(x0, x1))) → A__SEL(s(mark(y0)), a__fib1(mark(x0), mark(x1)))
MARK(sel(fib(y0), fib(x0))) → A__SEL(a__fib(mark(y0)), a__fib(mark(x0)))
MARK(sel(sel(y0, y1), fib1(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), a__fib1(mark(x0), mark(x1)))
MARK(sel(add(y0, y1), add(x0, x1))) → A__SEL(a__add(mark(y0), mark(y1)), a__add(mark(x0), mark(x1)))
MARK(sel(s(y0), add(x0, x1))) → A__SEL(s(mark(y0)), a__add(mark(x0), mark(x1)))
A__SEL(s(s(y0)), cons(y1, add(x0, x1))) → A__SEL(s(mark(y0)), a__add(mark(x0), mark(x1)))
MARK(sel(0, sel(x0, x1))) → A__SEL(0, a__sel(mark(x0), mark(x1)))
A__SEL(s(add(y0, y1)), cons(y2, sel(x0, x1))) → A__SEL(a__add(mark(y0), mark(y1)), a__sel(mark(x0), mark(x1)))
A__SEL(s(sel(y0, y1)), cons(y2, add(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), a__add(mark(x0), mark(x1)))
A__FIB(add(x0, x1)) → A__SEL(a__add(mark(x0), mark(x1)), a__fib1(s(0), s(0)))
A__SEL(s(0), cons(y0, fib1(x0, x1))) → A__SEL(0, a__fib1(mark(x0), mark(x1)))
MARK(sel(X1, X2)) → MARK(X2)

The TRS R consists of the following rules:

a__fib(N) → a__sel(mark(N), a__fib1(s(0), s(0)))
a__fib1(X, Y) → cons(mark(X), fib1(Y, add(X, Y)))
a__add(0, X) → mark(X)
a__add(s(X), Y) → s(a__add(mark(X), mark(Y)))
a__sel(0, cons(X, XS)) → mark(X)
a__sel(s(N), cons(X, XS)) → a__sel(mark(N), mark(XS))
mark(fib(X)) → a__fib(mark(X))
mark(sel(X1, X2)) → a__sel(mark(X1), mark(X2))
mark(fib1(X1, X2)) → a__fib1(mark(X1), mark(X2))
mark(add(X1, X2)) → a__add(mark(X1), mark(X2))
mark(s(X)) → s(mark(X))
mark(0) → 0
mark(cons(X1, X2)) → cons(mark(X1), X2)
a__fib(X) → fib(X)
a__sel(X1, X2) → sel(X1, X2)
a__fib1(X1, X2) → fib1(X1, X2)
a__add(X1, X2) → add(X1, X2)

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