Termination w.r.t. Q of the following Term Rewriting System could not be shown:
Q restricted rewrite system:
The TRS R consists of the following rules:
f(x, f(y, a)) → f(a, f(f(x, a), y))
Q is empty.
↳ QTRS
↳ Overlay + Local Confluence
Q restricted rewrite system:
The TRS R consists of the following rules:
f(x, f(y, a)) → f(a, f(f(x, a), y))
Q is empty.
The TRS is overlay and locally confluent. By [19] we can switch to innermost.
↳ QTRS
↳ Overlay + Local Confluence
↳ QTRS
↳ DependencyPairsProof
Q restricted rewrite system:
The TRS R consists of the following rules:
f(x, f(y, a)) → f(a, f(f(x, a), y))
The set Q consists of the following terms:
f(x0, f(x1, a))
Using Dependency Pairs [1,15] we result in the following initial DP problem:
Q DP problem:
The TRS P consists of the following rules:
F(x, f(y, a)) → F(x, a)
F(x, f(y, a)) → F(a, f(f(x, a), y))
F(x, f(y, a)) → F(f(x, a), y)
The TRS R consists of the following rules:
f(x, f(y, a)) → f(a, f(f(x, a), y))
The set Q consists of the following terms:
f(x0, f(x1, a))
We have to consider all minimal (P,Q,R)-chains.
↳ QTRS
↳ Overlay + Local Confluence
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
Q DP problem:
The TRS P consists of the following rules:
F(x, f(y, a)) → F(x, a)
F(x, f(y, a)) → F(a, f(f(x, a), y))
F(x, f(y, a)) → F(f(x, a), y)
The TRS R consists of the following rules:
f(x, f(y, a)) → f(a, f(f(x, a), y))
The set Q consists of the following terms:
f(x0, f(x1, a))
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 1 SCC with 1 less node.
↳ QTRS
↳ Overlay + Local Confluence
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Instantiation
↳ MNOCProof
Q DP problem:
The TRS P consists of the following rules:
F(x, f(y, a)) → F(a, f(f(x, a), y))
F(x, f(y, a)) → F(f(x, a), y)
The TRS R consists of the following rules:
f(x, f(y, a)) → f(a, f(f(x, a), y))
The set Q consists of the following terms:
f(x0, f(x1, a))
We have to consider all minimal (P,Q,R)-chains.
By instantiating [15] the rule F(x, f(y, a)) → F(f(x, a), y) we obtained the following new rules:
F(f(z0, a), f(x1, a)) → F(f(f(z0, a), a), x1)
F(a, f(x1, a)) → F(f(a, a), x1)
↳ QTRS
↳ Overlay + Local Confluence
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ MNOCProof
Q DP problem:
The TRS P consists of the following rules:
F(f(z0, a), f(x1, a)) → F(f(f(z0, a), a), x1)
F(a, f(x1, a)) → F(f(a, a), x1)
F(x, f(y, a)) → F(a, f(f(x, a), y))
The TRS R consists of the following rules:
f(x, f(y, a)) → f(a, f(f(x, a), y))
The set Q consists of the following terms:
f(x0, f(x1, a))
We have to consider all minimal (P,Q,R)-chains.
By instantiating [15] the rule F(x, f(y, a)) → F(a, f(f(x, a), y)) we obtained the following new rules:
F(f(f(z0, a), a), f(x1, a)) → F(a, f(f(f(f(z0, a), a), a), x1))
F(a, f(x1, a)) → F(a, f(f(a, a), x1))
F(f(a, a), f(x1, a)) → F(a, f(f(f(a, a), a), x1))
↳ QTRS
↳ Overlay + Local Confluence
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ MNOCProof
Q DP problem:
The TRS P consists of the following rules:
F(f(f(z0, a), a), f(x1, a)) → F(a, f(f(f(f(z0, a), a), a), x1))
F(f(z0, a), f(x1, a)) → F(f(f(z0, a), a), x1)
F(a, f(x1, a)) → F(f(a, a), x1)
F(a, f(x1, a)) → F(a, f(f(a, a), x1))
F(f(a, a), f(x1, a)) → F(a, f(f(f(a, a), a), x1))
The TRS R consists of the following rules:
f(x, f(y, a)) → f(a, f(f(x, a), y))
The set Q consists of the following terms:
f(x0, f(x1, a))
We have to consider all minimal (P,Q,R)-chains.
By instantiating [15] the rule F(f(z0, a), f(x1, a)) → F(f(f(z0, a), a), x1) we obtained the following new rules:
F(f(a, a), f(x1, a)) → F(f(f(a, a), a), x1)
F(f(f(z0, a), a), f(x1, a)) → F(f(f(f(z0, a), a), a), x1)
↳ QTRS
↳ Overlay + Local Confluence
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ MNOCProof
Q DP problem:
The TRS P consists of the following rules:
F(f(f(z0, a), a), f(x1, a)) → F(a, f(f(f(f(z0, a), a), a), x1))
F(f(a, a), f(x1, a)) → F(f(f(a, a), a), x1)
F(a, f(x1, a)) → F(f(a, a), x1)
F(f(f(z0, a), a), f(x1, a)) → F(f(f(f(z0, a), a), a), x1)
F(a, f(x1, a)) → F(a, f(f(a, a), x1))
F(f(a, a), f(x1, a)) → F(a, f(f(f(a, a), a), x1))
The TRS R consists of the following rules:
f(x, f(y, a)) → f(a, f(f(x, a), y))
The set Q consists of the following terms:
f(x0, f(x1, a))
We have to consider all minimal (P,Q,R)-chains.
By instantiating [15] the rule F(f(f(z0, a), a), f(x1, a)) → F(a, f(f(f(f(z0, a), a), a), x1)) we obtained the following new rules:
F(f(f(f(z0, a), a), a), f(x1, a)) → F(a, f(f(f(f(f(z0, a), a), a), a), x1))
F(f(f(a, a), a), f(x1, a)) → F(a, f(f(f(f(a, a), a), a), x1))
↳ QTRS
↳ Overlay + Local Confluence
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ ForwardInstantiation
↳ MNOCProof
Q DP problem:
The TRS P consists of the following rules:
F(f(f(f(z0, a), a), a), f(x1, a)) → F(a, f(f(f(f(f(z0, a), a), a), a), x1))
F(f(a, a), f(x1, a)) → F(f(f(a, a), a), x1)
F(a, f(x1, a)) → F(f(a, a), x1)
F(f(f(a, a), a), f(x1, a)) → F(a, f(f(f(f(a, a), a), a), x1))
F(f(f(z0, a), a), f(x1, a)) → F(f(f(f(z0, a), a), a), x1)
F(a, f(x1, a)) → F(a, f(f(a, a), x1))
F(f(a, a), f(x1, a)) → F(a, f(f(f(a, a), a), x1))
The TRS R consists of the following rules:
f(x, f(y, a)) → f(a, f(f(x, a), y))
The set Q consists of the following terms:
f(x0, f(x1, a))
We have to consider all minimal (P,Q,R)-chains.
By forward instantiating [14] the rule F(a, f(x1, a)) → F(f(a, a), x1) we obtained the following new rules:
F(a, f(f(y_0, a), a)) → F(f(a, a), f(y_0, a))
↳ QTRS
↳ Overlay + Local Confluence
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ ForwardInstantiation
↳ MNOCProof
Q DP problem:
The TRS P consists of the following rules:
F(f(f(f(z0, a), a), a), f(x1, a)) → F(a, f(f(f(f(f(z0, a), a), a), a), x1))
F(f(a, a), f(x1, a)) → F(f(f(a, a), a), x1)
F(a, f(f(y_0, a), a)) → F(f(a, a), f(y_0, a))
F(f(f(a, a), a), f(x1, a)) → F(a, f(f(f(f(a, a), a), a), x1))
F(f(f(z0, a), a), f(x1, a)) → F(f(f(f(z0, a), a), a), x1)
F(a, f(x1, a)) → F(a, f(f(a, a), x1))
F(f(a, a), f(x1, a)) → F(a, f(f(f(a, a), a), x1))
The TRS R consists of the following rules:
f(x, f(y, a)) → f(a, f(f(x, a), y))
The set Q consists of the following terms:
f(x0, f(x1, a))
We have to consider all minimal (P,Q,R)-chains.
By forward instantiating [14] the rule F(a, f(x1, a)) → F(a, f(f(a, a), x1)) we obtained the following new rules:
F(a, f(a, a)) → F(a, f(f(a, a), a))
↳ QTRS
↳ Overlay + Local Confluence
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ DependencyGraphProof
↳ MNOCProof
Q DP problem:
The TRS P consists of the following rules:
F(f(f(f(z0, a), a), a), f(x1, a)) → F(a, f(f(f(f(f(z0, a), a), a), a), x1))
F(f(a, a), f(x1, a)) → F(f(f(a, a), a), x1)
F(a, f(f(y_0, a), a)) → F(f(a, a), f(y_0, a))
F(f(f(a, a), a), f(x1, a)) → F(a, f(f(f(f(a, a), a), a), x1))
F(f(f(z0, a), a), f(x1, a)) → F(f(f(f(z0, a), a), a), x1)
F(a, f(a, a)) → F(a, f(f(a, a), a))
F(f(a, a), f(x1, a)) → F(a, f(f(f(a, a), a), x1))
The TRS R consists of the following rules:
f(x, f(y, a)) → f(a, f(f(x, a), y))
The set Q consists of the following terms:
f(x0, f(x1, a))
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 1 SCC with 1 less node.
↳ QTRS
↳ Overlay + Local Confluence
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ ForwardInstantiation
↳ MNOCProof
Q DP problem:
The TRS P consists of the following rules:
F(f(f(f(z0, a), a), a), f(x1, a)) → F(a, f(f(f(f(f(z0, a), a), a), a), x1))
F(f(a, a), f(x1, a)) → F(f(f(a, a), a), x1)
F(a, f(f(y_0, a), a)) → F(f(a, a), f(y_0, a))
F(f(f(a, a), a), f(x1, a)) → F(a, f(f(f(f(a, a), a), a), x1))
F(f(f(z0, a), a), f(x1, a)) → F(f(f(f(z0, a), a), a), x1)
F(f(a, a), f(x1, a)) → F(a, f(f(f(a, a), a), x1))
The TRS R consists of the following rules:
f(x, f(y, a)) → f(a, f(f(x, a), y))
The set Q consists of the following terms:
f(x0, f(x1, a))
We have to consider all minimal (P,Q,R)-chains.
By forward instantiating [14] the rule F(f(a, a), f(x1, a)) → F(a, f(f(f(a, a), a), x1)) we obtained the following new rules:
F(f(a, a), f(a, a)) → F(a, f(f(f(a, a), a), a))
↳ QTRS
↳ Overlay + Local Confluence
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ ForwardInstantiation
↳ MNOCProof
Q DP problem:
The TRS P consists of the following rules:
F(f(f(f(z0, a), a), a), f(x1, a)) → F(a, f(f(f(f(f(z0, a), a), a), a), x1))
F(f(a, a), f(x1, a)) → F(f(f(a, a), a), x1)
F(a, f(f(y_0, a), a)) → F(f(a, a), f(y_0, a))
F(f(f(a, a), a), f(x1, a)) → F(a, f(f(f(f(a, a), a), a), x1))
F(f(f(z0, a), a), f(x1, a)) → F(f(f(f(z0, a), a), a), x1)
F(f(a, a), f(a, a)) → F(a, f(f(f(a, a), a), a))
The TRS R consists of the following rules:
f(x, f(y, a)) → f(a, f(f(x, a), y))
The set Q consists of the following terms:
f(x0, f(x1, a))
We have to consider all minimal (P,Q,R)-chains.
By forward instantiating [14] the rule F(f(a, a), f(x1, a)) → F(f(f(a, a), a), x1) we obtained the following new rules:
F(f(a, a), f(f(y_1, a), a)) → F(f(f(a, a), a), f(y_1, a))
↳ QTRS
↳ Overlay + Local Confluence
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ ForwardInstantiation
↳ MNOCProof
Q DP problem:
The TRS P consists of the following rules:
F(f(f(f(z0, a), a), a), f(x1, a)) → F(a, f(f(f(f(f(z0, a), a), a), a), x1))
F(f(a, a), f(f(y_1, a), a)) → F(f(f(a, a), a), f(y_1, a))
F(a, f(f(y_0, a), a)) → F(f(a, a), f(y_0, a))
F(f(f(a, a), a), f(x1, a)) → F(a, f(f(f(f(a, a), a), a), x1))
F(f(f(z0, a), a), f(x1, a)) → F(f(f(f(z0, a), a), a), x1)
F(f(a, a), f(a, a)) → F(a, f(f(f(a, a), a), a))
The TRS R consists of the following rules:
f(x, f(y, a)) → f(a, f(f(x, a), y))
The set Q consists of the following terms:
f(x0, f(x1, a))
We have to consider all minimal (P,Q,R)-chains.
By forward instantiating [14] the rule F(a, f(f(y_0, a), a)) → F(f(a, a), f(y_0, a)) we obtained the following new rules:
F(a, f(f(f(y_0, a), a), a)) → F(f(a, a), f(f(y_0, a), a))
F(a, f(f(a, a), a)) → F(f(a, a), f(a, a))
↳ QTRS
↳ Overlay + Local Confluence
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ DependencyGraphProof
↳ MNOCProof
Q DP problem:
The TRS P consists of the following rules:
F(f(f(f(z0, a), a), a), f(x1, a)) → F(a, f(f(f(f(f(z0, a), a), a), a), x1))
F(f(a, a), f(f(y_1, a), a)) → F(f(f(a, a), a), f(y_1, a))
F(a, f(f(f(y_0, a), a), a)) → F(f(a, a), f(f(y_0, a), a))
F(f(f(a, a), a), f(x1, a)) → F(a, f(f(f(f(a, a), a), a), x1))
F(f(f(z0, a), a), f(x1, a)) → F(f(f(f(z0, a), a), a), x1)
F(a, f(f(a, a), a)) → F(f(a, a), f(a, a))
F(f(a, a), f(a, a)) → F(a, f(f(f(a, a), a), a))
The TRS R consists of the following rules:
f(x, f(y, a)) → f(a, f(f(x, a), y))
The set Q consists of the following terms:
f(x0, f(x1, a))
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 1 SCC with 2 less nodes.
↳ QTRS
↳ Overlay + Local Confluence
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ ForwardInstantiation
↳ MNOCProof
Q DP problem:
The TRS P consists of the following rules:
F(f(f(f(z0, a), a), a), f(x1, a)) → F(a, f(f(f(f(f(z0, a), a), a), a), x1))
F(f(a, a), f(f(y_1, a), a)) → F(f(f(a, a), a), f(y_1, a))
F(f(f(a, a), a), f(x1, a)) → F(a, f(f(f(f(a, a), a), a), x1))
F(a, f(f(f(y_0, a), a), a)) → F(f(a, a), f(f(y_0, a), a))
F(f(f(z0, a), a), f(x1, a)) → F(f(f(f(z0, a), a), a), x1)
The TRS R consists of the following rules:
f(x, f(y, a)) → f(a, f(f(x, a), y))
The set Q consists of the following terms:
f(x0, f(x1, a))
We have to consider all minimal (P,Q,R)-chains.
By forward instantiating [14] the rule F(f(f(z0, a), a), f(x1, a)) → F(f(f(f(z0, a), a), a), x1) we obtained the following new rules:
F(f(f(x0, a), a), f(f(y_1, a), a)) → F(f(f(f(x0, a), a), a), f(y_1, a))
↳ QTRS
↳ Overlay + Local Confluence
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ ForwardInstantiation
↳ MNOCProof
Q DP problem:
The TRS P consists of the following rules:
F(f(f(f(z0, a), a), a), f(x1, a)) → F(a, f(f(f(f(f(z0, a), a), a), a), x1))
F(f(a, a), f(f(y_1, a), a)) → F(f(f(a, a), a), f(y_1, a))
F(a, f(f(f(y_0, a), a), a)) → F(f(a, a), f(f(y_0, a), a))
F(f(f(a, a), a), f(x1, a)) → F(a, f(f(f(f(a, a), a), a), x1))
F(f(f(x0, a), a), f(f(y_1, a), a)) → F(f(f(f(x0, a), a), a), f(y_1, a))
The TRS R consists of the following rules:
f(x, f(y, a)) → f(a, f(f(x, a), y))
The set Q consists of the following terms:
f(x0, f(x1, a))
We have to consider all minimal (P,Q,R)-chains.
By forward instantiating [14] the rule F(f(f(f(z0, a), a), a), f(x1, a)) → F(a, f(f(f(f(f(z0, a), a), a), a), x1)) we obtained the following new rules:
F(f(f(f(x0, a), a), a), f(a, a)) → F(a, f(f(f(f(f(x0, a), a), a), a), a))
↳ QTRS
↳ Overlay + Local Confluence
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ ForwardInstantiation
↳ MNOCProof
Q DP problem:
The TRS P consists of the following rules:
F(f(a, a), f(f(y_1, a), a)) → F(f(f(a, a), a), f(y_1, a))
F(f(f(f(x0, a), a), a), f(a, a)) → F(a, f(f(f(f(f(x0, a), a), a), a), a))
F(f(f(a, a), a), f(x1, a)) → F(a, f(f(f(f(a, a), a), a), x1))
F(a, f(f(f(y_0, a), a), a)) → F(f(a, a), f(f(y_0, a), a))
F(f(f(x0, a), a), f(f(y_1, a), a)) → F(f(f(f(x0, a), a), a), f(y_1, a))
The TRS R consists of the following rules:
f(x, f(y, a)) → f(a, f(f(x, a), y))
The set Q consists of the following terms:
f(x0, f(x1, a))
We have to consider all minimal (P,Q,R)-chains.
By forward instantiating [14] the rule F(f(f(a, a), a), f(x1, a)) → F(a, f(f(f(f(a, a), a), a), x1)) we obtained the following new rules:
F(f(f(a, a), a), f(a, a)) → F(a, f(f(f(f(a, a), a), a), a))
↳ QTRS
↳ Overlay + Local Confluence
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ UsableRulesProof
↳ MNOCProof
Q DP problem:
The TRS P consists of the following rules:
F(f(a, a), f(f(y_1, a), a)) → F(f(f(a, a), a), f(y_1, a))
F(f(f(a, a), a), f(a, a)) → F(a, f(f(f(f(a, a), a), a), a))
F(f(f(f(x0, a), a), a), f(a, a)) → F(a, f(f(f(f(f(x0, a), a), a), a), a))
F(a, f(f(f(y_0, a), a), a)) → F(f(a, a), f(f(y_0, a), a))
F(f(f(x0, a), a), f(f(y_1, a), a)) → F(f(f(f(x0, a), a), a), f(y_1, a))
The TRS R consists of the following rules:
f(x, f(y, a)) → f(a, f(f(x, a), y))
The set Q consists of the following terms:
f(x0, f(x1, a))
We have to consider all minimal (P,Q,R)-chains.
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [15] we can delete all non-usable rules [17] from R.
↳ QTRS
↳ Overlay + Local Confluence
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Instantiation
↳ MNOCProof
Q DP problem:
The TRS P consists of the following rules:
F(f(a, a), f(f(y_1, a), a)) → F(f(f(a, a), a), f(y_1, a))
F(f(f(a, a), a), f(a, a)) → F(a, f(f(f(f(a, a), a), a), a))
F(f(f(f(x0, a), a), a), f(a, a)) → F(a, f(f(f(f(f(x0, a), a), a), a), a))
F(a, f(f(f(y_0, a), a), a)) → F(f(a, a), f(f(y_0, a), a))
F(f(f(x0, a), a), f(f(y_1, a), a)) → F(f(f(f(x0, a), a), a), f(y_1, a))
R is empty.
The set Q consists of the following terms:
f(x0, f(x1, a))
We have to consider all minimal (P,Q,R)-chains.
By instantiating [15] the rule F(a, f(f(f(y_0, a), a), a)) → F(f(a, a), f(f(y_0, a), a)) we obtained the following new rules:
F(a, f(f(f(f(f(z0, a), a), a), a), a)) → F(f(a, a), f(f(f(f(z0, a), a), a), a))
F(a, f(f(f(f(a, a), a), a), a)) → F(f(a, a), f(f(f(a, a), a), a))
↳ QTRS
↳ Overlay + Local Confluence
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ MNOCProof
Q DP problem:
The TRS P consists of the following rules:
F(a, f(f(f(f(f(z0, a), a), a), a), a)) → F(f(a, a), f(f(f(f(z0, a), a), a), a))
F(a, f(f(f(f(a, a), a), a), a)) → F(f(a, a), f(f(f(a, a), a), a))
F(f(a, a), f(f(y_1, a), a)) → F(f(f(a, a), a), f(y_1, a))
F(f(f(a, a), a), f(a, a)) → F(a, f(f(f(f(a, a), a), a), a))
F(f(f(f(x0, a), a), a), f(a, a)) → F(a, f(f(f(f(f(x0, a), a), a), a), a))
F(f(f(x0, a), a), f(f(y_1, a), a)) → F(f(f(f(x0, a), a), a), f(y_1, a))
R is empty.
The set Q consists of the following terms:
f(x0, f(x1, a))
We have to consider all minimal (P,Q,R)-chains.
By instantiating [15] the rule F(f(a, a), f(f(y_1, a), a)) → F(f(f(a, a), a), f(y_1, a)) we obtained the following new rules:
F(f(a, a), f(f(f(a, a), a), a)) → F(f(f(a, a), a), f(f(a, a), a))
F(f(a, a), f(f(f(f(z0, a), a), a), a)) → F(f(f(a, a), a), f(f(f(z0, a), a), a))
↳ QTRS
↳ Overlay + Local Confluence
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ DependencyGraphProof
↳ MNOCProof
Q DP problem:
The TRS P consists of the following rules:
F(a, f(f(f(f(f(z0, a), a), a), a), a)) → F(f(a, a), f(f(f(f(z0, a), a), a), a))
F(a, f(f(f(f(a, a), a), a), a)) → F(f(a, a), f(f(f(a, a), a), a))
F(f(f(a, a), a), f(a, a)) → F(a, f(f(f(f(a, a), a), a), a))
F(f(f(f(x0, a), a), a), f(a, a)) → F(a, f(f(f(f(f(x0, a), a), a), a), a))
F(f(a, a), f(f(f(a, a), a), a)) → F(f(f(a, a), a), f(f(a, a), a))
F(f(a, a), f(f(f(f(z0, a), a), a), a)) → F(f(f(a, a), a), f(f(f(z0, a), a), a))
F(f(f(x0, a), a), f(f(y_1, a), a)) → F(f(f(f(x0, a), a), a), f(y_1, a))
R is empty.
The set Q consists of the following terms:
f(x0, f(x1, a))
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 1 SCC with 3 less nodes.
↳ QTRS
↳ Overlay + Local Confluence
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ MNOCProof
Q DP problem:
The TRS P consists of the following rules:
F(a, f(f(f(f(f(z0, a), a), a), a), a)) → F(f(a, a), f(f(f(f(z0, a), a), a), a))
F(f(f(f(x0, a), a), a), f(a, a)) → F(a, f(f(f(f(f(x0, a), a), a), a), a))
F(f(a, a), f(f(f(f(z0, a), a), a), a)) → F(f(f(a, a), a), f(f(f(z0, a), a), a))
F(f(f(x0, a), a), f(f(y_1, a), a)) → F(f(f(f(x0, a), a), a), f(y_1, a))
R is empty.
The set Q consists of the following terms:
f(x0, f(x1, a))
We have to consider all minimal (P,Q,R)-chains.
We use the modular non-overlap check [17] to decrease Q to the empty set.
↳ QTRS
↳ Overlay + Local Confluence
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Instantiation
↳ MNOCProof
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
F(x, f(y, a)) → F(a, f(f(x, a), y))
F(x, f(y, a)) → F(f(x, a), y)
The TRS R consists of the following rules:
f(x, f(y, a)) → f(a, f(f(x, a), y))
Q is empty.
We have to consider all (P,Q,R)-chains.