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:
a(a(b(x1))) → a(c(c(a(a(a(x1))))))
a(c(x1)) → b(x1)
Q is empty.
↳ QTRS
↳ QTRS Reverse
↳ Overlay + Local Confluence
↳ QTRS Reverse
Q restricted rewrite system:
The TRS R consists of the following rules:
a(a(b(x1))) → a(c(c(a(a(a(x1))))))
a(c(x1)) → b(x1)
Q is empty.
We have reversed the following QTRS:
The set of rules R is
a(a(b(x1))) → a(c(c(a(a(a(x1))))))
a(c(x1)) → b(x1)
The set Q is empty.
We have obtained the following QTRS:
b(a(a(x))) → a(a(a(c(c(a(x))))))
c(a(x)) → b(x)
The set Q is empty.
↳ QTRS
↳ QTRS Reverse
↳ QTRS
↳ Overlay + Local Confluence
↳ QTRS Reverse
Q restricted rewrite system:
The TRS R consists of the following rules:
b(a(a(x))) → a(a(a(c(c(a(x))))))
c(a(x)) → b(x)
Q is empty.
The TRS is overlay and locally confluent. By [19] we can switch to innermost.
↳ QTRS
↳ QTRS Reverse
↳ Overlay + Local Confluence
↳ QTRS
↳ DependencyPairsProof
↳ QTRS Reverse
Q restricted rewrite system:
The TRS R consists of the following rules:
a(a(b(x1))) → a(c(c(a(a(a(x1))))))
a(c(x1)) → b(x1)
The set Q consists of the following terms:
a(a(b(x0)))
a(c(x0))
Using Dependency Pairs [1,15] we result in the following initial DP problem:
Q DP problem:
The TRS P consists of the following rules:
A(a(b(x1))) → A(x1)
A(a(b(x1))) → A(a(x1))
A(a(b(x1))) → A(a(a(x1)))
A(a(b(x1))) → A(c(c(a(a(a(x1))))))
The TRS R consists of the following rules:
a(a(b(x1))) → a(c(c(a(a(a(x1))))))
a(c(x1)) → b(x1)
The set Q consists of the following terms:
a(a(b(x0)))
a(c(x0))
We have to consider all minimal (P,Q,R)-chains.
↳ QTRS
↳ QTRS Reverse
↳ Overlay + Local Confluence
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QTRS Reverse
Q DP problem:
The TRS P consists of the following rules:
A(a(b(x1))) → A(x1)
A(a(b(x1))) → A(a(x1))
A(a(b(x1))) → A(a(a(x1)))
A(a(b(x1))) → A(c(c(a(a(a(x1))))))
The TRS R consists of the following rules:
a(a(b(x1))) → a(c(c(a(a(a(x1))))))
a(c(x1)) → b(x1)
The set Q consists of the following terms:
a(a(b(x0)))
a(c(x0))
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
↳ QTRS Reverse
↳ Overlay + Local Confluence
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ MNOCProof
↳ Narrowing
↳ QTRS Reverse
Q DP problem:
The TRS P consists of the following rules:
A(a(b(x1))) → A(x1)
A(a(b(x1))) → A(a(x1))
A(a(b(x1))) → A(a(a(x1)))
The TRS R consists of the following rules:
a(a(b(x1))) → a(c(c(a(a(a(x1))))))
a(c(x1)) → b(x1)
The set Q consists of the following terms:
a(a(b(x0)))
a(c(x0))
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
↳ QTRS Reverse
↳ Overlay + Local Confluence
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ MNOCProof
↳ QDP
↳ Narrowing
↳ QTRS Reverse
Q DP problem:
The TRS P consists of the following rules:
A(a(b(x1))) → A(x1)
A(a(b(x1))) → A(a(x1))
A(a(b(x1))) → A(a(a(x1)))
The TRS R consists of the following rules:
a(a(b(x1))) → a(c(c(a(a(a(x1))))))
a(c(x1)) → b(x1)
Q is empty.
We have to consider all (P,Q,R)-chains.
By narrowing [15] the rule A(a(b(x1))) → A(a(a(x1))) at position [0] we obtained the following new rules:
A(a(b(a(b(x0))))) → A(a(a(c(c(a(a(a(x0))))))))
A(a(b(b(x0)))) → A(a(c(c(a(a(a(x0)))))))
A(a(b(c(x0)))) → A(a(b(x0)))
↳ QTRS
↳ QTRS Reverse
↳ Overlay + Local Confluence
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ MNOCProof
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QTRS Reverse
Q DP problem:
The TRS P consists of the following rules:
A(a(b(c(x0)))) → A(a(b(x0)))
A(a(b(b(x0)))) → A(a(c(c(a(a(a(x0)))))))
A(a(b(a(b(x0))))) → A(a(a(c(c(a(a(a(x0))))))))
A(a(b(x1))) → A(x1)
A(a(b(x1))) → A(a(x1))
The TRS R consists of the following rules:
a(a(b(x1))) → a(c(c(a(a(a(x1))))))
a(c(x1)) → b(x1)
The set Q consists of the following terms:
a(a(b(x0)))
a(c(x0))
We have to consider all minimal (P,Q,R)-chains.
By rewriting [15] the rule A(a(b(a(b(x0))))) → A(a(a(c(c(a(a(a(x0)))))))) at position [0,0] we obtained the following new rules:
A(a(b(a(b(x0))))) → A(a(b(c(a(a(a(x0)))))))
↳ QTRS
↳ QTRS Reverse
↳ Overlay + Local Confluence
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ MNOCProof
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QTRS Reverse
Q DP problem:
The TRS P consists of the following rules:
A(a(b(c(x0)))) → A(a(b(x0)))
A(a(b(a(b(x0))))) → A(a(b(c(a(a(a(x0)))))))
A(a(b(x1))) → A(x1)
A(a(b(b(x0)))) → A(a(c(c(a(a(a(x0)))))))
A(a(b(x1))) → A(a(x1))
The TRS R consists of the following rules:
a(a(b(x1))) → a(c(c(a(a(a(x1))))))
a(c(x1)) → b(x1)
The set Q consists of the following terms:
a(a(b(x0)))
a(c(x0))
We have to consider all minimal (P,Q,R)-chains.
By rewriting [15] the rule A(a(b(b(x0)))) → A(a(c(c(a(a(a(x0))))))) at position [0] we obtained the following new rules:
A(a(b(b(x0)))) → A(b(c(a(a(a(x0))))))
↳ QTRS
↳ QTRS Reverse
↳ Overlay + Local Confluence
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ MNOCProof
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ DependencyGraphProof
↳ QTRS Reverse
Q DP problem:
The TRS P consists of the following rules:
A(a(b(c(x0)))) → A(a(b(x0)))
A(a(b(b(x0)))) → A(b(c(a(a(a(x0))))))
A(a(b(a(b(x0))))) → A(a(b(c(a(a(a(x0)))))))
A(a(b(x1))) → A(x1)
A(a(b(x1))) → A(a(x1))
The TRS R consists of the following rules:
a(a(b(x1))) → a(c(c(a(a(a(x1))))))
a(c(x1)) → b(x1)
The set Q consists of the following terms:
a(a(b(x0)))
a(c(x0))
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
↳ QTRS Reverse
↳ Overlay + Local Confluence
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ MNOCProof
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ ForwardInstantiation
↳ QTRS Reverse
Q DP problem:
The TRS P consists of the following rules:
A(a(b(c(x0)))) → A(a(b(x0)))
A(a(b(a(b(x0))))) → A(a(b(c(a(a(a(x0)))))))
A(a(b(x1))) → A(x1)
A(a(b(x1))) → A(a(x1))
The TRS R consists of the following rules:
a(a(b(x1))) → a(c(c(a(a(a(x1))))))
a(c(x1)) → b(x1)
The set Q consists of the following terms:
a(a(b(x0)))
a(c(x0))
We have to consider all minimal (P,Q,R)-chains.
By forward instantiating [14] the rule A(a(b(x1))) → A(x1) we obtained the following new rules:
A(a(b(a(b(y_0))))) → A(a(b(y_0)))
A(a(b(a(b(a(b(y_0))))))) → A(a(b(a(b(y_0)))))
A(a(b(a(b(c(y_0)))))) → A(a(b(c(y_0))))
↳ QTRS
↳ QTRS Reverse
↳ Overlay + Local Confluence
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ MNOCProof
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ QDPToSRSProof
↳ QTRS Reverse
Q DP problem:
The TRS P consists of the following rules:
A(a(b(a(b(y_0))))) → A(a(b(y_0)))
A(a(b(c(x0)))) → A(a(b(x0)))
A(a(b(a(b(x0))))) → A(a(b(c(a(a(a(x0)))))))
A(a(b(a(b(a(b(y_0))))))) → A(a(b(a(b(y_0)))))
A(a(b(x1))) → A(a(x1))
A(a(b(a(b(c(y_0)))))) → A(a(b(c(y_0))))
The TRS R consists of the following rules:
a(a(b(x1))) → a(c(c(a(a(a(x1))))))
a(c(x1)) → b(x1)
The set Q consists of the following terms:
a(a(b(x0)))
a(c(x0))
We have to consider all minimal (P,Q,R)-chains.
The finiteness of this DP problem is implied by strong termination of a SRS due to [12].
↳ QTRS
↳ QTRS Reverse
↳ Overlay + Local Confluence
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ MNOCProof
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ QDPToSRSProof
↳ QTRS
↳ QTRS Reverse
↳ QTRS Reverse
Q restricted rewrite system:
The TRS R consists of the following rules:
a(a(b(x1))) → a(c(c(a(a(a(x1))))))
a(c(x1)) → b(x1)
A(a(b(a(b(y_0))))) → A(a(b(y_0)))
A(a(b(c(x0)))) → A(a(b(x0)))
A(a(b(a(b(x0))))) → A(a(b(c(a(a(a(x0)))))))
A(a(b(a(b(a(b(y_0))))))) → A(a(b(a(b(y_0)))))
A(a(b(x1))) → A(a(x1))
A(a(b(a(b(c(y_0)))))) → A(a(b(c(y_0))))
The set Q consists of the following terms:
a(a(b(x0)))
a(c(x0))
We have reversed the following QTRS:
The set of rules R is
a(a(b(x1))) → a(c(c(a(a(a(x1))))))
a(c(x1)) → b(x1)
A(a(b(a(b(y_0))))) → A(a(b(y_0)))
A(a(b(c(x0)))) → A(a(b(x0)))
A(a(b(a(b(x0))))) → A(a(b(c(a(a(a(x0)))))))
A(a(b(a(b(a(b(y_0))))))) → A(a(b(a(b(y_0)))))
A(a(b(x1))) → A(a(x1))
A(a(b(a(b(c(y_0)))))) → A(a(b(c(y_0))))
The set Q is {a(a(b(x0))), a(c(x0))}.
We have obtained the following QTRS:
b(a(a(x))) → a(a(a(c(c(a(x))))))
c(a(x)) → b(x)
b(a(b(a(A(x))))) → b(a(A(x)))
c(b(a(A(x)))) → b(a(A(x)))
b(a(b(a(A(x))))) → a(a(a(c(b(a(A(x)))))))
b(a(b(a(b(a(A(x))))))) → b(a(b(a(A(x)))))
b(a(A(x))) → a(A(x))
c(b(a(b(a(A(x)))))) → c(b(a(A(x))))
The set Q is empty.
↳ QTRS
↳ QTRS Reverse
↳ Overlay + Local Confluence
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ MNOCProof
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ QDPToSRSProof
↳ QTRS
↳ QTRS Reverse
↳ QTRS
↳ QTRS Reverse
↳ DependencyPairsProof
↳ QTRS Reverse
↳ QTRS Reverse
Q restricted rewrite system:
The TRS R consists of the following rules:
b(a(a(x))) → a(a(a(c(c(a(x))))))
c(a(x)) → b(x)
b(a(b(a(A(x))))) → b(a(A(x)))
c(b(a(A(x)))) → b(a(A(x)))
b(a(b(a(A(x))))) → a(a(a(c(b(a(A(x)))))))
b(a(b(a(b(a(A(x))))))) → b(a(b(a(A(x)))))
b(a(A(x))) → a(A(x))
c(b(a(b(a(A(x)))))) → c(b(a(A(x))))
Q is empty.
We have reversed the following QTRS:
The set of rules R is
b(a(a(x))) → a(a(a(c(c(a(x))))))
c(a(x)) → b(x)
b(a(b(a(A(x))))) → b(a(A(x)))
c(b(a(A(x)))) → b(a(A(x)))
b(a(b(a(A(x))))) → a(a(a(c(b(a(A(x)))))))
b(a(b(a(b(a(A(x))))))) → b(a(b(a(A(x)))))
b(a(A(x))) → a(A(x))
c(b(a(b(a(A(x)))))) → c(b(a(A(x))))
The set Q is empty.
We have obtained the following QTRS:
a(a(b(x))) → a(c(c(a(a(a(x))))))
a(c(x)) → b(x)
A(a(b(a(b(x))))) → A(a(b(x)))
A(a(b(c(x)))) → A(a(b(x)))
A(a(b(a(b(x))))) → A(a(b(c(a(a(a(x)))))))
A(a(b(a(b(a(b(x))))))) → A(a(b(a(b(x)))))
A(a(b(x))) → A(a(x))
A(a(b(a(b(c(x)))))) → A(a(b(c(x))))
The set Q is empty.
↳ QTRS
↳ QTRS Reverse
↳ Overlay + Local Confluence
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ MNOCProof
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ QDPToSRSProof
↳ QTRS
↳ QTRS Reverse
↳ QTRS
↳ QTRS Reverse
↳ QTRS
↳ DependencyPairsProof
↳ QTRS Reverse
↳ QTRS Reverse
Q restricted rewrite system:
The TRS R consists of the following rules:
a(a(b(x))) → a(c(c(a(a(a(x))))))
a(c(x)) → b(x)
A(a(b(a(b(x))))) → A(a(b(x)))
A(a(b(c(x)))) → A(a(b(x)))
A(a(b(a(b(x))))) → A(a(b(c(a(a(a(x)))))))
A(a(b(a(b(a(b(x))))))) → A(a(b(a(b(x)))))
A(a(b(x))) → A(a(x))
A(a(b(a(b(c(x)))))) → A(a(b(c(x))))
Q is empty.
Using Dependency Pairs [1,15] we result in the following initial DP problem:
Q DP problem:
The TRS P consists of the following rules:
C(b(a(b(a(A(x)))))) → C(b(a(A(x))))
C(a(x)) → B(x)
B(a(a(x))) → C(a(x))
B(a(a(x))) → C(c(a(x)))
B(a(b(a(A(x))))) → C(b(a(A(x))))
The TRS R consists of the following rules:
b(a(a(x))) → a(a(a(c(c(a(x))))))
c(a(x)) → b(x)
b(a(b(a(A(x))))) → b(a(A(x)))
c(b(a(A(x)))) → b(a(A(x)))
b(a(b(a(A(x))))) → a(a(a(c(b(a(A(x)))))))
b(a(b(a(b(a(A(x))))))) → b(a(b(a(A(x)))))
b(a(A(x))) → a(A(x))
c(b(a(b(a(A(x)))))) → c(b(a(A(x))))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
↳ QTRS
↳ QTRS Reverse
↳ Overlay + Local Confluence
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ MNOCProof
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ QDPToSRSProof
↳ QTRS
↳ QTRS Reverse
↳ QTRS
↳ QTRS Reverse
↳ DependencyPairsProof
↳ QDP
↳ Narrowing
↳ QTRS Reverse
↳ QTRS Reverse
Q DP problem:
The TRS P consists of the following rules:
C(b(a(b(a(A(x)))))) → C(b(a(A(x))))
C(a(x)) → B(x)
B(a(a(x))) → C(a(x))
B(a(a(x))) → C(c(a(x)))
B(a(b(a(A(x))))) → C(b(a(A(x))))
The TRS R consists of the following rules:
b(a(a(x))) → a(a(a(c(c(a(x))))))
c(a(x)) → b(x)
b(a(b(a(A(x))))) → b(a(A(x)))
c(b(a(A(x)))) → b(a(A(x)))
b(a(b(a(A(x))))) → a(a(a(c(b(a(A(x)))))))
b(a(b(a(b(a(A(x))))))) → b(a(b(a(A(x)))))
b(a(A(x))) → a(A(x))
c(b(a(b(a(A(x)))))) → c(b(a(A(x))))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule B(a(a(x))) → C(c(a(x))) at position [0] we obtained the following new rules:
B(a(a(x0))) → C(b(x0))
↳ QTRS
↳ QTRS Reverse
↳ Overlay + Local Confluence
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ MNOCProof
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ QDPToSRSProof
↳ QTRS
↳ QTRS Reverse
↳ QTRS
↳ QTRS Reverse
↳ DependencyPairsProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QTRS Reverse
↳ QTRS Reverse
Q DP problem:
The TRS P consists of the following rules:
C(b(a(b(a(A(x)))))) → C(b(a(A(x))))
C(a(x)) → B(x)
B(a(a(x))) → C(a(x))
B(a(a(x0))) → C(b(x0))
B(a(b(a(A(x))))) → C(b(a(A(x))))
The TRS R consists of the following rules:
b(a(a(x))) → a(a(a(c(c(a(x))))))
c(a(x)) → b(x)
b(a(b(a(A(x))))) → b(a(A(x)))
c(b(a(A(x)))) → b(a(A(x)))
b(a(b(a(A(x))))) → a(a(a(c(b(a(A(x)))))))
b(a(b(a(b(a(A(x))))))) → b(a(b(a(A(x)))))
b(a(A(x))) → a(A(x))
c(b(a(b(a(A(x)))))) → c(b(a(A(x))))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule B(a(b(a(A(x))))) → C(b(a(A(x)))) at position [0] we obtained the following new rules:
B(a(b(a(A(x0))))) → C(a(A(x0)))
↳ QTRS
↳ QTRS Reverse
↳ Overlay + Local Confluence
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ MNOCProof
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ QDPToSRSProof
↳ QTRS
↳ QTRS Reverse
↳ QTRS
↳ QTRS Reverse
↳ DependencyPairsProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QTRS Reverse
↳ QTRS Reverse
Q DP problem:
The TRS P consists of the following rules:
C(b(a(b(a(A(x)))))) → C(b(a(A(x))))
C(a(x)) → B(x)
B(a(a(x))) → C(a(x))
B(a(a(x0))) → C(b(x0))
B(a(b(a(A(x0))))) → C(a(A(x0)))
The TRS R consists of the following rules:
b(a(a(x))) → a(a(a(c(c(a(x))))))
c(a(x)) → b(x)
b(a(b(a(A(x))))) → b(a(A(x)))
c(b(a(A(x)))) → b(a(A(x)))
b(a(b(a(A(x))))) → a(a(a(c(b(a(A(x)))))))
b(a(b(a(b(a(A(x))))))) → b(a(b(a(A(x)))))
b(a(A(x))) → a(A(x))
c(b(a(b(a(A(x)))))) → c(b(a(A(x))))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule C(b(a(b(a(A(x)))))) → C(b(a(A(x)))) at position [0] we obtained the following new rules:
C(b(a(b(a(A(x0)))))) → C(a(A(x0)))
↳ QTRS
↳ QTRS Reverse
↳ Overlay + Local Confluence
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ MNOCProof
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ QDPToSRSProof
↳ QTRS
↳ QTRS Reverse
↳ QTRS
↳ QTRS Reverse
↳ DependencyPairsProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ SemLabProof
↳ SemLabProof2
↳ QTRS Reverse
↳ QTRS Reverse
Q DP problem:
The TRS P consists of the following rules:
C(a(x)) → B(x)
B(a(a(x))) → C(a(x))
B(a(a(x0))) → C(b(x0))
C(b(a(b(a(A(x0)))))) → C(a(A(x0)))
B(a(b(a(A(x0))))) → C(a(A(x0)))
The TRS R consists of the following rules:
b(a(a(x))) → a(a(a(c(c(a(x))))))
c(a(x)) → b(x)
b(a(b(a(A(x))))) → b(a(A(x)))
c(b(a(A(x)))) → b(a(A(x)))
b(a(b(a(A(x))))) → a(a(a(c(b(a(A(x)))))))
b(a(b(a(b(a(A(x))))))) → b(a(b(a(A(x)))))
b(a(A(x))) → a(A(x))
c(b(a(b(a(A(x)))))) → c(b(a(A(x))))
Q is empty.
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.C: 0
c: 0
B: 0
a: 0
A: 1
b: 0
By semantic labelling [33] we obtain the following labelled TRS:Q DP problem:
The TRS P consists of the following rules:
C.0(b.0(a.0(b.0(a.1(A.0(x0)))))) → C.0(a.1(A.0(x0)))
B.0(a.0(a.1(x0))) → C.0(b.1(x0))
B.0(a.0(b.0(a.1(A.1(x0))))) → C.0(a.1(A.1(x0)))
C.0(a.0(x)) → B.0(x)
B.0(a.0(a.1(x))) → C.0(a.1(x))
B.0(a.0(a.0(x0))) → C.0(b.0(x0))
C.0(a.1(x)) → B.1(x)
C.0(b.0(a.0(b.0(a.1(A.1(x0)))))) → C.0(a.1(A.1(x0)))
B.0(a.0(b.0(a.1(A.0(x0))))) → C.0(a.1(A.0(x0)))
B.0(a.0(a.0(x))) → C.0(a.0(x))
The TRS R consists of the following rules:
c.0(b.0(a.0(b.0(a.1(A.1(x)))))) → c.0(b.0(a.1(A.1(x))))
c.0(a.1(x)) → b.1(x)
b.0(a.0(a.1(x))) → a.0(a.0(a.0(c.0(c.0(a.1(x))))))
b.0(a.0(b.0(a.1(A.0(x))))) → b.0(a.1(A.0(x)))
c.0(b.0(a.0(b.0(a.1(A.0(x)))))) → c.0(b.0(a.1(A.0(x))))
b.0(a.0(a.0(x))) → a.0(a.0(a.0(c.0(c.0(a.0(x))))))
c.0(b.0(a.1(A.1(x)))) → b.0(a.1(A.1(x)))
c.0(b.0(a.1(A.0(x)))) → b.0(a.1(A.0(x)))
b.0(a.0(b.0(a.0(b.0(a.1(A.1(x))))))) → b.0(a.0(b.0(a.1(A.1(x)))))
b.0(a.0(b.0(a.1(A.1(x))))) → a.0(a.0(a.0(c.0(b.0(a.1(A.1(x)))))))
b.0(a.1(A.0(x))) → a.1(A.0(x))
b.0(a.1(A.1(x))) → a.1(A.1(x))
b.0(a.0(b.0(a.0(b.0(a.1(A.0(x))))))) → b.0(a.0(b.0(a.1(A.0(x)))))
b.0(a.0(b.0(a.1(A.0(x))))) → a.0(a.0(a.0(c.0(b.0(a.1(A.0(x)))))))
c.0(a.0(x)) → b.0(x)
b.0(a.0(b.0(a.1(A.1(x))))) → b.0(a.1(A.1(x)))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
↳ QTRS
↳ QTRS Reverse
↳ Overlay + Local Confluence
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ MNOCProof
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ QDPToSRSProof
↳ QTRS
↳ QTRS Reverse
↳ QTRS
↳ QTRS Reverse
↳ DependencyPairsProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ SemLabProof
↳ QDP
↳ DependencyGraphProof
↳ SemLabProof2
↳ QTRS Reverse
↳ QTRS Reverse
Q DP problem:
The TRS P consists of the following rules:
C.0(b.0(a.0(b.0(a.1(A.0(x0)))))) → C.0(a.1(A.0(x0)))
B.0(a.0(a.1(x0))) → C.0(b.1(x0))
B.0(a.0(b.0(a.1(A.1(x0))))) → C.0(a.1(A.1(x0)))
C.0(a.0(x)) → B.0(x)
B.0(a.0(a.1(x))) → C.0(a.1(x))
B.0(a.0(a.0(x0))) → C.0(b.0(x0))
C.0(a.1(x)) → B.1(x)
C.0(b.0(a.0(b.0(a.1(A.1(x0)))))) → C.0(a.1(A.1(x0)))
B.0(a.0(b.0(a.1(A.0(x0))))) → C.0(a.1(A.0(x0)))
B.0(a.0(a.0(x))) → C.0(a.0(x))
The TRS R consists of the following rules:
c.0(b.0(a.0(b.0(a.1(A.1(x)))))) → c.0(b.0(a.1(A.1(x))))
c.0(a.1(x)) → b.1(x)
b.0(a.0(a.1(x))) → a.0(a.0(a.0(c.0(c.0(a.1(x))))))
b.0(a.0(b.0(a.1(A.0(x))))) → b.0(a.1(A.0(x)))
c.0(b.0(a.0(b.0(a.1(A.0(x)))))) → c.0(b.0(a.1(A.0(x))))
b.0(a.0(a.0(x))) → a.0(a.0(a.0(c.0(c.0(a.0(x))))))
c.0(b.0(a.1(A.1(x)))) → b.0(a.1(A.1(x)))
c.0(b.0(a.1(A.0(x)))) → b.0(a.1(A.0(x)))
b.0(a.0(b.0(a.0(b.0(a.1(A.1(x))))))) → b.0(a.0(b.0(a.1(A.1(x)))))
b.0(a.0(b.0(a.1(A.1(x))))) → a.0(a.0(a.0(c.0(b.0(a.1(A.1(x)))))))
b.0(a.1(A.0(x))) → a.1(A.0(x))
b.0(a.1(A.1(x))) → a.1(A.1(x))
b.0(a.0(b.0(a.0(b.0(a.1(A.0(x))))))) → b.0(a.0(b.0(a.1(A.0(x)))))
b.0(a.0(b.0(a.1(A.0(x))))) → a.0(a.0(a.0(c.0(b.0(a.1(A.0(x)))))))
c.0(a.0(x)) → b.0(x)
b.0(a.0(b.0(a.1(A.1(x))))) → b.0(a.1(A.1(x)))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 1 SCC with 7 less nodes.
↳ QTRS
↳ QTRS Reverse
↳ Overlay + Local Confluence
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ MNOCProof
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ QDPToSRSProof
↳ QTRS
↳ QTRS Reverse
↳ QTRS
↳ QTRS Reverse
↳ DependencyPairsProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ SemLabProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ RuleRemovalProof
↳ SemLabProof2
↳ QTRS Reverse
↳ QTRS Reverse
Q DP problem:
The TRS P consists of the following rules:
C.0(a.0(x)) → B.0(x)
B.0(a.0(a.0(x0))) → C.0(b.0(x0))
B.0(a.0(a.0(x))) → C.0(a.0(x))
The TRS R consists of the following rules:
c.0(b.0(a.0(b.0(a.1(A.1(x)))))) → c.0(b.0(a.1(A.1(x))))
c.0(a.1(x)) → b.1(x)
b.0(a.0(a.1(x))) → a.0(a.0(a.0(c.0(c.0(a.1(x))))))
b.0(a.0(b.0(a.1(A.0(x))))) → b.0(a.1(A.0(x)))
c.0(b.0(a.0(b.0(a.1(A.0(x)))))) → c.0(b.0(a.1(A.0(x))))
b.0(a.0(a.0(x))) → a.0(a.0(a.0(c.0(c.0(a.0(x))))))
c.0(b.0(a.1(A.1(x)))) → b.0(a.1(A.1(x)))
c.0(b.0(a.1(A.0(x)))) → b.0(a.1(A.0(x)))
b.0(a.0(b.0(a.0(b.0(a.1(A.1(x))))))) → b.0(a.0(b.0(a.1(A.1(x)))))
b.0(a.0(b.0(a.1(A.1(x))))) → a.0(a.0(a.0(c.0(b.0(a.1(A.1(x)))))))
b.0(a.1(A.0(x))) → a.1(A.0(x))
b.0(a.1(A.1(x))) → a.1(A.1(x))
b.0(a.0(b.0(a.0(b.0(a.1(A.0(x))))))) → b.0(a.0(b.0(a.1(A.0(x)))))
b.0(a.0(b.0(a.1(A.0(x))))) → a.0(a.0(a.0(c.0(b.0(a.1(A.0(x)))))))
c.0(a.0(x)) → b.0(x)
b.0(a.0(b.0(a.1(A.1(x))))) → b.0(a.1(A.1(x)))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By using the rule removal processor [15] with the following polynomial ordering [25], at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented.
Strictly oriented rules of the TRS R:
c.0(a.1(x)) → b.1(x)
Used ordering: POLO with Polynomial interpretation [25]:
POL(A.0(x1)) = x1
POL(A.1(x1)) = x1
POL(B.0(x1)) = x1
POL(C.0(x1)) = x1
POL(a.0(x1)) = x1
POL(a.1(x1)) = 1 + x1
POL(b.0(x1)) = x1
POL(b.1(x1)) = x1
POL(c.0(x1)) = x1
↳ QTRS
↳ QTRS Reverse
↳ Overlay + Local Confluence
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ MNOCProof
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ QDPToSRSProof
↳ QTRS
↳ QTRS Reverse
↳ QTRS
↳ QTRS Reverse
↳ DependencyPairsProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ SemLabProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ RuleRemovalProof
↳ QDP
↳ SemLabProof2
↳ QTRS Reverse
↳ QTRS Reverse
Q DP problem:
The TRS P consists of the following rules:
C.0(a.0(x)) → B.0(x)
B.0(a.0(a.0(x0))) → C.0(b.0(x0))
B.0(a.0(a.0(x))) → C.0(a.0(x))
The TRS R consists of the following rules:
c.0(b.0(a.0(b.0(a.1(A.1(x)))))) → c.0(b.0(a.1(A.1(x))))
b.0(a.0(a.1(x))) → a.0(a.0(a.0(c.0(c.0(a.1(x))))))
b.0(a.0(b.0(a.1(A.0(x))))) → b.0(a.1(A.0(x)))
c.0(b.0(a.0(b.0(a.1(A.0(x)))))) → c.0(b.0(a.1(A.0(x))))
b.0(a.0(a.0(x))) → a.0(a.0(a.0(c.0(c.0(a.0(x))))))
c.0(b.0(a.1(A.1(x)))) → b.0(a.1(A.1(x)))
c.0(b.0(a.1(A.0(x)))) → b.0(a.1(A.0(x)))
b.0(a.0(b.0(a.0(b.0(a.1(A.1(x))))))) → b.0(a.0(b.0(a.1(A.1(x)))))
b.0(a.0(b.0(a.1(A.1(x))))) → a.0(a.0(a.0(c.0(b.0(a.1(A.1(x)))))))
b.0(a.1(A.0(x))) → a.1(A.0(x))
b.0(a.1(A.1(x))) → a.1(A.1(x))
b.0(a.0(b.0(a.0(b.0(a.1(A.0(x))))))) → b.0(a.0(b.0(a.1(A.0(x)))))
b.0(a.0(b.0(a.1(A.0(x))))) → a.0(a.0(a.0(c.0(b.0(a.1(A.0(x)))))))
c.0(a.0(x)) → b.0(x)
b.0(a.0(b.0(a.1(A.1(x))))) → b.0(a.1(A.1(x)))
Q is empty.
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
↳ QTRS Reverse
↳ Overlay + Local Confluence
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ MNOCProof
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ QDPToSRSProof
↳ QTRS
↳ QTRS Reverse
↳ QTRS
↳ QTRS Reverse
↳ DependencyPairsProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ SemLabProof
↳ SemLabProof2
↳ QDP
↳ QTRS Reverse
↳ QTRS Reverse
Q DP problem:
The TRS P consists of the following rules:
C(a(x)) → B(x)
B(a(a(x))) → C(a(x))
B(a(a(x0))) → C(b(x0))
The TRS R consists of the following rules:
b(a(a(x))) → a(a(a(c(c(a(x))))))
c(a(x)) → b(x)
b(a(b(a(A(x))))) → b(a(A(x)))
c(b(a(A(x)))) → b(a(A(x)))
b(a(b(a(A(x))))) → a(a(a(c(b(a(A(x)))))))
b(a(b(a(b(a(A(x))))))) → b(a(b(a(A(x)))))
b(a(A(x))) → a(A(x))
c(b(a(b(a(A(x)))))) → c(b(a(A(x))))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We have reversed the following QTRS:
The set of rules R is
b(a(a(x))) → a(a(a(c(c(a(x))))))
c(a(x)) → b(x)
b(a(b(a(A(x))))) → b(a(A(x)))
c(b(a(A(x)))) → b(a(A(x)))
b(a(b(a(A(x))))) → a(a(a(c(b(a(A(x)))))))
b(a(b(a(b(a(A(x))))))) → b(a(b(a(A(x)))))
b(a(A(x))) → a(A(x))
c(b(a(b(a(A(x)))))) → c(b(a(A(x))))
The set Q is empty.
We have obtained the following QTRS:
a(a(b(x))) → a(c(c(a(a(a(x))))))
a(c(x)) → b(x)
A(a(b(a(b(x))))) → A(a(b(x)))
A(a(b(c(x)))) → A(a(b(x)))
A(a(b(a(b(x))))) → A(a(b(c(a(a(a(x)))))))
A(a(b(a(b(a(b(x))))))) → A(a(b(a(b(x)))))
A(a(b(x))) → A(a(x))
A(a(b(a(b(c(x)))))) → A(a(b(c(x))))
The set Q is empty.
↳ QTRS
↳ QTRS Reverse
↳ Overlay + Local Confluence
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ MNOCProof
↳ Narrowing
↳ QDP
↳ Rewriting
↳ QDP
↳ Rewriting
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ QDPToSRSProof
↳ QTRS
↳ QTRS Reverse
↳ QTRS
↳ QTRS Reverse
↳ DependencyPairsProof
↳ QTRS Reverse
↳ QTRS
↳ QTRS Reverse
Q restricted rewrite system:
The TRS R consists of the following rules:
a(a(b(x))) → a(c(c(a(a(a(x))))))
a(c(x)) → b(x)
A(a(b(a(b(x))))) → A(a(b(x)))
A(a(b(c(x)))) → A(a(b(x)))
A(a(b(a(b(x))))) → A(a(b(c(a(a(a(x)))))))
A(a(b(a(b(a(b(x))))))) → A(a(b(a(b(x)))))
A(a(b(x))) → A(a(x))
A(a(b(a(b(c(x)))))) → A(a(b(c(x))))
Q is empty.
We have reversed the following QTRS:
The set of rules R is
a(a(b(x1))) → a(c(c(a(a(a(x1))))))
a(c(x1)) → b(x1)
The set Q is empty.
We have obtained the following QTRS:
b(a(a(x))) → a(a(a(c(c(a(x))))))
c(a(x)) → b(x)
The set Q is empty.
↳ QTRS
↳ QTRS Reverse
↳ Overlay + Local Confluence
↳ QTRS Reverse
↳ QTRS
Q restricted rewrite system:
The TRS R consists of the following rules:
b(a(a(x))) → a(a(a(c(c(a(x))))))
c(a(x)) → b(x)
Q is empty.