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(f(a, 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(f(a, 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(f(a, 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, a)
F(x, f(y, a)) → F(f(a, 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(f(a, 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, a)
F(x, f(y, a)) → F(f(a, 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(f(a, 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
↳ MNOCProof
Q DP problem:
The TRS P consists of the following rules:
F(x, f(y, a)) → F(f(a, 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(f(a, 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(f(a, a), f(x1, a)) → F(f(f(a, 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(f(a, a), f(x1, a)) → F(f(f(a, a), a), x1)
F(x, f(y, a)) → F(f(a, a), f(f(x, a), y))
The TRS R consists of the following rules:
f(x, f(y, a)) → f(f(a, 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(a, a), f(f(x, a), y)) we obtained the following new rules:
F(f(f(z0, a), a), f(x1, a)) → F(f(a, a), f(f(f(f(z0, a), a), a), x1))
F(f(a, a), f(x1, a)) → F(f(a, a), f(f(f(a, a), a), x1))
F(f(f(a, a), a), f(x1, a)) → F(f(a, a), f(f(f(f(a, 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(z0, a), f(x1, a)) → F(f(f(z0, a), a), x1)
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(a, a), f(f(f(f(z0, a), a), a), x1))
F(f(a, a), f(x1, a)) → F(f(a, a), f(f(f(a, a), a), x1))
F(f(f(a, a), a), f(x1, a)) → F(f(a, a), f(f(f(f(a, a), a), a), x1))
The TRS R consists of the following rules:
f(x, f(y, a)) → f(f(a, 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(a, a), a), f(x1, 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)
↳ 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(a, a), f(x1, a)) → F(f(f(a, a), a), x1)
F(f(f(a, a), a), f(x1, 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(f(z0, a), a), f(x1, a)) → F(f(a, a), f(f(f(f(z0, a), a), a), x1))
F(f(a, a), f(x1, a)) → F(f(a, a), f(f(f(a, a), a), x1))
F(f(f(a, a), a), f(x1, a)) → F(f(a, a), f(f(f(f(a, a), a), a), x1))
The TRS R consists of the following rules:
f(x, f(y, a)) → f(f(a, 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(f(a, a), f(f(f(f(z0, a), a), a), x1)) we obtained the following new rules:
F(f(f(f(a, a), a), a), f(x1, a)) → F(f(a, a), f(f(f(f(f(a, a), a), a), a), x1))
F(f(f(f(z0, a), a), a), f(x1, a)) → F(f(a, a), f(f(f(f(f(z0, a), a), a), a), x1))
F(f(f(a, a), a), f(x1, a)) → F(f(a, 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(a, a), a), a), f(x1, a)) → F(f(a, a), f(f(f(f(f(a, a), a), a), a), x1))
F(f(a, a), f(x1, a)) → F(f(f(a, a), a), x1)
F(f(f(a, a), a), f(x1, 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(f(a, a), f(f(f(a, a), a), x1))
F(f(f(f(z0, a), a), a), f(x1, a)) → F(f(a, a), f(f(f(f(f(z0, a), a), a), a), x1))
F(f(f(a, a), a), f(x1, a)) → F(f(a, a), f(f(f(f(a, a), a), a), x1))
The TRS R consists of the following rules:
f(x, f(y, a)) → f(f(a, 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_0, a), a)) → F(f(f(a, 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(a, a), a), a), f(x1, a)) → F(f(a, a), f(f(f(f(f(a, a), a), a), a), x1))
F(f(a, a), f(f(y_0, a), a)) → F(f(f(a, a), a), f(y_0, a))
F(f(f(a, a), a), f(x1, 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(f(a, a), f(f(f(a, a), a), x1))
F(f(f(a, a), a), f(x1, a)) → F(f(a, a), f(f(f(f(a, a), a), a), x1))
F(f(f(f(z0, a), a), a), f(x1, a)) → F(f(a, a), f(f(f(f(f(z0, a), a), a), a), x1))
The TRS R consists of the following rules:
f(x, f(y, a)) → f(f(a, 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(a, a), f(f(f(a, a), a), x1)) we obtained the following new rules:
F(f(a, a), f(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
↳ MNOCProof
Q DP problem:
The TRS P consists of the following rules:
F(f(f(f(a, a), a), a), f(x1, a)) → F(f(a, a), f(f(f(f(f(a, a), a), a), a), x1))
F(f(a, a), f(f(y_0, a), a)) → F(f(f(a, a), a), f(y_0, a))
F(f(a, a), f(a, a)) → F(f(a, a), f(f(f(a, a), a), a))
F(f(f(a, a), a), f(x1, 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(f(f(z0, a), a), a), f(x1, a)) → F(f(a, a), f(f(f(f(f(z0, a), a), a), a), x1))
F(f(f(a, a), a), f(x1, a)) → F(f(a, a), f(f(f(f(a, a), a), a), x1))
The TRS R consists of the following rules:
f(x, f(y, a)) → f(f(a, 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(a, a), a), a), f(x1, a)) → F(f(a, a), f(f(f(f(f(a, a), a), a), a), x1))
F(f(a, a), f(f(y_0, a), a)) → F(f(f(a, a), a), f(y_0, a))
F(f(f(a, a), a), f(x1, 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(f(f(z0, a), a), a), f(x1, a)) → F(f(a, a), f(f(f(f(f(z0, a), a), a), a), x1))
F(f(f(a, a), a), f(x1, a)) → F(f(a, a), f(f(f(f(a, a), a), a), x1))
The TRS R consists of the following rules:
f(x, f(y, a)) → f(f(a, 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(f(a, 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(f(a, 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
↳ MNOCProof
Q DP problem:
The TRS P consists of the following rules:
F(f(f(f(a, a), a), a), f(x1, a)) → F(f(a, a), f(f(f(f(f(a, a), a), a), a), x1))
F(f(a, a), f(f(y_0, a), a)) → F(f(f(a, a), a), f(y_0, a))
F(f(f(a, a), a), f(a, a)) → F(f(a, a), f(f(f(f(a, a), a), a), a))
F(f(f(a, a), a), f(x1, 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(f(f(z0, a), a), a), f(x1, a)) → F(f(a, a), f(f(f(f(f(z0, a), a), a), a), x1))
The TRS R consists of the following rules:
f(x, f(y, a)) → f(f(a, 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(f(f(f(a, a), a), a), x1) we obtained the following new rules:
F(f(f(a, a), a), f(f(y_1, a), a)) → F(f(f(f(a, 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(a, a), a), a), f(x1, a)) → F(f(a, a), f(f(f(f(f(a, a), a), a), a), x1))
F(f(a, a), f(f(y_0, a), a)) → F(f(f(a, a), a), f(y_0, a))
F(f(f(a, a), a), f(a, a)) → F(f(a, a), f(f(f(f(a, a), a), a), a))
F(f(f(z0, a), a), f(x1, a)) → F(f(f(f(z0, a), a), a), x1)
F(f(f(a, a), a), f(f(y_1, a), a)) → F(f(f(f(a, a), a), a), f(y_1, a))
F(f(f(f(z0, a), a), a), f(x1, a)) → F(f(a, a), f(f(f(f(f(z0, a), a), a), a), x1))
The TRS R consists of the following rules:
f(x, f(y, a)) → f(f(a, 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(a, a), a), f(f(y_0, a), a)) → F(f(f(f(a, a), a), a), f(y_0, 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))
↳ 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
↳ ForwardInstantiation
↳ MNOCProof
Q DP problem:
The TRS P consists of the following rules:
F(f(f(f(a, a), a), a), f(x1, a)) → F(f(a, a), f(f(f(f(f(a, a), a), a), a), x1))
F(f(a, a), f(f(y_0, a), a)) → F(f(f(a, a), a), f(y_0, a))
F(f(f(a, a), a), f(a, a)) → F(f(a, a), f(f(f(f(a, a), a), a), a))
F(f(f(a, a), a), f(f(y_1, a), a)) → F(f(f(f(a, a), a), a), f(y_1, 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))
F(f(f(f(z0, a), a), a), f(x1, a)) → F(f(a, a), f(f(f(f(f(z0, a), a), a), a), x1))
The TRS R consists of the following rules:
f(x, f(y, a)) → f(f(a, 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(f(y_0, a), a)) → F(f(f(a, a), a), f(y_0, a)) we obtained the following new rules:
F(f(a, a), f(f(f(y_0, a), a), a)) → F(f(f(a, a), a), f(f(y_0, a), a))
F(f(a, a), f(f(a, a), a)) → F(f(f(a, 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
↳ ForwardInstantiation
↳ QDP
↳ DependencyGraphProof
↳ MNOCProof
Q DP problem:
The TRS P consists of the following rules:
F(f(f(f(a, a), a), a), f(x1, a)) → F(f(a, a), f(f(f(f(f(a, a), a), a), a), x1))
F(f(f(a, a), a), f(a, a)) → F(f(a, a), f(f(f(f(a, a), a), a), a))
F(f(a, a), f(f(f(y_0, a), a), a)) → F(f(f(a, a), a), f(f(y_0, a), a))
F(f(a, a), f(f(a, a), a)) → F(f(f(a, a), a), f(a, a))
F(f(f(a, a), a), f(f(y_1, a), a)) → F(f(f(f(a, a), a), a), f(y_1, 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))
F(f(f(f(z0, a), a), a), f(x1, a)) → F(f(a, a), f(f(f(f(f(z0, a), a), a), a), x1))
The TRS R consists of the following rules:
f(x, f(y, a)) → f(f(a, 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
↳ ForwardInstantiation
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ ForwardInstantiation
↳ MNOCProof
Q DP problem:
The TRS P consists of the following rules:
F(f(f(f(a, a), a), a), f(x1, a)) → F(f(a, a), f(f(f(f(f(a, a), a), a), a), x1))
F(f(a, a), f(f(f(y_0, a), a), a)) → F(f(f(a, a), a), f(f(y_0, a), a))
F(f(f(a, a), a), f(f(y_1, a), a)) → F(f(f(f(a, a), a), a), f(y_1, 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))
F(f(f(f(z0, a), a), a), f(x1, a)) → F(f(a, a), f(f(f(f(f(z0, a), a), a), a), x1))
The TRS R consists of the following rules:
f(x, f(y, a)) → f(f(a, 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(a, a), a), a), f(x1, a)) → F(f(a, a), f(f(f(f(f(a, a), a), a), a), x1)) we obtained the following new rules:
F(f(f(f(a, a), a), a), f(a, a)) → F(f(a, a), f(f(f(f(f(a, 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
↳ ForwardInstantiation
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ ForwardInstantiation
↳ MNOCProof
Q DP problem:
The TRS P consists of the following rules:
F(f(a, a), f(f(f(y_0, a), a), a)) → F(f(f(a, a), a), f(f(y_0, a), a))
F(f(f(a, a), a), f(f(y_1, a), a)) → F(f(f(f(a, a), a), a), f(y_1, 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))
F(f(f(f(a, a), a), a), f(a, a)) → F(f(a, a), f(f(f(f(f(a, a), a), a), a), a))
F(f(f(f(z0, a), a), a), f(x1, a)) → F(f(a, a), f(f(f(f(f(z0, a), a), a), a), x1))
The TRS R consists of the following rules:
f(x, f(y, a)) → f(f(a, 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(f(a, 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(f(a, 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
↳ ForwardInstantiation
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ UsableRulesProof
↳ MNOCProof
Q DP problem:
The TRS P consists of the following rules:
F(f(f(f(x0, a), a), a), f(a, a)) → F(f(a, a), f(f(f(f(f(x0, a), a), a), a), a))
F(f(a, a), f(f(f(y_0, a), a), a)) → F(f(f(a, a), a), f(f(y_0, a), a))
F(f(f(a, a), a), f(f(y_1, a), a)) → F(f(f(f(a, a), a), a), f(y_1, 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))
F(f(f(f(a, a), a), a), f(a, a)) → F(f(a, a), f(f(f(f(f(a, a), a), a), a), a))
The TRS R consists of the following rules:
f(x, f(y, a)) → f(f(a, 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
↳ ForwardInstantiation
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Instantiation
↳ MNOCProof
Q DP problem:
The TRS P consists of the following rules:
F(f(f(f(x0, a), a), a), f(a, a)) → F(f(a, a), f(f(f(f(f(x0, a), a), a), a), a))
F(f(a, a), f(f(f(y_0, a), a), a)) → F(f(f(a, a), a), f(f(y_0, a), a))
F(f(f(a, a), a), f(f(y_1, a), a)) → F(f(f(f(a, a), a), a), f(y_1, 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))
F(f(f(f(a, a), a), a), f(a, a)) → F(f(a, a), f(f(f(f(f(a, a), a), a), a), 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(f(y_0, a), a), a)) → F(f(f(a, a), a), f(f(y_0, a), a)) we obtained the following new rules:
F(f(a, a), f(f(f(f(f(z0, a), a), a), a), a)) → F(f(f(a, a), a), f(f(f(f(z0, a), a), a), a))
F(f(a, a), f(f(f(f(f(a, a), a), a), a), a)) → F(f(f(a, a), 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
↳ ForwardInstantiation
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Instantiation
↳ QDP
↳ SemLabProof
↳ SemLabProof2
↳ MNOCProof
Q DP problem:
The TRS P consists of the following rules:
F(f(f(f(x0, a), a), a), f(a, a)) → F(f(a, a), f(f(f(f(f(x0, a), a), a), a), a))
F(f(a, a), f(f(f(f(f(z0, a), a), a), a), a)) → F(f(f(a, a), a), f(f(f(f(z0, a), a), a), a))
F(f(f(a, a), a), f(f(y_1, a), a)) → F(f(f(f(a, a), a), a), f(y_1, 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))
F(f(a, a), f(f(f(f(f(a, a), a), a), a), a)) → F(f(f(a, a), a), f(f(f(f(a, a), a), a), a))
F(f(f(f(a, a), a), a), f(a, a)) → F(f(a, a), f(f(f(f(f(a, a), a), a), a), 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 found the following model for the rules of the TRS R.
Interpretation over the domain with elements from 0 to 1.a: 0
f: 1
F: 0
By semantic labelling [33] we obtain the following labelled TRS:Q DP problem:
The TRS P consists of the following rules:
F.1-1(f.1-0(f.1-0(f.0-0(a., a.), a.), a.), f.0-0(a., a.)) → F.1-1(f.0-0(a., a.), f.1-0(f.1-0(f.1-0(f.1-0(f.0-0(a., a.), a.), a.), a.), a.))
F.1-1(f.1-0(f.0-0(a., a.), a.), f.1-0(f.0-0(y_1, a.), a.)) → F.1-1(f.1-0(f.1-0(f.0-0(a., a.), a.), a.), f.0-0(y_1, a.))
F.1-1(f.1-0(f.1-0(f.0-0(x0, a.), a.), a.), f.0-0(a., a.)) → F.1-1(f.0-0(a., a.), f.1-0(f.1-0(f.1-0(f.1-0(f.0-0(x0, a.), a.), a.), a.), a.))
F.1-1(f.0-0(a., a.), f.1-0(f.1-0(f.1-0(f.1-0(f.1-0(z0, a.), a.), a.), a.), a.)) → F.1-1(f.1-0(f.0-0(a., a.), a.), f.1-0(f.1-0(f.1-0(f.1-0(z0, a.), a.), a.), a.))
F.1-1(f.1-0(f.1-0(f.1-0(x0, a.), a.), a.), f.0-0(a., a.)) → F.1-1(f.0-0(a., a.), f.1-0(f.1-0(f.1-0(f.1-0(f.1-0(x0, a.), a.), a.), a.), a.))
F.1-1(f.1-0(f.0-0(a., a.), a.), f.1-0(f.1-0(y_1, a.), a.)) → F.1-1(f.1-0(f.1-0(f.0-0(a., a.), a.), a.), f.1-0(y_1, a.))
F.1-1(f.1-0(f.1-0(x0, a.), a.), f.1-0(f.0-0(y_1, a.), a.)) → F.1-1(f.1-0(f.1-0(f.1-0(x0, a.), a.), a.), f.0-0(y_1, a.))
F.1-1(f.1-0(f.0-0(x0, a.), a.), f.1-0(f.0-0(y_1, a.), a.)) → F.1-1(f.1-0(f.1-0(f.0-0(x0, a.), a.), a.), f.0-0(y_1, a.))
F.1-1(f.0-0(a., a.), f.1-0(f.1-0(f.1-0(f.1-0(f.0-0(a., a.), a.), a.), a.), a.)) → F.1-1(f.1-0(f.0-0(a., a.), a.), f.1-0(f.1-0(f.1-0(f.0-0(a., a.), a.), a.), a.))
F.1-1(f.1-0(f.0-0(x0, a.), a.), f.1-0(f.1-0(y_1, a.), a.)) → F.1-1(f.1-0(f.1-0(f.0-0(x0, a.), a.), a.), f.1-0(y_1, a.))
F.1-1(f.0-0(a., a.), f.1-0(f.1-0(f.1-0(f.1-0(f.0-0(z0, a.), a.), a.), a.), a.)) → F.1-1(f.1-0(f.0-0(a., a.), a.), f.1-0(f.1-0(f.1-0(f.0-0(z0, a.), a.), a.), a.))
F.1-1(f.1-0(f.1-0(x0, a.), a.), f.1-0(f.1-0(y_1, a.), a.)) → F.1-1(f.1-0(f.1-0(f.1-0(x0, a.), a.), a.), f.1-0(y_1, a.))
R is empty.
The set Q consists of the following terms:
f.0-1(x0, f.0-0(x1, a.))
f.0-1(x0, f.1-0(x1, a.))
f.1-1(x0, f.0-0(x1, a.))
f.1-1(x0, f.1-0(x1, a.))
We have to consider all minimal (P,Q,R)-chains.
↳ 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
↳ ForwardInstantiation
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Instantiation
↳ QDP
↳ SemLabProof
↳ QDP
↳ DependencyGraphProof
↳ SemLabProof2
↳ MNOCProof
Q DP problem:
The TRS P consists of the following rules:
F.1-1(f.1-0(f.1-0(f.0-0(a., a.), a.), a.), f.0-0(a., a.)) → F.1-1(f.0-0(a., a.), f.1-0(f.1-0(f.1-0(f.1-0(f.0-0(a., a.), a.), a.), a.), a.))
F.1-1(f.1-0(f.0-0(a., a.), a.), f.1-0(f.0-0(y_1, a.), a.)) → F.1-1(f.1-0(f.1-0(f.0-0(a., a.), a.), a.), f.0-0(y_1, a.))
F.1-1(f.1-0(f.1-0(f.0-0(x0, a.), a.), a.), f.0-0(a., a.)) → F.1-1(f.0-0(a., a.), f.1-0(f.1-0(f.1-0(f.1-0(f.0-0(x0, a.), a.), a.), a.), a.))
F.1-1(f.0-0(a., a.), f.1-0(f.1-0(f.1-0(f.1-0(f.1-0(z0, a.), a.), a.), a.), a.)) → F.1-1(f.1-0(f.0-0(a., a.), a.), f.1-0(f.1-0(f.1-0(f.1-0(z0, a.), a.), a.), a.))
F.1-1(f.1-0(f.1-0(f.1-0(x0, a.), a.), a.), f.0-0(a., a.)) → F.1-1(f.0-0(a., a.), f.1-0(f.1-0(f.1-0(f.1-0(f.1-0(x0, a.), a.), a.), a.), a.))
F.1-1(f.1-0(f.0-0(a., a.), a.), f.1-0(f.1-0(y_1, a.), a.)) → F.1-1(f.1-0(f.1-0(f.0-0(a., a.), a.), a.), f.1-0(y_1, a.))
F.1-1(f.1-0(f.1-0(x0, a.), a.), f.1-0(f.0-0(y_1, a.), a.)) → F.1-1(f.1-0(f.1-0(f.1-0(x0, a.), a.), a.), f.0-0(y_1, a.))
F.1-1(f.1-0(f.0-0(x0, a.), a.), f.1-0(f.0-0(y_1, a.), a.)) → F.1-1(f.1-0(f.1-0(f.0-0(x0, a.), a.), a.), f.0-0(y_1, a.))
F.1-1(f.0-0(a., a.), f.1-0(f.1-0(f.1-0(f.1-0(f.0-0(a., a.), a.), a.), a.), a.)) → F.1-1(f.1-0(f.0-0(a., a.), a.), f.1-0(f.1-0(f.1-0(f.0-0(a., a.), a.), a.), a.))
F.1-1(f.1-0(f.0-0(x0, a.), a.), f.1-0(f.1-0(y_1, a.), a.)) → F.1-1(f.1-0(f.1-0(f.0-0(x0, a.), a.), a.), f.1-0(y_1, a.))
F.1-1(f.0-0(a., a.), f.1-0(f.1-0(f.1-0(f.1-0(f.0-0(z0, a.), a.), a.), a.), a.)) → F.1-1(f.1-0(f.0-0(a., a.), a.), f.1-0(f.1-0(f.1-0(f.0-0(z0, a.), a.), a.), a.))
F.1-1(f.1-0(f.1-0(x0, a.), a.), f.1-0(f.1-0(y_1, a.), a.)) → F.1-1(f.1-0(f.1-0(f.1-0(x0, a.), a.), a.), f.1-0(y_1, a.))
R is empty.
The set Q consists of the following terms:
f.0-1(x0, f.0-0(x1, a.))
f.0-1(x0, f.1-0(x1, a.))
f.1-1(x0, f.0-0(x1, a.))
f.1-1(x0, f.1-0(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 6 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
↳ ForwardInstantiation
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Instantiation
↳ QDP
↳ SemLabProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ SemLabProof2
↳ MNOCProof
Q DP problem:
The TRS P consists of the following rules:
F.1-1(f.1-0(f.0-0(a., a.), a.), f.1-0(f.1-0(y_1, a.), a.)) → F.1-1(f.1-0(f.1-0(f.0-0(a., a.), a.), a.), f.1-0(y_1, a.))
F.1-1(f.0-0(a., a.), f.1-0(f.1-0(f.1-0(f.1-0(f.1-0(z0, a.), a.), a.), a.), a.)) → F.1-1(f.1-0(f.0-0(a., a.), a.), f.1-0(f.1-0(f.1-0(f.1-0(z0, a.), a.), a.), a.))
F.1-1(f.1-0(f.1-0(f.1-0(x0, a.), a.), a.), f.0-0(a., a.)) → F.1-1(f.0-0(a., a.), f.1-0(f.1-0(f.1-0(f.1-0(f.1-0(x0, a.), a.), a.), a.), a.))
F.1-1(f.1-0(f.1-0(x0, a.), a.), f.1-0(f.0-0(y_1, a.), a.)) → F.1-1(f.1-0(f.1-0(f.1-0(x0, a.), a.), a.), f.0-0(y_1, a.))
F.1-1(f.1-0(f.0-0(x0, a.), a.), f.1-0(f.1-0(y_1, a.), a.)) → F.1-1(f.1-0(f.1-0(f.0-0(x0, a.), a.), a.), f.1-0(y_1, a.))
F.1-1(f.1-0(f.1-0(x0, a.), a.), f.1-0(f.1-0(y_1, a.), a.)) → F.1-1(f.1-0(f.1-0(f.1-0(x0, a.), a.), a.), f.1-0(y_1, a.))
R is empty.
The set Q consists of the following terms:
f.0-1(x0, f.0-0(x1, a.))
f.0-1(x0, f.1-0(x1, a.))
f.1-1(x0, f.0-0(x1, a.))
f.1-1(x0, f.1-0(x1, a.))
We have to consider all minimal (P,Q,R)-chains.
As can be seen after transforming the QDP problem by semantic labelling [33] and then some rule deleting processors, only certain labelled rules and pairs can be used.
Hence, we only have to consider all unlabelled pairs and rules (without the decreasing rules for quasi-models).
↳ 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
↳ ForwardInstantiation
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ Instantiation
↳ QDP
↳ SemLabProof
↳ SemLabProof2
↳ QDP
↳ MNOCProof
Q DP problem:
The TRS P consists of the following rules:
F(f(f(f(x0, a), a), a), f(a, a)) → F(f(a, a), f(f(f(f(f(x0, a), a), a), a), a))
F(f(a, a), f(f(f(f(f(z0, a), a), a), a), a)) → F(f(f(a, a), a), f(f(f(f(z0, a), a), a), a))
F(f(f(a, a), a), f(f(y_1, a), a)) → F(f(f(f(a, a), a), a), f(y_1, 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(f(a, 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(f(a, a), f(f(x, a), y))
Q is empty.
We have to consider all (P,Q,R)-chains.