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__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.

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__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.