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(f2(a, f2(x, a)), a) -> f2(a, f2(f2(x, a), a))
Q is empty.
↳ QTRS
↳ DependencyPairsProof
Q restricted rewrite system:
The TRS R consists of the following rules:
f2(f2(a, f2(x, a)), a) -> f2(a, f2(f2(x, a), a))
Q is empty.
Q DP problem:
The TRS P consists of the following rules:
F2(f2(a, f2(x, a)), a) -> F2(a, f2(f2(x, a), a))
F2(f2(a, f2(x, a)), a) -> F2(f2(x, a), a)
The TRS R consists of the following rules:
f2(f2(a, f2(x, a)), a) -> f2(a, f2(f2(x, a), a))
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(f2(a, f2(x, a)), a) -> F2(a, f2(f2(x, a), a))
F2(f2(a, f2(x, a)), a) -> F2(f2(x, a), a)
The TRS R consists of the following rules:
f2(f2(a, f2(x, a)), a) -> f2(a, f2(f2(x, a), a))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph contains 1 SCC with 1 less node.
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPAfsSolverProof
Q DP problem:
The TRS P consists of the following rules:
F2(f2(a, f2(x, a)), a) -> F2(f2(x, a), a)
The TRS R consists of the following rules:
f2(f2(a, f2(x, a)), a) -> f2(a, f2(f2(x, a), a))
Q is empty.
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.
F2(f2(a, f2(x, a)), a) -> F2(f2(x, a), a)
Used argument filtering: F2(x1, x2) = x1
f2(x1, x2) = f2(x1, x2)
a = a
Used ordering: Precedence:
trivial
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPAfsSolverProof
↳ QDP
↳ PisEmptyProof
Q DP problem:
P is empty.
The TRS R consists of the following rules:
f2(f2(a, f2(x, a)), a) -> f2(a, f2(f2(x, a), a))
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.