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:
f(a, f(f(a, x), a)) → f(f(a, f(a, x)), a)
Q is empty.
↳ QTRS
↳ DependencyPairsProof
Q restricted rewrite system:
The TRS R consists of the following rules:
f(a, f(f(a, x), a)) → f(f(a, f(a, x)), a)
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:
F(a, f(f(a, x), a)) → F(f(a, f(a, x)), a)
F(a, f(f(a, x), a)) → F(a, f(a, x))
The TRS R consists of the following rules:
f(a, f(f(a, x), a)) → f(f(a, f(a, x)), a)
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ EdgeDeletionProof
Q DP problem:
The TRS P consists of the following rules:
F(a, f(f(a, x), a)) → F(f(a, f(a, x)), a)
F(a, f(f(a, x), a)) → F(a, f(a, x))
The TRS R consists of the following rules:
f(a, f(f(a, x), a)) → f(f(a, f(a, x)), a)
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We deleted some edges using various graph approximations
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ EdgeDeletionProof
↳ QDP
↳ DependencyGraphProof
Q DP problem:
The TRS P consists of the following rules:
F(a, f(f(a, x), a)) → F(f(a, f(a, x)), a)
F(a, f(f(a, x), a)) → F(a, f(a, x))
The TRS R consists of the following rules:
f(a, f(f(a, x), a)) → f(f(a, f(a, x)), a)
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 1 less node.
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ EdgeDeletionProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
Q DP problem:
The TRS P consists of the following rules:
F(a, f(f(a, x), a)) → F(a, f(a, x))
The TRS R consists of the following rules:
f(a, f(f(a, x), a)) → f(f(a, f(a, x)), a)
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.
F(a, f(f(a, x), a)) → F(a, f(a, x))
The remaining pairs can at least be oriented weakly.
none
Used ordering: Combined order from the following AFS and order.
F(x1, x2) = F(x1, x2)
a = a
f(x1, x2) = f(x1, x2)
Lexicographic path order with status [19].
Quasi-Precedence:
a > F2 > f2
Status: a: multiset
f2: [2,1]
F2: [2,1]
The following usable rules [14] were oriented:
f(a, f(f(a, x), a)) → f(f(a, f(a, x)), a)
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ EdgeDeletionProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ PisEmptyProof
Q DP problem:
P is empty.
The TRS R consists of the following rules:
f(a, f(f(a, x), a)) → f(f(a, f(a, x)), 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.