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(f(a, x), y) → h(h(f(f(x, f(a, y)), a)))
Q is empty.
↳ QTRS
↳ Overlay + Local Confluence
Q restricted rewrite system:
The TRS R consists of the following rules:
f(f(a, x), y) → h(h(f(f(x, f(a, y)), a)))
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(f(a, x), y) → h(h(f(f(x, f(a, y)), a)))
The set Q consists of the following terms:
f(f(a, x0), x1)
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(f(a, x), y) → F(a, y)
F(f(a, x), y) → F(f(x, f(a, y)), a)
F(f(a, x), y) → F(x, f(a, y))
The TRS R consists of the following rules:
f(f(a, x), y) → h(h(f(f(x, f(a, y)), a)))
The set Q consists of the following terms:
f(f(a, x0), x1)
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(f(a, x), y) → F(a, y)
F(f(a, x), y) → F(f(x, f(a, y)), a)
F(f(a, x), y) → F(x, f(a, y))
The TRS R consists of the following rules:
f(f(a, x), y) → h(h(f(f(x, f(a, y)), a)))
The set Q consists of the following terms:
f(f(a, x0), x1)
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(f(a, x), y) → F(f(x, f(a, y)), a)
F(f(a, x), y) → F(x, f(a, y))
The TRS R consists of the following rules:
f(f(a, x), y) → h(h(f(f(x, f(a, y)), a)))
The set Q consists of the following terms:
f(f(a, x0), x1)
We have to consider all minimal (P,Q,R)-chains.
By instantiating [15] the rule F(f(a, x), y) → F(x, f(a, y)) we obtained the following new rules:
F(f(a, x0), a) → F(x0, f(a, a))
F(f(a, x0), f(a, z1)) → F(x0, f(a, f(a, z1)))
↳ 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(a, x0), a) → F(x0, f(a, a))
F(f(a, x0), f(a, z1)) → F(x0, f(a, f(a, z1)))
F(f(a, x), y) → F(f(x, f(a, y)), a)
The TRS R consists of the following rules:
f(f(a, x), y) → h(h(f(f(x, f(a, y)), a)))
The set Q consists of the following terms:
f(f(a, x0), x1)
We have to consider all minimal (P,Q,R)-chains.
By instantiating [15] the rule F(f(a, x), y) → F(f(x, f(a, y)), a) we obtained the following new rules:
F(f(a, x0), a) → F(f(x0, f(a, a)), a)
F(f(a, x0), f(a, f(a, z1))) → F(f(x0, f(a, f(a, f(a, z1)))), a)
F(f(a, x0), f(a, a)) → F(f(x0, f(a, f(a, a))), a)
↳ 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(a, x0), a) → F(x0, f(a, a))
F(f(a, x0), f(a, z1)) → F(x0, f(a, f(a, z1)))
F(f(a, x0), a) → F(f(x0, f(a, a)), a)
F(f(a, x0), f(a, f(a, z1))) → F(f(x0, f(a, f(a, f(a, z1)))), a)
F(f(a, x0), f(a, a)) → F(f(x0, f(a, f(a, a))), a)
The TRS R consists of the following rules:
f(f(a, x), y) → h(h(f(f(x, f(a, y)), a)))
The set Q consists of the following terms:
f(f(a, x0), x1)
We have to consider all minimal (P,Q,R)-chains.
By instantiating [15] the rule F(f(a, x0), f(a, z1)) → F(x0, f(a, f(a, z1))) we obtained the following new rules:
F(f(a, x0), f(a, f(a, z1))) → F(x0, f(a, f(a, f(a, z1))))
F(f(a, x0), f(a, a)) → F(x0, f(a, f(a, a)))
↳ 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, x0), f(a, f(a, z1))) → F(x0, f(a, f(a, f(a, z1))))
F(f(a, x0), f(a, a)) → F(x0, f(a, f(a, a)))
F(f(a, x0), a) → F(x0, f(a, a))
F(f(a, x0), a) → F(f(x0, f(a, a)), a)
F(f(a, x0), f(a, f(a, z1))) → F(f(x0, f(a, f(a, f(a, z1)))), a)
F(f(a, x0), f(a, a)) → F(f(x0, f(a, f(a, a))), a)
The TRS R consists of the following rules:
f(f(a, x), y) → h(h(f(f(x, f(a, y)), a)))
The set Q consists of the following terms:
f(f(a, x0), x1)
We have to consider all minimal (P,Q,R)-chains.
By instantiating [15] the rule F(f(a, x0), f(a, f(a, z1))) → F(f(x0, f(a, f(a, f(a, z1)))), a) we obtained the following new rules:
F(f(a, x0), f(a, f(a, f(a, z1)))) → F(f(x0, f(a, f(a, f(a, f(a, z1))))), a)
F(f(a, x0), f(a, f(a, a))) → F(f(x0, f(a, f(a, f(a, a)))), a)
↳ 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(a, x0), f(a, f(a, f(a, z1)))) → F(f(x0, f(a, f(a, f(a, f(a, z1))))), a)
F(f(a, x0), f(a, f(a, z1))) → F(x0, f(a, f(a, f(a, z1))))
F(f(a, x0), f(a, a)) → F(x0, f(a, f(a, a)))
F(f(a, x0), a) → F(x0, f(a, a))
F(f(a, x0), f(a, f(a, a))) → F(f(x0, f(a, f(a, f(a, a)))), a)
F(f(a, x0), a) → F(f(x0, f(a, a)), a)
F(f(a, x0), f(a, a)) → F(f(x0, f(a, f(a, a))), a)
The TRS R consists of the following rules:
f(f(a, x), y) → h(h(f(f(x, f(a, y)), a)))
The set Q consists of the following terms:
f(f(a, x0), x1)
We have to consider all minimal (P,Q,R)-chains.
By forward instantiating [14] the rule F(f(a, x0), a) → F(x0, f(a, a)) we obtained the following new rules:
F(f(a, f(a, y_0)), a) → F(f(a, y_0), f(a, 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(a, x0), f(a, f(a, f(a, z1)))) → F(f(x0, f(a, f(a, f(a, f(a, z1))))), a)
F(f(a, x0), f(a, f(a, z1))) → F(x0, f(a, f(a, f(a, z1))))
F(f(a, x0), f(a, a)) → F(x0, f(a, f(a, a)))
F(f(a, f(a, y_0)), a) → F(f(a, y_0), f(a, a))
F(f(a, x0), f(a, f(a, a))) → F(f(x0, f(a, f(a, f(a, a)))), a)
F(f(a, x0), a) → F(f(x0, f(a, a)), a)
F(f(a, x0), f(a, a)) → F(f(x0, f(a, f(a, a))), a)
The TRS R consists of the following rules:
f(f(a, x), y) → h(h(f(f(x, f(a, y)), a)))
The set Q consists of the following terms:
f(f(a, x0), x1)
We have to consider all minimal (P,Q,R)-chains.
By forward instantiating [14] the rule F(f(a, x0), a) → F(f(x0, f(a, a)), a) we obtained the following new rules:
F(f(a, a), a) → F(f(a, 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(a, x0), f(a, f(a, f(a, z1)))) → F(f(x0, f(a, f(a, f(a, f(a, z1))))), a)
F(f(a, x0), f(a, f(a, z1))) → F(x0, f(a, f(a, f(a, z1))))
F(f(a, x0), f(a, a)) → F(x0, f(a, f(a, a)))
F(f(a, f(a, y_0)), a) → F(f(a, y_0), f(a, a))
F(f(a, x0), f(a, f(a, a))) → F(f(x0, f(a, f(a, f(a, a)))), a)
F(f(a, a), a) → F(f(a, f(a, a)), a)
F(f(a, x0), f(a, a)) → F(f(x0, f(a, f(a, a))), a)
The TRS R consists of the following rules:
f(f(a, x), y) → h(h(f(f(x, f(a, y)), a)))
The set Q consists of the following terms:
f(f(a, x0), x1)
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(a, x0), f(a, f(a, f(a, z1)))) → F(f(x0, f(a, f(a, f(a, f(a, z1))))), a)
F(f(a, x0), f(a, f(a, z1))) → F(x0, f(a, f(a, f(a, z1))))
F(f(a, x0), f(a, a)) → F(x0, f(a, f(a, a)))
F(f(a, f(a, y_0)), a) → F(f(a, y_0), f(a, a))
F(f(a, x0), f(a, f(a, a))) → F(f(x0, f(a, f(a, f(a, a)))), a)
F(f(a, x0), f(a, a)) → F(f(x0, f(a, f(a, a))), a)
The TRS R consists of the following rules:
f(f(a, x), y) → h(h(f(f(x, f(a, y)), a)))
The set Q consists of the following terms:
f(f(a, x0), x1)
We have to consider all minimal (P,Q,R)-chains.
By forward instantiating [14] the rule F(f(a, x0), f(a, a)) → F(f(x0, f(a, f(a, a))), a) we obtained the following new rules:
F(f(a, a), f(a, a)) → F(f(a, f(a, 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
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ ForwardInstantiation
↳ MNOCProof
Q DP problem:
The TRS P consists of the following rules:
F(f(a, x0), f(a, f(a, f(a, z1)))) → F(f(x0, f(a, f(a, f(a, f(a, z1))))), a)
F(f(a, x0), f(a, f(a, z1))) → F(x0, f(a, f(a, f(a, z1))))
F(f(a, x0), f(a, a)) → F(x0, f(a, f(a, a)))
F(f(a, f(a, y_0)), a) → F(f(a, y_0), f(a, a))
F(f(a, x0), f(a, f(a, a))) → F(f(x0, f(a, f(a, f(a, a)))), a)
F(f(a, a), f(a, a)) → F(f(a, f(a, f(a, a))), a)
The TRS R consists of the following rules:
f(f(a, x), y) → h(h(f(f(x, f(a, y)), a)))
The set Q consists of the following terms:
f(f(a, x0), x1)
We have to consider all minimal (P,Q,R)-chains.
By forward instantiating [14] the rule F(f(a, x0), f(a, a)) → F(x0, f(a, f(a, a))) we obtained the following new rules:
F(f(a, f(a, y_0)), f(a, a)) → F(f(a, y_0), f(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
↳ MNOCProof
Q DP problem:
The TRS P consists of the following rules:
F(f(a, x0), f(a, f(a, f(a, z1)))) → F(f(x0, f(a, f(a, f(a, f(a, z1))))), a)
F(f(a, x0), f(a, f(a, z1))) → F(x0, f(a, f(a, f(a, z1))))
F(f(a, f(a, y_0)), f(a, a)) → F(f(a, y_0), f(a, f(a, a)))
F(f(a, f(a, y_0)), a) → F(f(a, y_0), f(a, a))
F(f(a, x0), f(a, f(a, a))) → F(f(x0, f(a, f(a, f(a, a)))), a)
F(f(a, a), f(a, a)) → F(f(a, f(a, f(a, a))), a)
The TRS R consists of the following rules:
f(f(a, x), y) → h(h(f(f(x, f(a, y)), a)))
The set Q consists of the following terms:
f(f(a, x0), x1)
We have to consider all minimal (P,Q,R)-chains.
By forward instantiating [14] the rule F(f(a, f(a, y_0)), a) → F(f(a, y_0), f(a, a)) we obtained the following new rules:
F(f(a, f(a, f(a, y_0))), a) → F(f(a, f(a, y_0)), f(a, a))
F(f(a, 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(a, x0), f(a, f(a, f(a, z1)))) → F(f(x0, f(a, f(a, f(a, f(a, z1))))), a)
F(f(a, x0), f(a, f(a, z1))) → F(x0, f(a, f(a, f(a, z1))))
F(f(a, f(a, y_0)), f(a, a)) → F(f(a, y_0), f(a, f(a, a)))
F(f(a, f(a, f(a, y_0))), a) → F(f(a, f(a, y_0)), f(a, a))
F(f(a, x0), f(a, f(a, a))) → F(f(x0, f(a, f(a, f(a, a)))), a)
F(f(a, f(a, a)), a) → F(f(a, a), f(a, a))
F(f(a, a), f(a, a)) → F(f(a, f(a, f(a, a))), a)
The TRS R consists of the following rules:
f(f(a, x), y) → h(h(f(f(x, f(a, y)), a)))
The set Q consists of the following terms:
f(f(a, x0), x1)
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(a, x0), f(a, f(a, f(a, z1)))) → F(f(x0, f(a, f(a, f(a, f(a, z1))))), a)
F(f(a, x0), f(a, f(a, z1))) → F(x0, f(a, f(a, f(a, z1))))
F(f(a, f(a, y_0)), f(a, a)) → F(f(a, y_0), f(a, f(a, a)))
F(f(a, x0), f(a, f(a, a))) → F(f(x0, f(a, f(a, f(a, a)))), a)
F(f(a, f(a, f(a, y_0))), a) → F(f(a, f(a, y_0)), f(a, a))
The TRS R consists of the following rules:
f(f(a, x), y) → h(h(f(f(x, f(a, y)), a)))
The set Q consists of the following terms:
f(f(a, x0), x1)
We have to consider all minimal (P,Q,R)-chains.
By forward instantiating [14] the rule F(f(a, x0), f(a, f(a, z1))) → F(x0, f(a, f(a, f(a, z1)))) we obtained the following new rules:
F(f(a, f(a, y_0)), f(a, f(a, x1))) → F(f(a, y_0), f(a, f(a, f(a, x1))))
↳ 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(a, x0), f(a, f(a, f(a, z1)))) → F(f(x0, f(a, f(a, f(a, f(a, z1))))), a)
F(f(a, f(a, y_0)), f(a, f(a, x1))) → F(f(a, y_0), f(a, f(a, f(a, x1))))
F(f(a, f(a, y_0)), f(a, a)) → F(f(a, y_0), f(a, f(a, a)))
F(f(a, f(a, f(a, y_0))), a) → F(f(a, f(a, y_0)), f(a, a))
F(f(a, x0), f(a, f(a, a))) → F(f(x0, f(a, f(a, f(a, a)))), a)
The TRS R consists of the following rules:
f(f(a, x), y) → h(h(f(f(x, f(a, y)), a)))
The set Q consists of the following terms:
f(f(a, x0), x1)
We have to consider all minimal (P,Q,R)-chains.
By forward instantiating [14] the rule F(f(a, x0), f(a, f(a, f(a, z1)))) → F(f(x0, f(a, f(a, f(a, f(a, z1))))), a) we obtained the following new rules:
F(f(a, a), f(a, f(a, f(a, x1)))) → F(f(a, f(a, f(a, f(a, f(a, x1))))), 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, f(a, y_0)), f(a, a)) → F(f(a, y_0), f(a, f(a, a)))
F(f(a, f(a, y_0)), f(a, f(a, x1))) → F(f(a, y_0), f(a, f(a, f(a, x1))))
F(f(a, a), f(a, f(a, f(a, x1)))) → F(f(a, f(a, f(a, f(a, f(a, x1))))), a)
F(f(a, x0), f(a, f(a, a))) → F(f(x0, f(a, f(a, f(a, a)))), a)
F(f(a, f(a, f(a, y_0))), a) → F(f(a, f(a, y_0)), f(a, a))
The TRS R consists of the following rules:
f(f(a, x), y) → h(h(f(f(x, f(a, y)), a)))
The set Q consists of the following terms:
f(f(a, x0), x1)
We have to consider all minimal (P,Q,R)-chains.
By forward instantiating [14] the rule F(f(a, x0), f(a, f(a, a))) → F(f(x0, f(a, f(a, f(a, a)))), a) we obtained the following new rules:
F(f(a, a), f(a, f(a, a))) → F(f(a, f(a, f(a, 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
↳ 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, f(a, y_0)), f(a, f(a, x1))) → F(f(a, y_0), f(a, f(a, f(a, x1))))
F(f(a, f(a, y_0)), f(a, a)) → F(f(a, y_0), f(a, f(a, a)))
F(f(a, a), f(a, f(a, a))) → F(f(a, f(a, f(a, f(a, a)))), a)
F(f(a, a), f(a, f(a, f(a, x1)))) → F(f(a, f(a, f(a, f(a, f(a, x1))))), a)
F(f(a, f(a, f(a, y_0))), a) → F(f(a, f(a, y_0)), f(a, a))
The TRS R consists of the following rules:
f(f(a, x), y) → h(h(f(f(x, f(a, y)), a)))
The set Q consists of the following terms:
f(f(a, x0), x1)
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, f(a, y_0)), f(a, f(a, x1))) → F(f(a, y_0), f(a, f(a, f(a, x1))))
F(f(a, f(a, y_0)), f(a, a)) → F(f(a, y_0), f(a, f(a, a)))
F(f(a, a), f(a, f(a, a))) → F(f(a, f(a, f(a, f(a, a)))), a)
F(f(a, a), f(a, f(a, f(a, x1)))) → F(f(a, f(a, f(a, f(a, f(a, x1))))), a)
F(f(a, f(a, f(a, y_0))), a) → F(f(a, f(a, y_0)), f(a, a))
R is empty.
The set Q consists of the following terms:
f(f(a, x0), x1)
We have to consider all minimal (P,Q,R)-chains.
By instantiating [15] the rule F(f(a, f(a, f(a, y_0))), a) → F(f(a, f(a, y_0)), f(a, a)) we obtained the following new rules:
F(f(a, f(a, f(a, f(a, f(a, z0))))), a) → F(f(a, f(a, f(a, f(a, z0)))), f(a, a))
F(f(a, f(a, f(a, f(a, a)))), a) → F(f(a, f(a, 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
↳ 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(f(a, f(a, f(a, f(a, f(a, z0))))), a) → F(f(a, f(a, f(a, f(a, z0)))), f(a, a))
F(f(a, f(a, f(a, f(a, a)))), a) → F(f(a, f(a, f(a, a))), f(a, a))
F(f(a, f(a, y_0)), f(a, a)) → F(f(a, y_0), f(a, f(a, a)))
F(f(a, f(a, y_0)), f(a, f(a, x1))) → F(f(a, y_0), f(a, f(a, f(a, x1))))
F(f(a, a), f(a, f(a, a))) → F(f(a, f(a, f(a, f(a, a)))), a)
F(f(a, a), f(a, f(a, f(a, x1)))) → F(f(a, f(a, f(a, f(a, f(a, x1))))), a)
R is empty.
The set Q consists of the following terms:
f(f(a, x0), x1)
We have to consider all minimal (P,Q,R)-chains.
By instantiating [15] the rule F(f(a, f(a, y_0)), f(a, a)) → F(f(a, y_0), f(a, f(a, a))) we obtained the following new rules:
F(f(a, f(a, f(a, a))), f(a, a)) → F(f(a, f(a, a)), f(a, f(a, a)))
F(f(a, f(a, f(a, f(a, z0)))), f(a, a)) → F(f(a, f(a, f(a, z0))), f(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
↳ 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(f(a, f(a, f(a, f(a, f(a, z0))))), a) → F(f(a, f(a, f(a, f(a, z0)))), f(a, a))
F(f(a, f(a, y_0)), f(a, f(a, x1))) → F(f(a, y_0), f(a, f(a, f(a, x1))))
F(f(a, f(a, f(a, f(a, a)))), a) → F(f(a, f(a, f(a, a))), f(a, a))
F(f(a, a), f(a, f(a, a))) → F(f(a, f(a, f(a, f(a, a)))), a)
F(f(a, a), f(a, f(a, f(a, x1)))) → F(f(a, f(a, f(a, f(a, f(a, x1))))), a)
F(f(a, f(a, f(a, a))), f(a, a)) → F(f(a, f(a, a)), f(a, f(a, a)))
F(f(a, f(a, f(a, f(a, z0)))), f(a, a)) → F(f(a, f(a, f(a, z0))), f(a, f(a, a)))
R is empty.
The set Q consists of the following terms:
f(f(a, x0), x1)
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(f(a, f(a, f(a, f(a, f(a, z0))))), a) → F(f(a, f(a, f(a, f(a, z0)))), f(a, a))
F(f(a, f(a, y_0)), f(a, f(a, x1))) → F(f(a, y_0), f(a, f(a, f(a, x1))))
F(f(a, a), f(a, f(a, f(a, x1)))) → F(f(a, f(a, f(a, f(a, f(a, x1))))), a)
F(f(a, f(a, f(a, f(a, z0)))), f(a, a)) → F(f(a, f(a, f(a, z0))), f(a, f(a, a)))
R is empty.
The set Q consists of the following terms:
f(f(a, x0), x1)
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(f(a, x), y) → F(f(x, f(a, y)), a)
F(f(a, x), y) → F(x, f(a, y))
The TRS R consists of the following rules:
f(f(a, x), y) → h(h(f(f(x, f(a, y)), a)))
Q is empty.
We have to consider all (P,Q,R)-chains.