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:
f2(a, f2(x, a)) -> f2(a, f2(f2(a, a), f2(a, x)))
Q is empty.
↳ QTRS
↳ DependencyPairsProof
Q restricted rewrite system:
The TRS R consists of the following rules:
f2(a, f2(x, a)) -> f2(a, f2(f2(a, a), f2(a, 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:
F2(a, f2(x, a)) -> F2(a, f2(f2(a, a), f2(a, x)))
F2(a, f2(x, a)) -> F2(a, a)
F2(a, f2(x, a)) -> F2(f2(a, a), f2(a, x))
F2(a, f2(x, a)) -> F2(a, x)
The TRS R consists of the following rules:
f2(a, f2(x, a)) -> f2(a, f2(f2(a, a), f2(a, x)))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
Q DP problem:
The TRS P consists of the following rules:
F2(a, f2(x, a)) -> F2(a, f2(f2(a, a), f2(a, x)))
F2(a, f2(x, a)) -> F2(a, a)
F2(a, f2(x, a)) -> F2(f2(a, a), f2(a, x))
F2(a, f2(x, a)) -> F2(a, x)
The TRS R consists of the following rules:
f2(a, f2(x, a)) -> f2(a, f2(f2(a, a), f2(a, x)))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [13,14,18] contains 1 SCC with 3 less nodes.
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
Q DP problem:
The TRS P consists of the following rules:
F2(a, f2(x, a)) -> F2(a, x)
The TRS R consists of the following rules:
f2(a, f2(x, a)) -> f2(a, f2(f2(a, a), f2(a, 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.
F2(a, f2(x, a)) -> F2(a, x)
The remaining pairs can at least be oriented weakly.
none
Used ordering: Polynomial Order [17,21] with Interpretation:
POL( F2(x1, x2) ) = max{0, x2 - 3}
POL( f2(x1, x2) ) = x1 + x2 + 1
POL( a ) = 3
The following usable rules [14] were oriented:
none
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ PisEmptyProof
Q DP problem:
P is empty.
The TRS R consists of the following rules:
f2(a, f2(x, a)) -> f2(a, f2(f2(a, a), f2(a, 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.