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__fib1(N) -> a__sel2(mark1(N), a__fib12(s1(0), s1(0)))
a__fib12(X, Y) -> cons2(mark1(X), fib12(Y, add2(X, Y)))
a__add2(0, X) -> mark1(X)
a__add2(s1(X), Y) -> s1(a__add2(mark1(X), mark1(Y)))
a__sel2(0, cons2(X, XS)) -> mark1(X)
a__sel2(s1(N), cons2(X, XS)) -> a__sel2(mark1(N), mark1(XS))
mark1(fib1(X)) -> a__fib1(mark1(X))
mark1(sel2(X1, X2)) -> a__sel2(mark1(X1), mark1(X2))
mark1(fib12(X1, X2)) -> a__fib12(mark1(X1), mark1(X2))
mark1(add2(X1, X2)) -> a__add2(mark1(X1), mark1(X2))
mark1(s1(X)) -> s1(mark1(X))
mark1(0) -> 0
mark1(cons2(X1, X2)) -> cons2(mark1(X1), X2)
a__fib1(X) -> fib1(X)
a__sel2(X1, X2) -> sel2(X1, X2)
a__fib12(X1, X2) -> fib12(X1, X2)
a__add2(X1, X2) -> add2(X1, X2)

Q is empty.


QTRS
  ↳ DependencyPairsProof

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

a__fib1(N) -> a__sel2(mark1(N), a__fib12(s1(0), s1(0)))
a__fib12(X, Y) -> cons2(mark1(X), fib12(Y, add2(X, Y)))
a__add2(0, X) -> mark1(X)
a__add2(s1(X), Y) -> s1(a__add2(mark1(X), mark1(Y)))
a__sel2(0, cons2(X, XS)) -> mark1(X)
a__sel2(s1(N), cons2(X, XS)) -> a__sel2(mark1(N), mark1(XS))
mark1(fib1(X)) -> a__fib1(mark1(X))
mark1(sel2(X1, X2)) -> a__sel2(mark1(X1), mark1(X2))
mark1(fib12(X1, X2)) -> a__fib12(mark1(X1), mark1(X2))
mark1(add2(X1, X2)) -> a__add2(mark1(X1), mark1(X2))
mark1(s1(X)) -> s1(mark1(X))
mark1(0) -> 0
mark1(cons2(X1, X2)) -> cons2(mark1(X1), X2)
a__fib1(X) -> fib1(X)
a__sel2(X1, X2) -> sel2(X1, X2)
a__fib12(X1, X2) -> fib12(X1, X2)
a__add2(X1, X2) -> add2(X1, X2)

Q is empty.

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

A__SEL2(0, cons2(X, XS)) -> MARK1(X)
MARK1(add2(X1, X2)) -> MARK1(X2)
MARK1(fib1(X)) -> A__FIB1(mark1(X))
MARK1(sel2(X1, X2)) -> A__SEL2(mark1(X1), mark1(X2))
A__SEL2(s1(N), cons2(X, XS)) -> A__SEL2(mark1(N), mark1(XS))
MARK1(sel2(X1, X2)) -> MARK1(X1)
A__ADD2(0, X) -> MARK1(X)
A__FIB1(N) -> A__SEL2(mark1(N), a__fib12(s1(0), s1(0)))
A__ADD2(s1(X), Y) -> A__ADD2(mark1(X), mark1(Y))
MARK1(add2(X1, X2)) -> A__ADD2(mark1(X1), mark1(X2))
A__SEL2(s1(N), cons2(X, XS)) -> MARK1(N)
MARK1(cons2(X1, X2)) -> MARK1(X1)
MARK1(fib1(X)) -> MARK1(X)
MARK1(fib12(X1, X2)) -> MARK1(X1)
A__FIB12(X, Y) -> MARK1(X)
A__ADD2(s1(X), Y) -> MARK1(X)
MARK1(sel2(X1, X2)) -> MARK1(X2)
A__SEL2(s1(N), cons2(X, XS)) -> MARK1(XS)
A__FIB1(N) -> A__FIB12(s1(0), s1(0))
MARK1(fib12(X1, X2)) -> MARK1(X2)
MARK1(fib12(X1, X2)) -> A__FIB12(mark1(X1), mark1(X2))
MARK1(add2(X1, X2)) -> MARK1(X1)
MARK1(s1(X)) -> MARK1(X)
A__FIB1(N) -> MARK1(N)
A__ADD2(s1(X), Y) -> MARK1(Y)

The TRS R consists of the following rules:

a__fib1(N) -> a__sel2(mark1(N), a__fib12(s1(0), s1(0)))
a__fib12(X, Y) -> cons2(mark1(X), fib12(Y, add2(X, Y)))
a__add2(0, X) -> mark1(X)
a__add2(s1(X), Y) -> s1(a__add2(mark1(X), mark1(Y)))
a__sel2(0, cons2(X, XS)) -> mark1(X)
a__sel2(s1(N), cons2(X, XS)) -> a__sel2(mark1(N), mark1(XS))
mark1(fib1(X)) -> a__fib1(mark1(X))
mark1(sel2(X1, X2)) -> a__sel2(mark1(X1), mark1(X2))
mark1(fib12(X1, X2)) -> a__fib12(mark1(X1), mark1(X2))
mark1(add2(X1, X2)) -> a__add2(mark1(X1), mark1(X2))
mark1(s1(X)) -> s1(mark1(X))
mark1(0) -> 0
mark1(cons2(X1, X2)) -> cons2(mark1(X1), X2)
a__fib1(X) -> fib1(X)
a__sel2(X1, X2) -> sel2(X1, X2)
a__fib12(X1, X2) -> fib12(X1, X2)
a__add2(X1, X2) -> add2(X1, X2)

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

↳ QTRS
  ↳ DependencyPairsProof
QDP

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

A__SEL2(0, cons2(X, XS)) -> MARK1(X)
MARK1(add2(X1, X2)) -> MARK1(X2)
MARK1(fib1(X)) -> A__FIB1(mark1(X))
MARK1(sel2(X1, X2)) -> A__SEL2(mark1(X1), mark1(X2))
A__SEL2(s1(N), cons2(X, XS)) -> A__SEL2(mark1(N), mark1(XS))
MARK1(sel2(X1, X2)) -> MARK1(X1)
A__ADD2(0, X) -> MARK1(X)
A__FIB1(N) -> A__SEL2(mark1(N), a__fib12(s1(0), s1(0)))
A__ADD2(s1(X), Y) -> A__ADD2(mark1(X), mark1(Y))
MARK1(add2(X1, X2)) -> A__ADD2(mark1(X1), mark1(X2))
A__SEL2(s1(N), cons2(X, XS)) -> MARK1(N)
MARK1(cons2(X1, X2)) -> MARK1(X1)
MARK1(fib1(X)) -> MARK1(X)
MARK1(fib12(X1, X2)) -> MARK1(X1)
A__FIB12(X, Y) -> MARK1(X)
A__ADD2(s1(X), Y) -> MARK1(X)
MARK1(sel2(X1, X2)) -> MARK1(X2)
A__SEL2(s1(N), cons2(X, XS)) -> MARK1(XS)
A__FIB1(N) -> A__FIB12(s1(0), s1(0))
MARK1(fib12(X1, X2)) -> MARK1(X2)
MARK1(fib12(X1, X2)) -> A__FIB12(mark1(X1), mark1(X2))
MARK1(add2(X1, X2)) -> MARK1(X1)
MARK1(s1(X)) -> MARK1(X)
A__FIB1(N) -> MARK1(N)
A__ADD2(s1(X), Y) -> MARK1(Y)

The TRS R consists of the following rules:

a__fib1(N) -> a__sel2(mark1(N), a__fib12(s1(0), s1(0)))
a__fib12(X, Y) -> cons2(mark1(X), fib12(Y, add2(X, Y)))
a__add2(0, X) -> mark1(X)
a__add2(s1(X), Y) -> s1(a__add2(mark1(X), mark1(Y)))
a__sel2(0, cons2(X, XS)) -> mark1(X)
a__sel2(s1(N), cons2(X, XS)) -> a__sel2(mark1(N), mark1(XS))
mark1(fib1(X)) -> a__fib1(mark1(X))
mark1(sel2(X1, X2)) -> a__sel2(mark1(X1), mark1(X2))
mark1(fib12(X1, X2)) -> a__fib12(mark1(X1), mark1(X2))
mark1(add2(X1, X2)) -> a__add2(mark1(X1), mark1(X2))
mark1(s1(X)) -> s1(mark1(X))
mark1(0) -> 0
mark1(cons2(X1, X2)) -> cons2(mark1(X1), X2)
a__fib1(X) -> fib1(X)
a__sel2(X1, X2) -> sel2(X1, X2)
a__fib12(X1, X2) -> fib12(X1, X2)
a__add2(X1, X2) -> add2(X1, X2)

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