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:
app2(f, app2(f, x)) -> app2(g, app2(f, x))
app2(g, app2(g, x)) -> app2(f, x)
Q is empty.
↳ QTRS
↳ DependencyPairsProof
Q restricted rewrite system:
The TRS R consists of the following rules:
app2(f, app2(f, x)) -> app2(g, app2(f, x))
app2(g, app2(g, x)) -> app2(f, 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:
APP2(g, app2(g, x)) -> APP2(f, x)
APP2(f, app2(f, x)) -> APP2(g, app2(f, x))
The TRS R consists of the following rules:
app2(f, app2(f, x)) -> app2(g, app2(f, x))
app2(g, app2(g, x)) -> app2(f, 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:
APP2(g, app2(g, x)) -> APP2(f, x)
APP2(f, app2(f, x)) -> APP2(g, app2(f, x))
The TRS R consists of the following rules:
app2(f, app2(f, x)) -> app2(g, app2(f, x))
app2(g, app2(g, x)) -> app2(f, 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 strictly oriented and are deleted.
APP2(g, app2(g, x)) -> APP2(f, x)
APP2(f, app2(f, x)) -> APP2(g, app2(f, x))
The remaining pairs can at least by weakly be oriented.
none
Used ordering: Combined order from the following AFS and order.
APP2(x1, x2) = APP2(x1, x2)
g = g
app2(x1, x2) = app1(x2)
f = f
Lexicographic Path Order [19].
Precedence:
[app1, f] > APP2 > g
The following usable rules [14] were oriented:
app2(f, app2(f, x)) -> app2(g, app2(f, x))
app2(g, app2(g, x)) -> app2(f, x)
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ PisEmptyProof
Q DP problem:
P is empty.
The TRS R consists of the following rules:
app2(f, app2(f, x)) -> app2(g, app2(f, x))
app2(g, app2(g, x)) -> app2(f, 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.