Termination w.r.t. Q proof of /home/nowonder/forschung/aprove/satrung/aprove/lpar1/SK90SLASH2.24.trs

Termination w.r.t. Q of the following Term Rewriting System could be proven:

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

fib₁(0) -> 0
fib₁(s₁(0)) -> s₁(0)
fib₁(s₁(s₁(x))) -> +₂(fib₁(s₁(x)), fib₁(x))

Q is empty.

↳ QTRS

  ↳ Non-Overlap Check

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

fib₁(0) -> 0
fib₁(s₁(0)) -> s₁(0)
fib₁(s₁(s₁(x))) -> +₂(fib₁(s₁(x)), fib₁(x))

Q is empty.

The TRS is non-overlapping. Hence, we can switch to innermost.

↳ QTRS

  ↳ Non-Overlap Check

    ↳ QTRS

      ↳ DependencyPairsProof

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

fib₁(0) -> 0
fib₁(s₁(0)) -> s₁(0)
fib₁(s₁(s₁(x))) -> +₂(fib₁(s₁(x)), fib₁(x))

The set Q consists of the following terms:

fib₁(0)
fib₁(s₁(0))
fib₁(s₁(s₁(x0)))

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

FIB₁(s₁(s₁(x))) -> FIB₁(s₁(x))
FIB₁(s₁(s₁(x))) -> FIB₁(x)

The TRS R consists of the following rules:

fib₁(0) -> 0
fib₁(s₁(0)) -> s₁(0)
fib₁(s₁(s₁(x))) -> +₂(fib₁(s₁(x)), fib₁(x))

The set Q consists of the following terms:

fib₁(0)
fib₁(s₁(0))
fib₁(s₁(s₁(x0)))

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

↳ QTRS

  ↳ Non-Overlap Check

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ QDPAfsSolverProof

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

FIB₁(s₁(s₁(x))) -> FIB₁(s₁(x))
FIB₁(s₁(s₁(x))) -> FIB₁(x)

The TRS R consists of the following rules:

fib₁(0) -> 0
fib₁(s₁(0)) -> s₁(0)
fib₁(s₁(s₁(x))) -> +₂(fib₁(s₁(x)), fib₁(x))

The set Q consists of the following terms:

fib₁(0)
fib₁(s₁(0))
fib₁(s₁(s₁(x0)))

We have to consider all minimal (P,Q,R)-chains.
By using an argument filtering and a montonic ordering, at least one Dependency Pair of this SCC can be strictly oriented.

FIB₁(s₁(s₁(x))) -> FIB₁(s₁(x))
FIB₁(s₁(s₁(x))) -> FIB₁(x)

Used argument filtering: FIB₁(x1) = x1
s₁(x1) = s₁(x1)
Used ordering: Precedence:

trivial

↳ QTRS

  ↳ Non-Overlap Check

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ QDPAfsSolverProof

            ↳ QDP

              ↳ PisEmptyProof

Q DP problem:
P is empty.
The TRS R consists of the following rules:

fib₁(0) -> 0
fib₁(s₁(0)) -> s₁(0)
fib₁(s₁(s₁(x))) -> +₂(fib₁(s₁(x)), fib₁(x))

The set Q consists of the following terms:

fib₁(0)
fib₁(s₁(0))
fib₁(s₁(s₁(x0)))

We have to consider all minimal (P,Q,R)-chains.
The TRS P is empty. Hence, there is no (P,Q,R) chain.