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:

f1(0) -> s1(0)
f1(s1(0)) -> s1(0)
f1(s1(s1(x))) -> f1(f1(s1(x)))

Q is empty.


QTRS
  ↳ DependencyPairsProof

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

f1(0) -> s1(0)
f1(s1(0)) -> s1(0)
f1(s1(s1(x))) -> f1(f1(s1(x)))

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:

F1(s1(s1(x))) -> F1(f1(s1(x)))
F1(s1(s1(x))) -> F1(s1(x))

The TRS R consists of the following rules:

f1(0) -> s1(0)
f1(s1(0)) -> s1(0)
f1(s1(s1(x))) -> f1(f1(s1(x)))

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

↳ QTRS
  ↳ DependencyPairsProof
QDP
      ↳ QDPOrderProof

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

F1(s1(s1(x))) -> F1(f1(s1(x)))
F1(s1(s1(x))) -> F1(s1(x))

The TRS R consists of the following rules:

f1(0) -> s1(0)
f1(s1(0)) -> s1(0)
f1(s1(s1(x))) -> f1(f1(s1(x)))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [13].


The following pairs can be oriented strictly and are deleted.


F1(s1(s1(x))) -> F1(f1(s1(x)))
F1(s1(s1(x))) -> F1(s1(x))
The remaining pairs can at least be oriented weakly.
none
Used ordering: Polynomial Order [17,21] with Interpretation:

POL( 0 ) = max{0, -3}


POL( s1(x1) ) = 2x1 + 2


POL( F1(x1) ) = max{0, 2x1 - 2}


POL( f1(x1) ) = 2



The following usable rules [14] were oriented:

f1(s1(0)) -> s1(0)
f1(s1(s1(x))) -> f1(f1(s1(x)))
f1(0) -> s1(0)



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ QDPOrderProof
QDP
          ↳ PisEmptyProof

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

f1(0) -> s1(0)
f1(s1(0)) -> s1(0)
f1(s1(s1(x))) -> f1(f1(s1(x)))

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