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: Quasi 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.