Termination w.r.t. Q proof of TRS/TRCSREx8_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 [1,13] we result in the following initial DP problem:
Q DP problem:
The TRS P consists of the following rules:

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

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

      ↳ EdgeDeletionProof

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

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

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 deleted some edges using various graph approximations

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ EdgeDeletionProof

        ↳ QDP

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

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

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.