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(a(x1))) → b(b(b(x1)))
b(a(a(b(x1)))) → x1
b(a(a(b(x1)))) → b(a(a(a(b(x1)))))
Q is empty.
↳ QTRS
↳ DependencyPairsProof
↳ QTRS Reverse
↳ QTRS Reverse
Q restricted rewrite system:
The TRS R consists of the following rules:
a(a(a(x1))) → b(b(b(x1)))
b(a(a(b(x1)))) → x1
b(a(a(b(x1)))) → b(a(a(a(b(x1)))))
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:
A(a(a(x1))) → B(x1)
A(a(a(x1))) → B(b(x1))
B(a(a(b(x1)))) → A(a(a(b(x1))))
B(a(a(b(x1)))) → B(a(a(a(b(x1)))))
A(a(a(x1))) → B(b(b(x1)))
The TRS R consists of the following rules:
a(a(a(x1))) → b(b(b(x1)))
b(a(a(b(x1)))) → x1
b(a(a(b(x1)))) → b(a(a(a(b(x1)))))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ Narrowing
↳ QTRS Reverse
↳ QTRS Reverse
Q DP problem:
The TRS P consists of the following rules:
A(a(a(x1))) → B(x1)
A(a(a(x1))) → B(b(x1))
B(a(a(b(x1)))) → A(a(a(b(x1))))
B(a(a(b(x1)))) → B(a(a(a(b(x1)))))
A(a(a(x1))) → B(b(b(x1)))
The TRS R consists of the following rules:
a(a(a(x1))) → b(b(b(x1)))
b(a(a(b(x1)))) → x1
b(a(a(b(x1)))) → b(a(a(a(b(x1)))))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule A(a(a(x1))) → B(b(b(x1))) at position [0] we obtained the following new rules:
A(a(a(a(a(b(x0)))))) → B(b(b(a(a(a(b(x0)))))))
A(a(a(a(a(b(x0)))))) → B(b(x0))
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QTRS Reverse
↳ QTRS Reverse
Q DP problem:
The TRS P consists of the following rules:
A(a(a(x1))) → B(x1)
A(a(a(a(a(b(x0)))))) → B(b(b(a(a(a(b(x0)))))))
A(a(a(x1))) → B(b(x1))
B(a(a(b(x1)))) → B(a(a(a(b(x1)))))
B(a(a(b(x1)))) → A(a(a(b(x1))))
A(a(a(a(a(b(x0)))))) → B(b(x0))
The TRS R consists of the following rules:
a(a(a(x1))) → b(b(b(x1)))
b(a(a(b(x1)))) → x1
b(a(a(b(x1)))) → b(a(a(a(b(x1)))))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule A(a(a(x1))) → B(b(x1)) at position [0] we obtained the following new rules:
A(a(a(a(a(b(x0)))))) → B(b(a(a(a(b(x0))))))
A(a(a(a(a(b(x0)))))) → B(x0)
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QTRS Reverse
↳ QTRS Reverse
Q DP problem:
The TRS P consists of the following rules:
A(a(a(x1))) → B(x1)
A(a(a(a(a(b(x0)))))) → B(b(a(a(a(b(x0))))))
A(a(a(a(a(b(x0)))))) → B(x0)
A(a(a(a(a(b(x0)))))) → B(b(b(a(a(a(b(x0)))))))
B(a(a(b(x1)))) → A(a(a(b(x1))))
B(a(a(b(x1)))) → B(a(a(a(b(x1)))))
A(a(a(a(a(b(x0)))))) → B(b(x0))
The TRS R consists of the following rules:
a(a(a(x1))) → b(b(b(x1)))
b(a(a(b(x1)))) → x1
b(a(a(b(x1)))) → b(a(a(a(b(x1)))))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule B(a(a(b(x1)))) → B(a(a(a(b(x1))))) at position [0] we obtained the following new rules:
B(a(a(b(a(a(b(x0))))))) → B(a(a(a(b(a(a(a(b(x0)))))))))
B(a(a(b(a(a(b(x0))))))) → B(a(a(a(x0))))
B(a(a(b(y0)))) → B(b(b(b(b(y0)))))
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ QDPToSRSProof
↳ QTRS Reverse
↳ QTRS Reverse
Q DP problem:
The TRS P consists of the following rules:
A(a(a(x1))) → B(x1)
A(a(a(a(a(b(x0)))))) → B(b(a(a(a(b(x0))))))
B(a(a(b(a(a(b(x0))))))) → B(a(a(a(b(a(a(a(b(x0)))))))))
B(a(a(b(y0)))) → B(b(b(b(b(y0)))))
B(a(a(b(a(a(b(x0))))))) → B(a(a(a(x0))))
A(a(a(a(a(b(x0)))))) → B(b(b(a(a(a(b(x0)))))))
A(a(a(a(a(b(x0)))))) → B(x0)
B(a(a(b(x1)))) → A(a(a(b(x1))))
A(a(a(a(a(b(x0)))))) → B(b(x0))
The TRS R consists of the following rules:
a(a(a(x1))) → b(b(b(x1)))
b(a(a(b(x1)))) → x1
b(a(a(b(x1)))) → b(a(a(a(b(x1)))))
Q is empty.
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
↳ DependencyPairsProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ QDPToSRSProof
↳ QTRS
↳ QTRS Reverse
↳ QTRS Reverse
↳ QTRS Reverse
Q restricted rewrite system:
The TRS R consists of the following rules:
a(a(a(x1))) → b(b(b(x1)))
b(a(a(b(x1)))) → x1
b(a(a(b(x1)))) → b(a(a(a(b(x1)))))
A(a(a(x1))) → B(x1)
A(a(a(a(a(b(x0)))))) → B(b(a(a(a(b(x0))))))
B(a(a(b(a(a(b(x0))))))) → B(a(a(a(b(a(a(a(b(x0)))))))))
B(a(a(b(y0)))) → B(b(b(b(b(y0)))))
B(a(a(b(a(a(b(x0))))))) → B(a(a(a(x0))))
A(a(a(a(a(b(x0)))))) → B(b(b(a(a(a(b(x0)))))))
A(a(a(a(a(b(x0)))))) → B(x0)
B(a(a(b(x1)))) → A(a(a(b(x1))))
A(a(a(a(a(b(x0)))))) → B(b(x0))
Q is empty.
We have reversed the following QTRS:
The set of rules R is
a(a(a(x1))) → b(b(b(x1)))
b(a(a(b(x1)))) → x1
b(a(a(b(x1)))) → b(a(a(a(b(x1)))))
A(a(a(x1))) → B(x1)
A(a(a(a(a(b(x0)))))) → B(b(a(a(a(b(x0))))))
B(a(a(b(a(a(b(x0))))))) → B(a(a(a(b(a(a(a(b(x0)))))))))
B(a(a(b(y0)))) → B(b(b(b(b(y0)))))
B(a(a(b(a(a(b(x0))))))) → B(a(a(a(x0))))
A(a(a(a(a(b(x0)))))) → B(b(b(a(a(a(b(x0)))))))
A(a(a(a(a(b(x0)))))) → B(x0)
B(a(a(b(x1)))) → A(a(a(b(x1))))
A(a(a(a(a(b(x0)))))) → B(b(x0))
The set Q is empty.
We have obtained the following QTRS:
a(a(a(x))) → b(b(b(x)))
b(a(a(b(x)))) → x
b(a(a(b(x)))) → b(a(a(a(b(x)))))
a(a(A(x))) → B(x)
b(a(a(a(a(A(x)))))) → b(a(a(a(b(B(x))))))
b(a(a(b(a(a(B(x))))))) → b(a(a(a(b(a(a(a(B(x)))))))))
b(a(a(B(x)))) → b(b(b(b(B(x)))))
b(a(a(b(a(a(B(x))))))) → a(a(a(B(x))))
b(a(a(a(a(A(x)))))) → b(a(a(a(b(b(B(x)))))))
b(a(a(a(a(A(x)))))) → B(x)
b(a(a(B(x)))) → b(a(a(A(x))))
b(a(a(a(a(A(x)))))) → b(B(x))
The set Q is empty.
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ QDPToSRSProof
↳ QTRS
↳ QTRS Reverse
↳ QTRS
↳ QTRS Reverse
↳ QTRS Reverse
↳ DependencyPairsProof
↳ QTRS Reverse
↳ QTRS Reverse
Q restricted rewrite system:
The TRS R consists of the following rules:
a(a(a(x))) → b(b(b(x)))
b(a(a(b(x)))) → x
b(a(a(b(x)))) → b(a(a(a(b(x)))))
a(a(A(x))) → B(x)
b(a(a(a(a(A(x)))))) → b(a(a(a(b(B(x))))))
b(a(a(b(a(a(B(x))))))) → b(a(a(a(b(a(a(a(B(x)))))))))
b(a(a(B(x)))) → b(b(b(b(B(x)))))
b(a(a(b(a(a(B(x))))))) → a(a(a(B(x))))
b(a(a(a(a(A(x)))))) → b(a(a(a(b(b(B(x)))))))
b(a(a(a(a(A(x)))))) → B(x)
b(a(a(B(x)))) → b(a(a(A(x))))
b(a(a(a(a(A(x)))))) → b(B(x))
Q is empty.
We have reversed the following QTRS:
The set of rules R is
a(a(a(x))) → b(b(b(x)))
b(a(a(b(x)))) → x
b(a(a(b(x)))) → b(a(a(a(b(x)))))
a(a(A(x))) → B(x)
b(a(a(a(a(A(x)))))) → b(a(a(a(b(B(x))))))
b(a(a(b(a(a(B(x))))))) → b(a(a(a(b(a(a(a(B(x)))))))))
b(a(a(B(x)))) → b(b(b(b(B(x)))))
b(a(a(b(a(a(B(x))))))) → a(a(a(B(x))))
b(a(a(a(a(A(x)))))) → b(a(a(a(b(b(B(x)))))))
b(a(a(a(a(A(x)))))) → B(x)
b(a(a(B(x)))) → b(a(a(A(x))))
b(a(a(a(a(A(x)))))) → b(B(x))
The set Q is empty.
We have obtained the following QTRS:
a(a(a(x))) → b(b(b(x)))
b(a(a(b(x)))) → x
b(a(a(b(x)))) → b(a(a(a(b(x)))))
A(a(a(x))) → B(x)
A(a(a(a(a(b(x)))))) → B(b(a(a(a(b(x))))))
B(a(a(b(a(a(b(x))))))) → B(a(a(a(b(a(a(a(b(x)))))))))
B(a(a(b(x)))) → B(b(b(b(b(x)))))
B(a(a(b(a(a(b(x))))))) → B(a(a(a(x))))
A(a(a(a(a(b(x)))))) → B(b(b(a(a(a(b(x)))))))
A(a(a(a(a(b(x)))))) → B(x)
B(a(a(b(x)))) → A(a(a(b(x))))
A(a(a(a(a(b(x)))))) → B(b(x))
The set Q is empty.
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ QDPToSRSProof
↳ QTRS
↳ QTRS Reverse
↳ QTRS
↳ QTRS Reverse
↳ QTRS
↳ QTRS Reverse
↳ DependencyPairsProof
↳ QTRS Reverse
↳ QTRS Reverse
Q restricted rewrite system:
The TRS R consists of the following rules:
a(a(a(x))) → b(b(b(x)))
b(a(a(b(x)))) → x
b(a(a(b(x)))) → b(a(a(a(b(x)))))
A(a(a(x))) → B(x)
A(a(a(a(a(b(x)))))) → B(b(a(a(a(b(x))))))
B(a(a(b(a(a(b(x))))))) → B(a(a(a(b(a(a(a(b(x)))))))))
B(a(a(b(x)))) → B(b(b(b(b(x)))))
B(a(a(b(a(a(b(x))))))) → B(a(a(a(x))))
A(a(a(a(a(b(x)))))) → B(b(b(a(a(a(b(x)))))))
A(a(a(a(a(b(x)))))) → B(x)
B(a(a(b(x)))) → A(a(a(b(x))))
A(a(a(a(a(b(x)))))) → B(b(x))
Q is empty.
We have reversed the following QTRS:
The set of rules R is
a(a(a(x))) → b(b(b(x)))
b(a(a(b(x)))) → x
b(a(a(b(x)))) → b(a(a(a(b(x)))))
a(a(A(x))) → B(x)
b(a(a(a(a(A(x)))))) → b(a(a(a(b(B(x))))))
b(a(a(b(a(a(B(x))))))) → b(a(a(a(b(a(a(a(B(x)))))))))
b(a(a(B(x)))) → b(b(b(b(B(x)))))
b(a(a(b(a(a(B(x))))))) → a(a(a(B(x))))
b(a(a(a(a(A(x)))))) → b(a(a(a(b(b(B(x)))))))
b(a(a(a(a(A(x)))))) → B(x)
b(a(a(B(x)))) → b(a(a(A(x))))
b(a(a(a(a(A(x)))))) → b(B(x))
The set Q is empty.
We have obtained the following QTRS:
a(a(a(x))) → b(b(b(x)))
b(a(a(b(x)))) → x
b(a(a(b(x)))) → b(a(a(a(b(x)))))
A(a(a(x))) → B(x)
A(a(a(a(a(b(x)))))) → B(b(a(a(a(b(x))))))
B(a(a(b(a(a(b(x))))))) → B(a(a(a(b(a(a(a(b(x)))))))))
B(a(a(b(x)))) → B(b(b(b(b(x)))))
B(a(a(b(a(a(b(x))))))) → B(a(a(a(x))))
A(a(a(a(a(b(x)))))) → B(b(b(a(a(a(b(x)))))))
A(a(a(a(a(b(x)))))) → B(x)
B(a(a(b(x)))) → A(a(a(b(x))))
A(a(a(a(a(b(x)))))) → B(b(x))
The set Q is empty.
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ QDPToSRSProof
↳ QTRS
↳ QTRS Reverse
↳ QTRS
↳ QTRS Reverse
↳ QTRS Reverse
↳ QTRS
↳ DependencyPairsProof
↳ QTRS Reverse
↳ QTRS Reverse
Q restricted rewrite system:
The TRS R consists of the following rules:
a(a(a(x))) → b(b(b(x)))
b(a(a(b(x)))) → x
b(a(a(b(x)))) → b(a(a(a(b(x)))))
A(a(a(x))) → B(x)
A(a(a(a(a(b(x)))))) → B(b(a(a(a(b(x))))))
B(a(a(b(a(a(b(x))))))) → B(a(a(a(b(a(a(a(b(x)))))))))
B(a(a(b(x)))) → B(b(b(b(b(x)))))
B(a(a(b(a(a(b(x))))))) → B(a(a(a(x))))
A(a(a(a(a(b(x)))))) → B(b(b(a(a(a(b(x)))))))
A(a(a(a(a(b(x)))))) → B(x)
B(a(a(b(x)))) → A(a(a(b(x))))
A(a(a(a(a(b(x)))))) → B(b(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:
B1(a(a(B(x)))) → B1(b(b(b(B(x)))))
B1(a(a(a(a(A(x)))))) → B1(B(x))
B1(a(a(a(a(A(x)))))) → A1(a(b(B(x))))
B1(a(a(b(x)))) → B1(a(a(a(b(x)))))
B1(a(a(a(a(A(x)))))) → A1(a(a(b(b(B(x))))))
B1(a(a(B(x)))) → B1(B(x))
B1(a(a(b(a(a(B(x))))))) → A1(a(a(B(x))))
B1(a(a(a(a(A(x)))))) → A1(b(b(B(x))))
B1(a(a(B(x)))) → A1(a(A(x)))
B1(a(a(a(a(A(x)))))) → A1(b(B(x)))
B1(a(a(a(a(A(x)))))) → B1(a(a(a(b(b(B(x)))))))
B1(a(a(a(a(A(x)))))) → A1(a(b(b(B(x)))))
B1(a(a(a(a(A(x)))))) → A1(a(a(b(B(x)))))
B1(a(a(b(a(a(B(x))))))) → A1(b(a(a(a(B(x))))))
B1(a(a(B(x)))) → B1(a(a(A(x))))
B1(a(a(b(x)))) → A1(a(a(b(x))))
A1(a(a(x))) → B1(x)
B1(a(a(b(a(a(B(x))))))) → A1(a(a(b(a(a(a(B(x))))))))
B1(a(a(a(a(A(x)))))) → B1(a(a(a(b(B(x))))))
B1(a(a(b(a(a(B(x))))))) → A1(a(b(a(a(a(B(x)))))))
A1(a(a(x))) → B1(b(x))
B1(a(a(B(x)))) → A1(A(x))
B1(a(a(B(x)))) → B1(b(b(B(x))))
B1(a(a(B(x)))) → B1(b(B(x)))
B1(a(a(b(a(a(B(x))))))) → B1(a(a(a(b(a(a(a(B(x)))))))))
B1(a(a(b(a(a(B(x))))))) → B1(a(a(a(B(x)))))
A1(a(a(x))) → B1(b(b(x)))
B1(a(a(a(a(A(x)))))) → B1(b(B(x)))
The TRS R consists of the following rules:
a(a(a(x))) → b(b(b(x)))
b(a(a(b(x)))) → x
b(a(a(b(x)))) → b(a(a(a(b(x)))))
a(a(A(x))) → B(x)
b(a(a(a(a(A(x)))))) → b(a(a(a(b(B(x))))))
b(a(a(b(a(a(B(x))))))) → b(a(a(a(b(a(a(a(B(x)))))))))
b(a(a(B(x)))) → b(b(b(b(B(x)))))
b(a(a(b(a(a(B(x))))))) → a(a(a(B(x))))
b(a(a(a(a(A(x)))))) → b(a(a(a(b(b(B(x)))))))
b(a(a(a(a(A(x)))))) → B(x)
b(a(a(B(x)))) → b(a(a(A(x))))
b(a(a(a(a(A(x)))))) → b(B(x))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ QDPToSRSProof
↳ QTRS
↳ QTRS Reverse
↳ QTRS
↳ QTRS Reverse
↳ QTRS Reverse
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QTRS Reverse
↳ QTRS Reverse
Q DP problem:
The TRS P consists of the following rules:
B1(a(a(B(x)))) → B1(b(b(b(B(x)))))
B1(a(a(a(a(A(x)))))) → B1(B(x))
B1(a(a(a(a(A(x)))))) → A1(a(b(B(x))))
B1(a(a(b(x)))) → B1(a(a(a(b(x)))))
B1(a(a(a(a(A(x)))))) → A1(a(a(b(b(B(x))))))
B1(a(a(B(x)))) → B1(B(x))
B1(a(a(b(a(a(B(x))))))) → A1(a(a(B(x))))
B1(a(a(a(a(A(x)))))) → A1(b(b(B(x))))
B1(a(a(B(x)))) → A1(a(A(x)))
B1(a(a(a(a(A(x)))))) → A1(b(B(x)))
B1(a(a(a(a(A(x)))))) → B1(a(a(a(b(b(B(x)))))))
B1(a(a(a(a(A(x)))))) → A1(a(b(b(B(x)))))
B1(a(a(a(a(A(x)))))) → A1(a(a(b(B(x)))))
B1(a(a(b(a(a(B(x))))))) → A1(b(a(a(a(B(x))))))
B1(a(a(B(x)))) → B1(a(a(A(x))))
B1(a(a(b(x)))) → A1(a(a(b(x))))
A1(a(a(x))) → B1(x)
B1(a(a(b(a(a(B(x))))))) → A1(a(a(b(a(a(a(B(x))))))))
B1(a(a(a(a(A(x)))))) → B1(a(a(a(b(B(x))))))
B1(a(a(b(a(a(B(x))))))) → A1(a(b(a(a(a(B(x)))))))
A1(a(a(x))) → B1(b(x))
B1(a(a(B(x)))) → A1(A(x))
B1(a(a(B(x)))) → B1(b(b(B(x))))
B1(a(a(B(x)))) → B1(b(B(x)))
B1(a(a(b(a(a(B(x))))))) → B1(a(a(a(b(a(a(a(B(x)))))))))
B1(a(a(b(a(a(B(x))))))) → B1(a(a(a(B(x)))))
A1(a(a(x))) → B1(b(b(x)))
B1(a(a(a(a(A(x)))))) → B1(b(B(x)))
The TRS R consists of the following rules:
a(a(a(x))) → b(b(b(x)))
b(a(a(b(x)))) → x
b(a(a(b(x)))) → b(a(a(a(b(x)))))
a(a(A(x))) → B(x)
b(a(a(a(a(A(x)))))) → b(a(a(a(b(B(x))))))
b(a(a(b(a(a(B(x))))))) → b(a(a(a(b(a(a(a(B(x)))))))))
b(a(a(B(x)))) → b(b(b(b(B(x)))))
b(a(a(b(a(a(B(x))))))) → a(a(a(B(x))))
b(a(a(a(a(A(x)))))) → b(a(a(a(b(b(B(x)))))))
b(a(a(a(a(A(x)))))) → B(x)
b(a(a(B(x)))) → b(a(a(A(x))))
b(a(a(a(a(A(x)))))) → b(B(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 12 less nodes.
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ QDPToSRSProof
↳ QTRS
↳ QTRS Reverse
↳ QTRS
↳ QTRS Reverse
↳ QTRS Reverse
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QTRS Reverse
↳ QTRS Reverse
Q DP problem:
The TRS P consists of the following rules:
B1(a(a(b(a(a(B(x))))))) → A1(a(a(b(a(a(a(B(x))))))))
B1(a(a(a(a(A(x)))))) → B1(a(a(a(b(B(x))))))
B1(a(a(b(a(a(B(x))))))) → A1(a(b(a(a(a(B(x)))))))
A1(a(a(x))) → B1(b(x))
B1(a(a(b(x)))) → B1(a(a(a(b(x)))))
B1(a(a(a(a(A(x)))))) → A1(a(a(b(b(B(x))))))
B1(a(a(b(a(a(B(x))))))) → A1(a(a(B(x))))
B1(a(a(b(a(a(B(x))))))) → B1(a(a(a(b(a(a(a(B(x)))))))))
B1(a(a(b(a(a(B(x))))))) → B1(a(a(a(B(x)))))
A1(a(a(x))) → B1(b(b(x)))
B1(a(a(a(a(A(x)))))) → B1(a(a(a(b(b(B(x)))))))
B1(a(a(a(a(A(x)))))) → A1(a(a(b(B(x)))))
B1(a(a(b(a(a(B(x))))))) → A1(b(a(a(a(B(x))))))
B1(a(a(B(x)))) → B1(a(a(A(x))))
B1(a(a(b(x)))) → A1(a(a(b(x))))
A1(a(a(x))) → B1(x)
The TRS R consists of the following rules:
a(a(a(x))) → b(b(b(x)))
b(a(a(b(x)))) → x
b(a(a(b(x)))) → b(a(a(a(b(x)))))
a(a(A(x))) → B(x)
b(a(a(a(a(A(x)))))) → b(a(a(a(b(B(x))))))
b(a(a(b(a(a(B(x))))))) → b(a(a(a(b(a(a(a(B(x)))))))))
b(a(a(B(x)))) → b(b(b(b(B(x)))))
b(a(a(b(a(a(B(x))))))) → a(a(a(B(x))))
b(a(a(a(a(A(x)))))) → b(a(a(a(b(b(B(x)))))))
b(a(a(a(a(A(x)))))) → B(x)
b(a(a(B(x)))) → b(a(a(A(x))))
b(a(a(a(a(A(x)))))) → b(B(x))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule B1(a(a(a(a(A(x)))))) → B1(a(a(a(b(B(x)))))) at position [0] we obtained the following new rules:
B1(a(a(a(a(A(y0)))))) → B1(b(b(b(b(B(y0))))))
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ QDPToSRSProof
↳ QTRS
↳ QTRS Reverse
↳ QTRS
↳ QTRS Reverse
↳ QTRS Reverse
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QTRS Reverse
↳ QTRS Reverse
Q DP problem:
The TRS P consists of the following rules:
B1(a(a(b(a(a(B(x))))))) → A1(a(a(b(a(a(a(B(x))))))))
B1(a(a(b(a(a(B(x))))))) → A1(a(b(a(a(a(B(x)))))))
A1(a(a(x))) → B1(b(x))
B1(a(a(b(x)))) → B1(a(a(a(b(x)))))
B1(a(a(a(a(A(x)))))) → A1(a(a(b(b(B(x))))))
B1(a(a(b(a(a(B(x))))))) → A1(a(a(B(x))))
B1(a(a(a(a(A(y0)))))) → B1(b(b(b(b(B(y0))))))
B1(a(a(b(a(a(B(x))))))) → B1(a(a(a(B(x)))))
B1(a(a(b(a(a(B(x))))))) → B1(a(a(a(b(a(a(a(B(x)))))))))
A1(a(a(x))) → B1(b(b(x)))
B1(a(a(a(a(A(x)))))) → B1(a(a(a(b(b(B(x)))))))
B1(a(a(a(a(A(x)))))) → A1(a(a(b(B(x)))))
B1(a(a(b(a(a(B(x))))))) → A1(b(a(a(a(B(x))))))
B1(a(a(b(x)))) → A1(a(a(b(x))))
B1(a(a(B(x)))) → B1(a(a(A(x))))
A1(a(a(x))) → B1(x)
The TRS R consists of the following rules:
a(a(a(x))) → b(b(b(x)))
b(a(a(b(x)))) → x
b(a(a(b(x)))) → b(a(a(a(b(x)))))
a(a(A(x))) → B(x)
b(a(a(a(a(A(x)))))) → b(a(a(a(b(B(x))))))
b(a(a(b(a(a(B(x))))))) → b(a(a(a(b(a(a(a(B(x)))))))))
b(a(a(B(x)))) → b(b(b(b(B(x)))))
b(a(a(b(a(a(B(x))))))) → a(a(a(B(x))))
b(a(a(a(a(A(x)))))) → b(a(a(a(b(b(B(x)))))))
b(a(a(a(a(A(x)))))) → B(x)
b(a(a(B(x)))) → b(a(a(A(x))))
b(a(a(a(a(A(x)))))) → b(B(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 1 less node.
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ QDPToSRSProof
↳ QTRS
↳ QTRS Reverse
↳ QTRS
↳ QTRS Reverse
↳ QTRS Reverse
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QTRS Reverse
↳ QTRS Reverse
Q DP problem:
The TRS P consists of the following rules:
B1(a(a(b(a(a(B(x))))))) → A1(a(a(b(a(a(a(B(x))))))))
B1(a(a(b(a(a(B(x))))))) → A1(a(b(a(a(a(B(x)))))))
A1(a(a(x))) → B1(b(x))
B1(a(a(b(x)))) → B1(a(a(a(b(x)))))
B1(a(a(a(a(A(x)))))) → A1(a(a(b(b(B(x))))))
B1(a(a(b(a(a(B(x))))))) → A1(a(a(B(x))))
B1(a(a(b(a(a(B(x))))))) → B1(a(a(a(b(a(a(a(B(x)))))))))
B1(a(a(b(a(a(B(x))))))) → B1(a(a(a(B(x)))))
A1(a(a(x))) → B1(b(b(x)))
B1(a(a(a(a(A(x)))))) → B1(a(a(a(b(b(B(x)))))))
B1(a(a(a(a(A(x)))))) → A1(a(a(b(B(x)))))
B1(a(a(b(a(a(B(x))))))) → A1(b(a(a(a(B(x))))))
B1(a(a(B(x)))) → B1(a(a(A(x))))
B1(a(a(b(x)))) → A1(a(a(b(x))))
A1(a(a(x))) → B1(x)
The TRS R consists of the following rules:
a(a(a(x))) → b(b(b(x)))
b(a(a(b(x)))) → x
b(a(a(b(x)))) → b(a(a(a(b(x)))))
a(a(A(x))) → B(x)
b(a(a(a(a(A(x)))))) → b(a(a(a(b(B(x))))))
b(a(a(b(a(a(B(x))))))) → b(a(a(a(b(a(a(a(B(x)))))))))
b(a(a(B(x)))) → b(b(b(b(B(x)))))
b(a(a(b(a(a(B(x))))))) → a(a(a(B(x))))
b(a(a(a(a(A(x)))))) → b(a(a(a(b(b(B(x)))))))
b(a(a(a(a(A(x)))))) → B(x)
b(a(a(B(x)))) → b(a(a(A(x))))
b(a(a(a(a(A(x)))))) → b(B(x))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule B1(a(a(b(a(a(B(x))))))) → A1(a(b(a(a(a(B(x))))))) at position [0] we obtained the following new rules:
B1(a(a(b(a(a(B(y0))))))) → A1(a(b(b(b(b(B(y0)))))))
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ QDPToSRSProof
↳ QTRS
↳ QTRS Reverse
↳ QTRS
↳ QTRS Reverse
↳ QTRS Reverse
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QTRS Reverse
↳ QTRS Reverse
Q DP problem:
The TRS P consists of the following rules:
B1(a(a(b(a(a(B(x))))))) → A1(a(a(b(a(a(a(B(x))))))))
A1(a(a(x))) → B1(b(x))
B1(a(a(b(x)))) → B1(a(a(a(b(x)))))
B1(a(a(a(a(A(x)))))) → A1(a(a(b(b(B(x))))))
B1(a(a(b(a(a(B(x))))))) → A1(a(a(B(x))))
B1(a(a(b(a(a(B(x))))))) → B1(a(a(a(B(x)))))
B1(a(a(b(a(a(B(x))))))) → B1(a(a(a(b(a(a(a(B(x)))))))))
A1(a(a(x))) → B1(b(b(x)))
B1(a(a(a(a(A(x)))))) → B1(a(a(a(b(b(B(x)))))))
B1(a(a(a(a(A(x)))))) → A1(a(a(b(B(x)))))
B1(a(a(b(a(a(B(x))))))) → A1(b(a(a(a(B(x))))))
B1(a(a(b(x)))) → A1(a(a(b(x))))
B1(a(a(B(x)))) → B1(a(a(A(x))))
B1(a(a(b(a(a(B(y0))))))) → A1(a(b(b(b(b(B(y0)))))))
A1(a(a(x))) → B1(x)
The TRS R consists of the following rules:
a(a(a(x))) → b(b(b(x)))
b(a(a(b(x)))) → x
b(a(a(b(x)))) → b(a(a(a(b(x)))))
a(a(A(x))) → B(x)
b(a(a(a(a(A(x)))))) → b(a(a(a(b(B(x))))))
b(a(a(b(a(a(B(x))))))) → b(a(a(a(b(a(a(a(B(x)))))))))
b(a(a(B(x)))) → b(b(b(b(B(x)))))
b(a(a(b(a(a(B(x))))))) → a(a(a(B(x))))
b(a(a(a(a(A(x)))))) → b(a(a(a(b(b(B(x)))))))
b(a(a(a(a(A(x)))))) → B(x)
b(a(a(B(x)))) → b(a(a(A(x))))
b(a(a(a(a(A(x)))))) → b(B(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 1 less node.
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ QDPToSRSProof
↳ QTRS
↳ QTRS Reverse
↳ QTRS
↳ QTRS Reverse
↳ QTRS Reverse
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QTRS Reverse
↳ QTRS Reverse
Q DP problem:
The TRS P consists of the following rules:
B1(a(a(b(a(a(B(x))))))) → A1(a(a(b(a(a(a(B(x))))))))
A1(a(a(x))) → B1(b(x))
B1(a(a(b(x)))) → B1(a(a(a(b(x)))))
B1(a(a(a(a(A(x)))))) → A1(a(a(b(b(B(x))))))
B1(a(a(b(a(a(B(x))))))) → A1(a(a(B(x))))
B1(a(a(b(a(a(B(x))))))) → B1(a(a(a(b(a(a(a(B(x)))))))))
B1(a(a(b(a(a(B(x))))))) → B1(a(a(a(B(x)))))
A1(a(a(x))) → B1(b(b(x)))
B1(a(a(a(a(A(x)))))) → B1(a(a(a(b(b(B(x)))))))
B1(a(a(a(a(A(x)))))) → A1(a(a(b(B(x)))))
B1(a(a(b(a(a(B(x))))))) → A1(b(a(a(a(B(x))))))
B1(a(a(B(x)))) → B1(a(a(A(x))))
B1(a(a(b(x)))) → A1(a(a(b(x))))
A1(a(a(x))) → B1(x)
The TRS R consists of the following rules:
a(a(a(x))) → b(b(b(x)))
b(a(a(b(x)))) → x
b(a(a(b(x)))) → b(a(a(a(b(x)))))
a(a(A(x))) → B(x)
b(a(a(a(a(A(x)))))) → b(a(a(a(b(B(x))))))
b(a(a(b(a(a(B(x))))))) → b(a(a(a(b(a(a(a(B(x)))))))))
b(a(a(B(x)))) → b(b(b(b(B(x)))))
b(a(a(b(a(a(B(x))))))) → a(a(a(B(x))))
b(a(a(a(a(A(x)))))) → b(a(a(a(b(b(B(x)))))))
b(a(a(a(a(A(x)))))) → B(x)
b(a(a(B(x)))) → b(a(a(A(x))))
b(a(a(a(a(A(x)))))) → b(B(x))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule B1(a(a(b(a(a(B(x))))))) → A1(b(a(a(a(B(x)))))) at position [0] we obtained the following new rules:
B1(a(a(b(a(a(B(y0))))))) → A1(b(b(b(b(B(y0))))))
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ QDPToSRSProof
↳ QTRS
↳ QTRS Reverse
↳ QTRS
↳ QTRS Reverse
↳ QTRS Reverse
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QTRS Reverse
↳ QTRS Reverse
Q DP problem:
The TRS P consists of the following rules:
B1(a(a(b(a(a(B(x))))))) → A1(a(a(b(a(a(a(B(x))))))))
A1(a(a(x))) → B1(b(x))
B1(a(a(b(x)))) → B1(a(a(a(b(x)))))
B1(a(a(a(a(A(x)))))) → A1(a(a(b(b(B(x))))))
B1(a(a(b(a(a(B(x))))))) → A1(a(a(B(x))))
B1(a(a(b(a(a(B(x))))))) → B1(a(a(a(B(x)))))
B1(a(a(b(a(a(B(x))))))) → B1(a(a(a(b(a(a(a(B(x)))))))))
A1(a(a(x))) → B1(b(b(x)))
B1(a(a(a(a(A(x)))))) → B1(a(a(a(b(b(B(x)))))))
B1(a(a(a(a(A(x)))))) → A1(a(a(b(B(x)))))
B1(a(a(b(a(a(B(y0))))))) → A1(b(b(b(b(B(y0))))))
B1(a(a(b(x)))) → A1(a(a(b(x))))
B1(a(a(B(x)))) → B1(a(a(A(x))))
A1(a(a(x))) → B1(x)
The TRS R consists of the following rules:
a(a(a(x))) → b(b(b(x)))
b(a(a(b(x)))) → x
b(a(a(b(x)))) → b(a(a(a(b(x)))))
a(a(A(x))) → B(x)
b(a(a(a(a(A(x)))))) → b(a(a(a(b(B(x))))))
b(a(a(b(a(a(B(x))))))) → b(a(a(a(b(a(a(a(B(x)))))))))
b(a(a(B(x)))) → b(b(b(b(B(x)))))
b(a(a(b(a(a(B(x))))))) → a(a(a(B(x))))
b(a(a(a(a(A(x)))))) → b(a(a(a(b(b(B(x)))))))
b(a(a(a(a(A(x)))))) → B(x)
b(a(a(B(x)))) → b(a(a(A(x))))
b(a(a(a(a(A(x)))))) → b(B(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 1 less node.
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ QDPToSRSProof
↳ QTRS
↳ QTRS Reverse
↳ QTRS
↳ QTRS Reverse
↳ QTRS Reverse
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QTRS Reverse
↳ QTRS Reverse
Q DP problem:
The TRS P consists of the following rules:
B1(a(a(b(a(a(B(x))))))) → A1(a(a(b(a(a(a(B(x))))))))
A1(a(a(x))) → B1(b(x))
B1(a(a(b(x)))) → B1(a(a(a(b(x)))))
B1(a(a(a(a(A(x)))))) → A1(a(a(b(b(B(x))))))
B1(a(a(b(a(a(B(x))))))) → A1(a(a(B(x))))
B1(a(a(b(a(a(B(x))))))) → B1(a(a(a(b(a(a(a(B(x)))))))))
B1(a(a(b(a(a(B(x))))))) → B1(a(a(a(B(x)))))
A1(a(a(x))) → B1(b(b(x)))
B1(a(a(a(a(A(x)))))) → B1(a(a(a(b(b(B(x)))))))
B1(a(a(a(a(A(x)))))) → A1(a(a(b(B(x)))))
B1(a(a(b(x)))) → A1(a(a(b(x))))
B1(a(a(B(x)))) → B1(a(a(A(x))))
A1(a(a(x))) → B1(x)
The TRS R consists of the following rules:
a(a(a(x))) → b(b(b(x)))
b(a(a(b(x)))) → x
b(a(a(b(x)))) → b(a(a(a(b(x)))))
a(a(A(x))) → B(x)
b(a(a(a(a(A(x)))))) → b(a(a(a(b(B(x))))))
b(a(a(b(a(a(B(x))))))) → b(a(a(a(b(a(a(a(B(x)))))))))
b(a(a(B(x)))) → b(b(b(b(B(x)))))
b(a(a(b(a(a(B(x))))))) → a(a(a(B(x))))
b(a(a(a(a(A(x)))))) → b(a(a(a(b(b(B(x)))))))
b(a(a(a(a(A(x)))))) → B(x)
b(a(a(B(x)))) → b(a(a(A(x))))
b(a(a(a(a(A(x)))))) → b(B(x))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule B1(a(a(b(a(a(B(x))))))) → B1(a(a(a(B(x))))) at position [0] we obtained the following new rules:
B1(a(a(b(a(a(B(y0))))))) → B1(b(b(b(B(y0)))))
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ QDPToSRSProof
↳ QTRS
↳ QTRS Reverse
↳ QTRS
↳ QTRS Reverse
↳ QTRS Reverse
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QTRS Reverse
↳ QTRS Reverse
Q DP problem:
The TRS P consists of the following rules:
B1(a(a(b(a(a(B(x))))))) → A1(a(a(b(a(a(a(B(x))))))))
A1(a(a(x))) → B1(b(x))
B1(a(a(b(x)))) → B1(a(a(a(b(x)))))
B1(a(a(b(a(a(B(y0))))))) → B1(b(b(b(B(y0)))))
B1(a(a(a(a(A(x)))))) → A1(a(a(b(b(B(x))))))
B1(a(a(b(a(a(B(x))))))) → A1(a(a(B(x))))
B1(a(a(b(a(a(B(x))))))) → B1(a(a(a(b(a(a(a(B(x)))))))))
A1(a(a(x))) → B1(b(b(x)))
B1(a(a(a(a(A(x)))))) → B1(a(a(a(b(b(B(x)))))))
B1(a(a(a(a(A(x)))))) → A1(a(a(b(B(x)))))
B1(a(a(B(x)))) → B1(a(a(A(x))))
B1(a(a(b(x)))) → A1(a(a(b(x))))
A1(a(a(x))) → B1(x)
The TRS R consists of the following rules:
a(a(a(x))) → b(b(b(x)))
b(a(a(b(x)))) → x
b(a(a(b(x)))) → b(a(a(a(b(x)))))
a(a(A(x))) → B(x)
b(a(a(a(a(A(x)))))) → b(a(a(a(b(B(x))))))
b(a(a(b(a(a(B(x))))))) → b(a(a(a(b(a(a(a(B(x)))))))))
b(a(a(B(x)))) → b(b(b(b(B(x)))))
b(a(a(b(a(a(B(x))))))) → a(a(a(B(x))))
b(a(a(a(a(A(x)))))) → b(a(a(a(b(b(B(x)))))))
b(a(a(a(a(A(x)))))) → B(x)
b(a(a(B(x)))) → b(a(a(A(x))))
b(a(a(a(a(A(x)))))) → b(B(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 1 less node.
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ QDPToSRSProof
↳ QTRS
↳ QTRS Reverse
↳ QTRS
↳ QTRS Reverse
↳ QTRS Reverse
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QTRS Reverse
↳ QTRS Reverse
Q DP problem:
The TRS P consists of the following rules:
B1(a(a(b(a(a(B(x))))))) → A1(a(a(b(a(a(a(B(x))))))))
B1(a(a(b(a(a(B(x))))))) → A1(a(a(B(x))))
B1(a(a(b(a(a(B(x))))))) → B1(a(a(a(b(a(a(a(B(x)))))))))
A1(a(a(x))) → B1(b(b(x)))
B1(a(a(a(a(A(x)))))) → B1(a(a(a(b(b(B(x)))))))
A1(a(a(x))) → B1(b(x))
B1(a(a(a(a(A(x)))))) → A1(a(a(b(B(x)))))
B1(a(a(b(x)))) → B1(a(a(a(b(x)))))
B1(a(a(B(x)))) → B1(a(a(A(x))))
B1(a(a(b(x)))) → A1(a(a(b(x))))
B1(a(a(a(a(A(x)))))) → A1(a(a(b(b(B(x))))))
A1(a(a(x))) → B1(x)
The TRS R consists of the following rules:
a(a(a(x))) → b(b(b(x)))
b(a(a(b(x)))) → x
b(a(a(b(x)))) → b(a(a(a(b(x)))))
a(a(A(x))) → B(x)
b(a(a(a(a(A(x)))))) → b(a(a(a(b(B(x))))))
b(a(a(b(a(a(B(x))))))) → b(a(a(a(b(a(a(a(B(x)))))))))
b(a(a(B(x)))) → b(b(b(b(B(x)))))
b(a(a(b(a(a(B(x))))))) → a(a(a(B(x))))
b(a(a(a(a(A(x)))))) → b(a(a(a(b(b(B(x)))))))
b(a(a(a(a(A(x)))))) → B(x)
b(a(a(B(x)))) → b(a(a(A(x))))
b(a(a(a(a(A(x)))))) → b(B(x))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule B1(a(a(a(a(A(x)))))) → B1(a(a(a(b(b(B(x))))))) at position [0] we obtained the following new rules:
B1(a(a(a(a(A(y0)))))) → B1(b(b(b(b(b(B(y0)))))))
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ QDPToSRSProof
↳ QTRS
↳ QTRS Reverse
↳ QTRS
↳ QTRS Reverse
↳ QTRS Reverse
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QTRS Reverse
↳ QTRS Reverse
Q DP problem:
The TRS P consists of the following rules:
B1(a(a(b(a(a(B(x))))))) → A1(a(a(b(a(a(a(B(x))))))))
A1(a(a(x))) → B1(b(x))
B1(a(a(a(a(A(y0)))))) → B1(b(b(b(b(b(B(y0)))))))
B1(a(a(b(x)))) → B1(a(a(a(b(x)))))
B1(a(a(a(a(A(x)))))) → A1(a(a(b(b(B(x))))))
B1(a(a(b(a(a(B(x))))))) → A1(a(a(B(x))))
B1(a(a(b(a(a(B(x))))))) → B1(a(a(a(b(a(a(a(B(x)))))))))
A1(a(a(x))) → B1(b(b(x)))
B1(a(a(a(a(A(x)))))) → A1(a(a(b(B(x)))))
B1(a(a(b(x)))) → A1(a(a(b(x))))
B1(a(a(B(x)))) → B1(a(a(A(x))))
A1(a(a(x))) → B1(x)
The TRS R consists of the following rules:
a(a(a(x))) → b(b(b(x)))
b(a(a(b(x)))) → x
b(a(a(b(x)))) → b(a(a(a(b(x)))))
a(a(A(x))) → B(x)
b(a(a(a(a(A(x)))))) → b(a(a(a(b(B(x))))))
b(a(a(b(a(a(B(x))))))) → b(a(a(a(b(a(a(a(B(x)))))))))
b(a(a(B(x)))) → b(b(b(b(B(x)))))
b(a(a(b(a(a(B(x))))))) → a(a(a(B(x))))
b(a(a(a(a(A(x)))))) → b(a(a(a(b(b(B(x)))))))
b(a(a(a(a(A(x)))))) → B(x)
b(a(a(B(x)))) → b(a(a(A(x))))
b(a(a(a(a(A(x)))))) → b(B(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 1 less node.
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ QDPToSRSProof
↳ QTRS
↳ QTRS Reverse
↳ QTRS
↳ QTRS Reverse
↳ QTRS Reverse
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QTRS Reverse
↳ QTRS Reverse
Q DP problem:
The TRS P consists of the following rules:
B1(a(a(b(a(a(B(x))))))) → A1(a(a(b(a(a(a(B(x))))))))
B1(a(a(b(a(a(B(x))))))) → A1(a(a(B(x))))
B1(a(a(b(a(a(B(x))))))) → B1(a(a(a(b(a(a(a(B(x)))))))))
A1(a(a(x))) → B1(b(b(x)))
A1(a(a(x))) → B1(b(x))
B1(a(a(a(a(A(x)))))) → A1(a(a(b(B(x)))))
B1(a(a(b(x)))) → B1(a(a(a(b(x)))))
B1(a(a(B(x)))) → B1(a(a(A(x))))
B1(a(a(b(x)))) → A1(a(a(b(x))))
B1(a(a(a(a(A(x)))))) → A1(a(a(b(b(B(x))))))
A1(a(a(x))) → B1(x)
The TRS R consists of the following rules:
a(a(a(x))) → b(b(b(x)))
b(a(a(b(x)))) → x
b(a(a(b(x)))) → b(a(a(a(b(x)))))
a(a(A(x))) → B(x)
b(a(a(a(a(A(x)))))) → b(a(a(a(b(B(x))))))
b(a(a(b(a(a(B(x))))))) → b(a(a(a(b(a(a(a(B(x)))))))))
b(a(a(B(x)))) → b(b(b(b(B(x)))))
b(a(a(b(a(a(B(x))))))) → a(a(a(B(x))))
b(a(a(a(a(A(x)))))) → b(a(a(a(b(b(B(x)))))))
b(a(a(a(a(A(x)))))) → B(x)
b(a(a(B(x)))) → b(a(a(A(x))))
b(a(a(a(a(A(x)))))) → b(B(x))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule B1(a(a(B(x)))) → B1(a(a(A(x)))) at position [0] we obtained the following new rules:
B1(a(a(B(x0)))) → B1(B(x0))
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ QDPToSRSProof
↳ QTRS
↳ QTRS Reverse
↳ QTRS
↳ QTRS Reverse
↳ QTRS Reverse
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QTRS Reverse
↳ QTRS Reverse
Q DP problem:
The TRS P consists of the following rules:
B1(a(a(B(x0)))) → B1(B(x0))
B1(a(a(b(a(a(B(x))))))) → A1(a(a(b(a(a(a(B(x))))))))
B1(a(a(b(a(a(B(x))))))) → A1(a(a(B(x))))
B1(a(a(b(a(a(B(x))))))) → B1(a(a(a(b(a(a(a(B(x)))))))))
A1(a(a(x))) → B1(b(b(x)))
A1(a(a(x))) → B1(b(x))
B1(a(a(a(a(A(x)))))) → A1(a(a(b(B(x)))))
B1(a(a(b(x)))) → B1(a(a(a(b(x)))))
B1(a(a(b(x)))) → A1(a(a(b(x))))
A1(a(a(x))) → B1(x)
B1(a(a(a(a(A(x)))))) → A1(a(a(b(b(B(x))))))
The TRS R consists of the following rules:
a(a(a(x))) → b(b(b(x)))
b(a(a(b(x)))) → x
b(a(a(b(x)))) → b(a(a(a(b(x)))))
a(a(A(x))) → B(x)
b(a(a(a(a(A(x)))))) → b(a(a(a(b(B(x))))))
b(a(a(b(a(a(B(x))))))) → b(a(a(a(b(a(a(a(B(x)))))))))
b(a(a(B(x)))) → b(b(b(b(B(x)))))
b(a(a(b(a(a(B(x))))))) → a(a(a(B(x))))
b(a(a(a(a(A(x)))))) → b(a(a(a(b(b(B(x)))))))
b(a(a(a(a(A(x)))))) → B(x)
b(a(a(B(x)))) → b(a(a(A(x))))
b(a(a(a(a(A(x)))))) → b(B(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 1 less node.
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ QDPToSRSProof
↳ QTRS
↳ QTRS Reverse
↳ QTRS
↳ QTRS Reverse
↳ QTRS Reverse
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QTRS Reverse
↳ QTRS Reverse
Q DP problem:
The TRS P consists of the following rules:
B1(a(a(b(a(a(B(x))))))) → A1(a(a(b(a(a(a(B(x))))))))
B1(a(a(b(a(a(B(x))))))) → A1(a(a(B(x))))
B1(a(a(b(a(a(B(x))))))) → B1(a(a(a(b(a(a(a(B(x)))))))))
A1(a(a(x))) → B1(b(b(x)))
A1(a(a(x))) → B1(b(x))
B1(a(a(a(a(A(x)))))) → A1(a(a(b(B(x)))))
B1(a(a(b(x)))) → B1(a(a(a(b(x)))))
B1(a(a(b(x)))) → A1(a(a(b(x))))
B1(a(a(a(a(A(x)))))) → A1(a(a(b(b(B(x))))))
A1(a(a(x))) → B1(x)
The TRS R consists of the following rules:
a(a(a(x))) → b(b(b(x)))
b(a(a(b(x)))) → x
b(a(a(b(x)))) → b(a(a(a(b(x)))))
a(a(A(x))) → B(x)
b(a(a(a(a(A(x)))))) → b(a(a(a(b(B(x))))))
b(a(a(b(a(a(B(x))))))) → b(a(a(a(b(a(a(a(B(x)))))))))
b(a(a(B(x)))) → b(b(b(b(B(x)))))
b(a(a(b(a(a(B(x))))))) → a(a(a(B(x))))
b(a(a(a(a(A(x)))))) → b(a(a(a(b(b(B(x)))))))
b(a(a(a(a(A(x)))))) → B(x)
b(a(a(B(x)))) → b(a(a(A(x))))
b(a(a(a(a(A(x)))))) → b(B(x))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule B1(a(a(b(x)))) → B1(a(a(a(b(x))))) at position [0] we obtained the following new rules:
B1(a(a(b(a(a(a(a(A(x0))))))))) → B1(a(a(a(B(x0)))))
B1(a(a(b(a(a(a(a(A(x0))))))))) → B1(a(a(a(b(a(a(a(b(b(B(x0)))))))))))
B1(a(a(b(a(a(a(a(A(x0))))))))) → B1(a(a(a(b(a(a(a(b(B(x0))))))))))
B1(a(a(b(a(a(a(a(A(x0))))))))) → B1(a(a(a(b(B(x0))))))
B1(a(a(b(a(a(b(x0))))))) → B1(a(a(a(x0))))
B1(a(a(b(a(a(b(a(a(B(x0)))))))))) → B1(a(a(a(b(a(a(a(b(a(a(a(B(x0)))))))))))))
B1(a(a(b(a(a(B(x0))))))) → B1(a(a(a(b(a(a(A(x0))))))))
B1(a(a(b(a(a(B(x0))))))) → B1(a(a(a(b(b(b(b(B(x0)))))))))
B1(a(a(b(a(a(b(a(a(B(x0)))))))))) → B1(a(a(a(a(a(a(B(x0))))))))
B1(a(a(b(a(a(b(x0))))))) → B1(a(a(a(b(a(a(a(b(x0)))))))))
B1(a(a(b(y0)))) → B1(b(b(b(b(y0)))))
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ QDPToSRSProof
↳ QTRS
↳ QTRS Reverse
↳ QTRS
↳ QTRS Reverse
↳ QTRS Reverse
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QTRS Reverse
↳ QTRS Reverse
Q DP problem:
The TRS P consists of the following rules:
B1(a(a(b(a(a(B(x))))))) → A1(a(a(b(a(a(a(B(x))))))))
B1(a(a(b(a(a(a(a(A(x0))))))))) → B1(a(a(a(b(a(a(a(b(B(x0))))))))))
B1(a(a(b(a(a(b(x0))))))) → B1(a(a(a(x0))))
A1(a(a(x))) → B1(b(x))
B1(a(a(b(a(a(B(x0))))))) → B1(a(a(a(b(b(b(b(B(x0)))))))))
B1(a(a(b(a(a(b(a(a(B(x0)))))))))) → B1(a(a(a(a(a(a(B(x0))))))))
B1(a(a(b(a(a(b(x0))))))) → B1(a(a(a(b(a(a(a(b(x0)))))))))
B1(a(a(b(y0)))) → B1(b(b(b(b(y0)))))
B1(a(a(a(a(A(x)))))) → A1(a(a(b(b(B(x))))))
B1(a(a(b(a(a(a(a(A(x0))))))))) → B1(a(a(a(B(x0)))))
B1(a(a(b(a(a(a(a(A(x0))))))))) → B1(a(a(a(b(a(a(a(b(b(B(x0)))))))))))
B1(a(a(b(a(a(B(x))))))) → A1(a(a(B(x))))
B1(a(a(b(a(a(a(a(A(x0))))))))) → B1(a(a(a(b(B(x0))))))
B1(a(a(b(a(a(B(x))))))) → B1(a(a(a(b(a(a(a(B(x)))))))))
A1(a(a(x))) → B1(b(b(x)))
B1(a(a(b(a(a(b(a(a(B(x0)))))))))) → B1(a(a(a(b(a(a(a(b(a(a(a(B(x0)))))))))))))
B1(a(a(a(a(A(x)))))) → A1(a(a(b(B(x)))))
B1(a(a(b(a(a(B(x0))))))) → B1(a(a(a(b(a(a(A(x0))))))))
B1(a(a(b(x)))) → A1(a(a(b(x))))
A1(a(a(x))) → B1(x)
The TRS R consists of the following rules:
a(a(a(x))) → b(b(b(x)))
b(a(a(b(x)))) → x
b(a(a(b(x)))) → b(a(a(a(b(x)))))
a(a(A(x))) → B(x)
b(a(a(a(a(A(x)))))) → b(a(a(a(b(B(x))))))
b(a(a(b(a(a(B(x))))))) → b(a(a(a(b(a(a(a(B(x)))))))))
b(a(a(B(x)))) → b(b(b(b(B(x)))))
b(a(a(b(a(a(B(x))))))) → a(a(a(B(x))))
b(a(a(a(a(A(x)))))) → b(a(a(a(b(b(B(x)))))))
b(a(a(a(a(A(x)))))) → B(x)
b(a(a(B(x)))) → b(a(a(A(x))))
b(a(a(a(a(A(x)))))) → b(B(x))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule A1(a(a(x))) → B1(b(b(x))) at position [0] we obtained the following new rules:
A1(a(a(a(a(b(a(a(B(x0))))))))) → B1(b(a(a(a(B(x0))))))
A1(a(a(a(a(a(a(A(x0)))))))) → B1(b(B(x0)))
A1(a(a(a(a(B(x0)))))) → B1(b(b(a(a(A(x0))))))
A1(a(a(a(a(a(a(A(x0)))))))) → B1(b(b(a(a(a(b(B(x0))))))))
A1(a(a(a(a(a(a(A(x0)))))))) → B1(b(b(B(x0))))
A1(a(a(a(a(b(x0)))))) → B1(b(x0))
A1(a(a(a(a(B(x0)))))) → B1(b(b(b(b(b(B(x0)))))))
A1(a(a(a(a(a(a(A(x0)))))))) → B1(b(b(a(a(a(b(b(B(x0)))))))))
A1(a(a(a(a(b(x0)))))) → B1(b(b(a(a(a(b(x0)))))))
A1(a(a(a(a(b(a(a(B(x0))))))))) → B1(b(b(a(a(a(b(a(a(a(B(x0)))))))))))
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ QDPToSRSProof
↳ QTRS
↳ QTRS Reverse
↳ QTRS
↳ QTRS Reverse
↳ QTRS Reverse
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QTRS Reverse
↳ QTRS Reverse
Q DP problem:
The TRS P consists of the following rules:
B1(a(a(b(a(a(b(x0))))))) → B1(a(a(a(x0))))
A1(a(a(a(a(B(x0)))))) → B1(b(b(a(a(A(x0))))))
A1(a(a(a(a(B(x0)))))) → B1(b(b(b(b(b(B(x0)))))))
B1(a(a(b(a(a(b(x0))))))) → B1(a(a(a(b(a(a(a(b(x0)))))))))
A1(a(a(a(a(b(a(a(B(x0))))))))) → B1(b(b(a(a(a(b(a(a(a(B(x0)))))))))))
A1(a(a(a(a(b(x0)))))) → B1(b(b(a(a(a(b(x0)))))))
B1(a(a(a(a(A(x)))))) → A1(a(a(b(b(B(x))))))
B1(a(a(b(a(a(a(a(A(x0))))))))) → B1(a(a(a(B(x0)))))
B1(a(a(b(a(a(a(a(A(x0))))))))) → B1(a(a(a(b(a(a(a(b(b(B(x0)))))))))))
B1(a(a(b(a(a(B(x))))))) → A1(a(a(B(x))))
B1(a(a(b(a(a(a(a(A(x0))))))))) → B1(a(a(a(b(B(x0))))))
B1(a(a(a(a(A(x)))))) → A1(a(a(b(B(x)))))
A1(a(a(a(a(a(a(A(x0)))))))) → B1(b(B(x0)))
A1(a(a(a(a(b(x0)))))) → B1(b(x0))
A1(a(a(a(a(a(a(A(x0)))))))) → B1(b(b(a(a(a(b(b(B(x0)))))))))
B1(a(a(b(x)))) → A1(a(a(b(x))))
A1(a(a(x))) → B1(x)
B1(a(a(b(a(a(B(x))))))) → A1(a(a(b(a(a(a(B(x))))))))
B1(a(a(b(a(a(a(a(A(x0))))))))) → B1(a(a(a(b(a(a(a(b(B(x0))))))))))
A1(a(a(a(a(b(a(a(B(x0))))))))) → B1(b(a(a(a(B(x0))))))
A1(a(a(x))) → B1(b(x))
B1(a(a(b(a(a(B(x0))))))) → B1(a(a(a(b(b(b(b(B(x0)))))))))
B1(a(a(b(a(a(b(a(a(B(x0)))))))))) → B1(a(a(a(a(a(a(B(x0))))))))
A1(a(a(a(a(a(a(A(x0)))))))) → B1(b(b(a(a(a(b(B(x0))))))))
B1(a(a(b(y0)))) → B1(b(b(b(b(y0)))))
B1(a(a(b(a(a(B(x))))))) → B1(a(a(a(b(a(a(a(B(x)))))))))
B1(a(a(b(a(a(b(a(a(B(x0)))))))))) → B1(a(a(a(b(a(a(a(b(a(a(a(B(x0)))))))))))))
B1(a(a(b(a(a(B(x0))))))) → B1(a(a(a(b(a(a(A(x0))))))))
A1(a(a(a(a(a(a(A(x0)))))))) → B1(b(b(B(x0))))
The TRS R consists of the following rules:
a(a(a(x))) → b(b(b(x)))
b(a(a(b(x)))) → x
b(a(a(b(x)))) → b(a(a(a(b(x)))))
a(a(A(x))) → B(x)
b(a(a(a(a(A(x)))))) → b(a(a(a(b(B(x))))))
b(a(a(b(a(a(B(x))))))) → b(a(a(a(b(a(a(a(B(x)))))))))
b(a(a(B(x)))) → b(b(b(b(B(x)))))
b(a(a(b(a(a(B(x))))))) → a(a(a(B(x))))
b(a(a(a(a(A(x)))))) → b(a(a(a(b(b(B(x)))))))
b(a(a(a(a(A(x)))))) → B(x)
b(a(a(B(x)))) → b(a(a(A(x))))
b(a(a(a(a(A(x)))))) → b(B(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 3 less nodes.
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ QDPToSRSProof
↳ QTRS
↳ QTRS Reverse
↳ QTRS
↳ QTRS Reverse
↳ QTRS Reverse
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QTRS Reverse
↳ QTRS Reverse
Q DP problem:
The TRS P consists of the following rules:
B1(a(a(b(a(a(b(x0))))))) → B1(a(a(a(x0))))
A1(a(a(a(a(B(x0)))))) → B1(b(b(a(a(A(x0))))))
B1(a(a(b(a(a(b(x0))))))) → B1(a(a(a(b(a(a(a(b(x0)))))))))
A1(a(a(a(a(b(a(a(B(x0))))))))) → B1(b(b(a(a(a(b(a(a(a(B(x0)))))))))))
B1(a(a(a(a(A(x)))))) → A1(a(a(b(b(B(x))))))
A1(a(a(a(a(b(x0)))))) → B1(b(b(a(a(a(b(x0)))))))
B1(a(a(b(a(a(a(a(A(x0))))))))) → B1(a(a(a(b(a(a(a(b(b(B(x0)))))))))))
B1(a(a(b(a(a(a(a(A(x0))))))))) → B1(a(a(a(B(x0)))))
B1(a(a(b(a(a(B(x))))))) → A1(a(a(B(x))))
B1(a(a(b(a(a(a(a(A(x0))))))))) → B1(a(a(a(b(B(x0))))))
B1(a(a(a(a(A(x)))))) → A1(a(a(b(B(x)))))
A1(a(a(a(a(b(x0)))))) → B1(b(x0))
B1(a(a(b(x)))) → A1(a(a(b(x))))
A1(a(a(a(a(a(a(A(x0)))))))) → B1(b(b(a(a(a(b(b(B(x0)))))))))
A1(a(a(x))) → B1(x)
B1(a(a(b(a(a(a(a(A(x0))))))))) → B1(a(a(a(b(a(a(a(b(B(x0))))))))))
B1(a(a(b(a(a(B(x))))))) → A1(a(a(b(a(a(a(B(x))))))))
A1(a(a(a(a(b(a(a(B(x0))))))))) → B1(b(a(a(a(B(x0))))))
A1(a(a(x))) → B1(b(x))
B1(a(a(b(a(a(B(x0))))))) → B1(a(a(a(b(b(b(b(B(x0)))))))))
B1(a(a(b(a(a(b(a(a(B(x0)))))))))) → B1(a(a(a(a(a(a(B(x0))))))))
A1(a(a(a(a(a(a(A(x0)))))))) → B1(b(b(a(a(a(b(B(x0))))))))
B1(a(a(b(y0)))) → B1(b(b(b(b(y0)))))
B1(a(a(b(a(a(B(x))))))) → B1(a(a(a(b(a(a(a(B(x)))))))))
B1(a(a(b(a(a(b(a(a(B(x0)))))))))) → B1(a(a(a(b(a(a(a(b(a(a(a(B(x0)))))))))))))
B1(a(a(b(a(a(B(x0))))))) → B1(a(a(a(b(a(a(A(x0))))))))
The TRS R consists of the following rules:
a(a(a(x))) → b(b(b(x)))
b(a(a(b(x)))) → x
b(a(a(b(x)))) → b(a(a(a(b(x)))))
a(a(A(x))) → B(x)
b(a(a(a(a(A(x)))))) → b(a(a(a(b(B(x))))))
b(a(a(b(a(a(B(x))))))) → b(a(a(a(b(a(a(a(B(x)))))))))
b(a(a(B(x)))) → b(b(b(b(B(x)))))
b(a(a(b(a(a(B(x))))))) → a(a(a(B(x))))
b(a(a(a(a(A(x)))))) → b(a(a(a(b(b(B(x)))))))
b(a(a(a(a(A(x)))))) → B(x)
b(a(a(B(x)))) → b(a(a(A(x))))
b(a(a(a(a(A(x)))))) → b(B(x))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule A1(a(a(x))) → B1(b(x)) at position [0] we obtained the following new rules:
A1(a(a(a(a(b(x0)))))) → B1(x0)
A1(a(a(a(a(b(x0)))))) → B1(b(a(a(a(b(x0))))))
A1(a(a(a(a(b(a(a(B(x0))))))))) → B1(b(a(a(a(b(a(a(a(B(x0))))))))))
A1(a(a(a(a(a(a(A(x0)))))))) → B1(b(B(x0)))
A1(a(a(a(a(B(x0)))))) → B1(b(a(a(A(x0)))))
A1(a(a(a(a(b(a(a(B(x0))))))))) → B1(a(a(a(B(x0)))))
A1(a(a(a(a(a(a(A(x0)))))))) → B1(b(a(a(a(b(b(B(x0))))))))
A1(a(a(a(a(a(a(A(x0)))))))) → B1(b(a(a(a(b(B(x0)))))))
A1(a(a(a(a(B(x0)))))) → B1(b(b(b(b(B(x0))))))
A1(a(a(a(a(a(a(A(x0)))))))) → B1(B(x0))
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ QDPToSRSProof
↳ QTRS
↳ QTRS Reverse
↳ QTRS
↳ QTRS Reverse
↳ QTRS Reverse
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QTRS Reverse
↳ QTRS Reverse
Q DP problem:
The TRS P consists of the following rules:
B1(a(a(b(a(a(b(x0))))))) → B1(a(a(a(x0))))
A1(a(a(a(a(b(a(a(B(x0))))))))) → B1(b(a(a(a(b(a(a(a(B(x0))))))))))
A1(a(a(a(a(B(x0)))))) → B1(b(b(a(a(A(x0))))))
A1(a(a(a(a(a(a(A(x0)))))))) → B1(b(a(a(a(b(B(x0)))))))
B1(a(a(b(a(a(b(x0))))))) → B1(a(a(a(b(a(a(a(b(x0)))))))))
A1(a(a(a(a(b(x0)))))) → B1(b(b(a(a(a(b(x0)))))))
B1(a(a(a(a(A(x)))))) → A1(a(a(b(b(B(x))))))
A1(a(a(a(a(b(a(a(B(x0))))))))) → B1(b(b(a(a(a(b(a(a(a(B(x0)))))))))))
B1(a(a(b(a(a(a(a(A(x0))))))))) → B1(a(a(a(B(x0)))))
B1(a(a(b(a(a(a(a(A(x0))))))))) → B1(a(a(a(b(a(a(a(b(b(B(x0)))))))))))
B1(a(a(b(a(a(B(x))))))) → A1(a(a(B(x))))
B1(a(a(b(a(a(a(a(A(x0))))))))) → B1(a(a(a(b(B(x0))))))
A1(a(a(a(a(b(x0)))))) → B1(b(a(a(a(b(x0))))))
B1(a(a(a(a(A(x)))))) → A1(a(a(b(B(x)))))
A1(a(a(a(a(a(a(A(x0)))))))) → B1(b(B(x0)))
A1(a(a(a(a(b(x0)))))) → B1(b(x0))
A1(a(a(a(a(B(x0)))))) → B1(b(b(b(b(B(x0))))))
A1(a(a(a(a(a(a(A(x0)))))))) → B1(b(b(a(a(a(b(b(B(x0)))))))))
B1(a(a(b(x)))) → A1(a(a(b(x))))
A1(a(a(x))) → B1(x)
B1(a(a(b(a(a(B(x))))))) → A1(a(a(b(a(a(a(B(x))))))))
B1(a(a(b(a(a(a(a(A(x0))))))))) → B1(a(a(a(b(a(a(a(b(B(x0))))))))))
A1(a(a(a(a(b(x0)))))) → B1(x0)
A1(a(a(a(a(b(a(a(B(x0))))))))) → B1(b(a(a(a(B(x0))))))
A1(a(a(a(a(b(a(a(B(x0))))))))) → B1(a(a(a(B(x0)))))
B1(a(a(b(a(a(B(x0))))))) → B1(a(a(a(b(b(b(b(B(x0)))))))))
A1(a(a(a(a(a(a(A(x0)))))))) → B1(b(a(a(a(b(b(B(x0))))))))
B1(a(a(b(a(a(b(a(a(B(x0)))))))))) → B1(a(a(a(a(a(a(B(x0))))))))
A1(a(a(a(a(a(a(A(x0)))))))) → B1(b(b(a(a(a(b(B(x0))))))))
B1(a(a(b(y0)))) → B1(b(b(b(b(y0)))))
B1(a(a(b(a(a(B(x))))))) → B1(a(a(a(b(a(a(a(B(x)))))))))
B1(a(a(b(a(a(b(a(a(B(x0)))))))))) → B1(a(a(a(b(a(a(a(b(a(a(a(B(x0)))))))))))))
A1(a(a(a(a(B(x0)))))) → B1(b(a(a(A(x0)))))
B1(a(a(b(a(a(B(x0))))))) → B1(a(a(a(b(a(a(A(x0))))))))
A1(a(a(a(a(a(a(A(x0)))))))) → B1(B(x0))
The TRS R consists of the following rules:
a(a(a(x))) → b(b(b(x)))
b(a(a(b(x)))) → x
b(a(a(b(x)))) → b(a(a(a(b(x)))))
a(a(A(x))) → B(x)
b(a(a(a(a(A(x)))))) → b(a(a(a(b(B(x))))))
b(a(a(b(a(a(B(x))))))) → b(a(a(a(b(a(a(a(B(x)))))))))
b(a(a(B(x)))) → b(b(b(b(B(x)))))
b(a(a(b(a(a(B(x))))))) → a(a(a(B(x))))
b(a(a(a(a(A(x)))))) → b(a(a(a(b(b(B(x)))))))
b(a(a(a(a(A(x)))))) → B(x)
b(a(a(B(x)))) → b(a(a(A(x))))
b(a(a(a(a(A(x)))))) → b(B(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 3 less nodes.
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ QDPToSRSProof
↳ QTRS
↳ QTRS Reverse
↳ QTRS
↳ QTRS Reverse
↳ QTRS Reverse
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QTRS Reverse
↳ QTRS Reverse
Q DP problem:
The TRS P consists of the following rules:
B1(a(a(b(a(a(b(x0))))))) → B1(a(a(a(x0))))
A1(a(a(a(a(b(a(a(B(x0))))))))) → B1(b(a(a(a(b(a(a(a(B(x0))))))))))
A1(a(a(a(a(B(x0)))))) → B1(b(b(a(a(A(x0))))))
A1(a(a(a(a(a(a(A(x0)))))))) → B1(b(a(a(a(b(B(x0)))))))
B1(a(a(b(a(a(b(x0))))))) → B1(a(a(a(b(a(a(a(b(x0)))))))))
B1(a(a(a(a(A(x)))))) → A1(a(a(b(b(B(x))))))
A1(a(a(a(a(b(x0)))))) → B1(b(b(a(a(a(b(x0)))))))
A1(a(a(a(a(b(a(a(B(x0))))))))) → B1(b(b(a(a(a(b(a(a(a(B(x0)))))))))))
B1(a(a(b(a(a(a(a(A(x0))))))))) → B1(a(a(a(b(a(a(a(b(b(B(x0)))))))))))
B1(a(a(b(a(a(a(a(A(x0))))))))) → B1(a(a(a(B(x0)))))
B1(a(a(b(a(a(B(x))))))) → A1(a(a(B(x))))
B1(a(a(b(a(a(a(a(A(x0))))))))) → B1(a(a(a(b(B(x0))))))
A1(a(a(a(a(b(x0)))))) → B1(b(a(a(a(b(x0))))))
B1(a(a(a(a(A(x)))))) → A1(a(a(b(B(x)))))
A1(a(a(a(a(b(x0)))))) → B1(b(x0))
B1(a(a(b(x)))) → A1(a(a(b(x))))
A1(a(a(a(a(a(a(A(x0)))))))) → B1(b(b(a(a(a(b(b(B(x0)))))))))
A1(a(a(x))) → B1(x)
B1(a(a(b(a(a(a(a(A(x0))))))))) → B1(a(a(a(b(a(a(a(b(B(x0))))))))))
B1(a(a(b(a(a(B(x))))))) → A1(a(a(b(a(a(a(B(x))))))))
A1(a(a(a(a(b(x0)))))) → B1(x0)
A1(a(a(a(a(b(a(a(B(x0))))))))) → B1(b(a(a(a(B(x0))))))
A1(a(a(a(a(b(a(a(B(x0))))))))) → B1(a(a(a(B(x0)))))
B1(a(a(b(a(a(B(x0))))))) → B1(a(a(a(b(b(b(b(B(x0)))))))))
A1(a(a(a(a(a(a(A(x0)))))))) → B1(b(a(a(a(b(b(B(x0))))))))
B1(a(a(b(a(a(b(a(a(B(x0)))))))))) → B1(a(a(a(a(a(a(B(x0))))))))
A1(a(a(a(a(a(a(A(x0)))))))) → B1(b(b(a(a(a(b(B(x0))))))))
B1(a(a(b(y0)))) → B1(b(b(b(b(y0)))))
B1(a(a(b(a(a(B(x))))))) → B1(a(a(a(b(a(a(a(B(x)))))))))
B1(a(a(b(a(a(b(a(a(B(x0)))))))))) → B1(a(a(a(b(a(a(a(b(a(a(a(B(x0)))))))))))))
A1(a(a(a(a(B(x0)))))) → B1(b(a(a(A(x0)))))
B1(a(a(b(a(a(B(x0))))))) → B1(a(a(a(b(a(a(A(x0))))))))
The TRS R consists of the following rules:
a(a(a(x))) → b(b(b(x)))
b(a(a(b(x)))) → x
b(a(a(b(x)))) → b(a(a(a(b(x)))))
a(a(A(x))) → B(x)
b(a(a(a(a(A(x)))))) → b(a(a(a(b(B(x))))))
b(a(a(b(a(a(B(x))))))) → b(a(a(a(b(a(a(a(B(x)))))))))
b(a(a(B(x)))) → b(b(b(b(B(x)))))
b(a(a(b(a(a(B(x))))))) → a(a(a(B(x))))
b(a(a(a(a(A(x)))))) → b(a(a(a(b(b(B(x)))))))
b(a(a(a(a(A(x)))))) → B(x)
b(a(a(B(x)))) → b(a(a(A(x))))
b(a(a(a(a(A(x)))))) → b(B(x))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule A1(a(a(a(a(b(a(a(B(x0))))))))) → B1(b(a(a(a(B(x0)))))) at position [0] we obtained the following new rules:
A1(a(a(a(a(b(a(a(B(y0))))))))) → B1(b(b(b(b(B(y0))))))
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ QDPToSRSProof
↳ QTRS
↳ QTRS Reverse
↳ QTRS
↳ QTRS Reverse
↳ QTRS Reverse
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QTRS Reverse
↳ QTRS Reverse
Q DP problem:
The TRS P consists of the following rules:
B1(a(a(b(a(a(b(x0))))))) → B1(a(a(a(x0))))
A1(a(a(a(a(b(a(a(B(x0))))))))) → B1(b(a(a(a(b(a(a(a(B(x0))))))))))
A1(a(a(a(a(B(x0)))))) → B1(b(b(a(a(A(x0))))))
A1(a(a(a(a(a(a(A(x0)))))))) → B1(b(a(a(a(b(B(x0)))))))
B1(a(a(b(a(a(b(x0))))))) → B1(a(a(a(b(a(a(a(b(x0)))))))))
A1(a(a(a(a(b(a(a(B(x0))))))))) → B1(b(b(a(a(a(b(a(a(a(B(x0)))))))))))
A1(a(a(a(a(b(x0)))))) → B1(b(b(a(a(a(b(x0)))))))
B1(a(a(a(a(A(x)))))) → A1(a(a(b(b(B(x))))))
B1(a(a(b(a(a(a(a(A(x0))))))))) → B1(a(a(a(B(x0)))))
B1(a(a(b(a(a(a(a(A(x0))))))))) → B1(a(a(a(b(a(a(a(b(b(B(x0)))))))))))
B1(a(a(b(a(a(B(x))))))) → A1(a(a(B(x))))
B1(a(a(b(a(a(a(a(A(x0))))))))) → B1(a(a(a(b(B(x0))))))
A1(a(a(a(a(b(x0)))))) → B1(b(a(a(a(b(x0))))))
B1(a(a(a(a(A(x)))))) → A1(a(a(b(B(x)))))
A1(a(a(a(a(b(x0)))))) → B1(b(x0))
A1(a(a(a(a(a(a(A(x0)))))))) → B1(b(b(a(a(a(b(b(B(x0)))))))))
B1(a(a(b(x)))) → A1(a(a(b(x))))
A1(a(a(x))) → B1(x)
B1(a(a(b(a(a(B(x))))))) → A1(a(a(b(a(a(a(B(x))))))))
B1(a(a(b(a(a(a(a(A(x0))))))))) → B1(a(a(a(b(a(a(a(b(B(x0))))))))))
A1(a(a(a(a(b(x0)))))) → B1(x0)
B1(a(a(b(a(a(B(x0))))))) → B1(a(a(a(b(b(b(b(B(x0)))))))))
A1(a(a(a(a(b(a(a(B(x0))))))))) → B1(a(a(a(B(x0)))))
B1(a(a(b(a(a(b(a(a(B(x0)))))))))) → B1(a(a(a(a(a(a(B(x0))))))))
A1(a(a(a(a(a(a(A(x0)))))))) → B1(b(a(a(a(b(b(B(x0))))))))
A1(a(a(a(a(a(a(A(x0)))))))) → B1(b(b(a(a(a(b(B(x0))))))))
B1(a(a(b(y0)))) → B1(b(b(b(b(y0)))))
B1(a(a(b(a(a(B(x))))))) → B1(a(a(a(b(a(a(a(B(x)))))))))
B1(a(a(b(a(a(b(a(a(B(x0)))))))))) → B1(a(a(a(b(a(a(a(b(a(a(a(B(x0)))))))))))))
B1(a(a(b(a(a(B(x0))))))) → B1(a(a(a(b(a(a(A(x0))))))))
A1(a(a(a(a(B(x0)))))) → B1(b(a(a(A(x0)))))
A1(a(a(a(a(b(a(a(B(y0))))))))) → B1(b(b(b(b(B(y0))))))
The TRS R consists of the following rules:
a(a(a(x))) → b(b(b(x)))
b(a(a(b(x)))) → x
b(a(a(b(x)))) → b(a(a(a(b(x)))))
a(a(A(x))) → B(x)
b(a(a(a(a(A(x)))))) → b(a(a(a(b(B(x))))))
b(a(a(b(a(a(B(x))))))) → b(a(a(a(b(a(a(a(B(x)))))))))
b(a(a(B(x)))) → b(b(b(b(B(x)))))
b(a(a(b(a(a(B(x))))))) → a(a(a(B(x))))
b(a(a(a(a(A(x)))))) → b(a(a(a(b(b(B(x)))))))
b(a(a(a(a(A(x)))))) → B(x)
b(a(a(B(x)))) → b(a(a(A(x))))
b(a(a(a(a(A(x)))))) → b(B(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 1 less node.
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ QDPToSRSProof
↳ QTRS
↳ QTRS Reverse
↳ QTRS
↳ QTRS Reverse
↳ QTRS Reverse
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QTRS Reverse
↳ QTRS Reverse
Q DP problem:
The TRS P consists of the following rules:
B1(a(a(b(a(a(b(x0))))))) → B1(a(a(a(x0))))
A1(a(a(a(a(b(a(a(B(x0))))))))) → B1(b(a(a(a(b(a(a(a(B(x0))))))))))
A1(a(a(a(a(B(x0)))))) → B1(b(b(a(a(A(x0))))))
A1(a(a(a(a(a(a(A(x0)))))))) → B1(b(a(a(a(b(B(x0)))))))
B1(a(a(b(a(a(b(x0))))))) → B1(a(a(a(b(a(a(a(b(x0)))))))))
B1(a(a(a(a(A(x)))))) → A1(a(a(b(b(B(x))))))
A1(a(a(a(a(b(x0)))))) → B1(b(b(a(a(a(b(x0)))))))
A1(a(a(a(a(b(a(a(B(x0))))))))) → B1(b(b(a(a(a(b(a(a(a(B(x0)))))))))))
B1(a(a(b(a(a(a(a(A(x0))))))))) → B1(a(a(a(b(a(a(a(b(b(B(x0)))))))))))
B1(a(a(b(a(a(a(a(A(x0))))))))) → B1(a(a(a(B(x0)))))
B1(a(a(b(a(a(B(x))))))) → A1(a(a(B(x))))
B1(a(a(b(a(a(a(a(A(x0))))))))) → B1(a(a(a(b(B(x0))))))
A1(a(a(a(a(b(x0)))))) → B1(b(a(a(a(b(x0))))))
B1(a(a(a(a(A(x)))))) → A1(a(a(b(B(x)))))
A1(a(a(a(a(b(x0)))))) → B1(b(x0))
B1(a(a(b(x)))) → A1(a(a(b(x))))
A1(a(a(a(a(a(a(A(x0)))))))) → B1(b(b(a(a(a(b(b(B(x0)))))))))
A1(a(a(x))) → B1(x)
B1(a(a(b(a(a(a(a(A(x0))))))))) → B1(a(a(a(b(a(a(a(b(B(x0))))))))))
B1(a(a(b(a(a(B(x))))))) → A1(a(a(b(a(a(a(B(x))))))))
A1(a(a(a(a(b(x0)))))) → B1(x0)
A1(a(a(a(a(b(a(a(B(x0))))))))) → B1(a(a(a(B(x0)))))
B1(a(a(b(a(a(B(x0))))))) → B1(a(a(a(b(b(b(b(B(x0)))))))))
A1(a(a(a(a(a(a(A(x0)))))))) → B1(b(a(a(a(b(b(B(x0))))))))
B1(a(a(b(a(a(b(a(a(B(x0)))))))))) → B1(a(a(a(a(a(a(B(x0))))))))
A1(a(a(a(a(a(a(A(x0)))))))) → B1(b(b(a(a(a(b(B(x0))))))))
B1(a(a(b(y0)))) → B1(b(b(b(b(y0)))))
B1(a(a(b(a(a(B(x))))))) → B1(a(a(a(b(a(a(a(B(x)))))))))
B1(a(a(b(a(a(b(a(a(B(x0)))))))))) → B1(a(a(a(b(a(a(a(b(a(a(a(B(x0)))))))))))))
A1(a(a(a(a(B(x0)))))) → B1(b(a(a(A(x0)))))
B1(a(a(b(a(a(B(x0))))))) → B1(a(a(a(b(a(a(A(x0))))))))
The TRS R consists of the following rules:
a(a(a(x))) → b(b(b(x)))
b(a(a(b(x)))) → x
b(a(a(b(x)))) → b(a(a(a(b(x)))))
a(a(A(x))) → B(x)
b(a(a(a(a(A(x)))))) → b(a(a(a(b(B(x))))))
b(a(a(b(a(a(B(x))))))) → b(a(a(a(b(a(a(a(B(x)))))))))
b(a(a(B(x)))) → b(b(b(b(B(x)))))
b(a(a(b(a(a(B(x))))))) → a(a(a(B(x))))
b(a(a(a(a(A(x)))))) → b(a(a(a(b(b(B(x)))))))
b(a(a(a(a(A(x)))))) → B(x)
b(a(a(B(x)))) → b(a(a(A(x))))
b(a(a(a(a(A(x)))))) → b(B(x))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule B1(a(a(b(a(a(B(x))))))) → B1(a(a(a(b(a(a(a(B(x))))))))) at position [0] we obtained the following new rules:
B1(a(a(b(a(a(B(y0))))))) → B1(a(a(a(b(b(b(b(B(y0)))))))))
B1(a(a(b(a(a(B(y0))))))) → B1(b(b(b(b(a(a(a(B(y0)))))))))
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ QDPToSRSProof
↳ QTRS
↳ QTRS Reverse
↳ QTRS
↳ QTRS Reverse
↳ QTRS Reverse
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QTRS Reverse
↳ QTRS Reverse
Q DP problem:
The TRS P consists of the following rules:
B1(a(a(b(a(a(b(x0))))))) → B1(a(a(a(x0))))
A1(a(a(a(a(b(a(a(B(x0))))))))) → B1(b(a(a(a(b(a(a(a(B(x0))))))))))
A1(a(a(a(a(B(x0)))))) → B1(b(b(a(a(A(x0))))))
A1(a(a(a(a(a(a(A(x0)))))))) → B1(b(a(a(a(b(B(x0)))))))
B1(a(a(b(a(a(b(x0))))))) → B1(a(a(a(b(a(a(a(b(x0)))))))))
A1(a(a(a(a(b(a(a(B(x0))))))))) → B1(b(b(a(a(a(b(a(a(a(B(x0)))))))))))
A1(a(a(a(a(b(x0)))))) → B1(b(b(a(a(a(b(x0)))))))
B1(a(a(a(a(A(x)))))) → A1(a(a(b(b(B(x))))))
B1(a(a(b(a(a(a(a(A(x0))))))))) → B1(a(a(a(B(x0)))))
B1(a(a(b(a(a(a(a(A(x0))))))))) → B1(a(a(a(b(a(a(a(b(b(B(x0)))))))))))
B1(a(a(b(a(a(B(x))))))) → A1(a(a(B(x))))
B1(a(a(b(a(a(B(y0))))))) → B1(b(b(b(b(a(a(a(B(y0)))))))))
B1(a(a(b(a(a(a(a(A(x0))))))))) → B1(a(a(a(b(B(x0))))))
A1(a(a(a(a(b(x0)))))) → B1(b(a(a(a(b(x0))))))
B1(a(a(a(a(A(x)))))) → A1(a(a(b(B(x)))))
A1(a(a(a(a(b(x0)))))) → B1(b(x0))
A1(a(a(a(a(a(a(A(x0)))))))) → B1(b(b(a(a(a(b(b(B(x0)))))))))
B1(a(a(b(x)))) → A1(a(a(b(x))))
A1(a(a(x))) → B1(x)
B1(a(a(b(a(a(B(x))))))) → A1(a(a(b(a(a(a(B(x))))))))
B1(a(a(b(a(a(a(a(A(x0))))))))) → B1(a(a(a(b(a(a(a(b(B(x0))))))))))
A1(a(a(a(a(b(x0)))))) → B1(x0)
B1(a(a(b(a(a(B(x0))))))) → B1(a(a(a(b(b(b(b(B(x0)))))))))
A1(a(a(a(a(b(a(a(B(x0))))))))) → B1(a(a(a(B(x0)))))
B1(a(a(b(a(a(b(a(a(B(x0)))))))))) → B1(a(a(a(a(a(a(B(x0))))))))
A1(a(a(a(a(a(a(A(x0)))))))) → B1(b(a(a(a(b(b(B(x0))))))))
A1(a(a(a(a(a(a(A(x0)))))))) → B1(b(b(a(a(a(b(B(x0))))))))
B1(a(a(b(y0)))) → B1(b(b(b(b(y0)))))
B1(a(a(b(a(a(b(a(a(B(x0)))))))))) → B1(a(a(a(b(a(a(a(b(a(a(a(B(x0)))))))))))))
B1(a(a(b(a(a(B(x0))))))) → B1(a(a(a(b(a(a(A(x0))))))))
A1(a(a(a(a(B(x0)))))) → B1(b(a(a(A(x0)))))
The TRS R consists of the following rules:
a(a(a(x))) → b(b(b(x)))
b(a(a(b(x)))) → x
b(a(a(b(x)))) → b(a(a(a(b(x)))))
a(a(A(x))) → B(x)
b(a(a(a(a(A(x)))))) → b(a(a(a(b(B(x))))))
b(a(a(b(a(a(B(x))))))) → b(a(a(a(b(a(a(a(B(x)))))))))
b(a(a(B(x)))) → b(b(b(b(B(x)))))
b(a(a(b(a(a(B(x))))))) → a(a(a(B(x))))
b(a(a(a(a(A(x)))))) → b(a(a(a(b(b(B(x)))))))
b(a(a(a(a(A(x)))))) → B(x)
b(a(a(B(x)))) → b(a(a(A(x))))
b(a(a(a(a(A(x)))))) → b(B(x))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule A1(a(a(a(a(B(x0)))))) → B1(b(b(a(a(A(x0)))))) at position [0] we obtained the following new rules:
A1(a(a(a(a(B(x0)))))) → B1(b(b(B(x0))))
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ QDPToSRSProof
↳ QTRS
↳ QTRS Reverse
↳ QTRS
↳ QTRS Reverse
↳ QTRS Reverse
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QTRS Reverse
↳ QTRS Reverse
Q DP problem:
The TRS P consists of the following rules:
A1(a(a(a(a(B(x0)))))) → B1(b(b(B(x0))))
B1(a(a(b(a(a(b(x0))))))) → B1(a(a(a(x0))))
A1(a(a(a(a(b(a(a(B(x0))))))))) → B1(b(a(a(a(b(a(a(a(B(x0))))))))))
A1(a(a(a(a(a(a(A(x0)))))))) → B1(b(a(a(a(b(B(x0)))))))
B1(a(a(b(a(a(b(x0))))))) → B1(a(a(a(b(a(a(a(b(x0)))))))))
B1(a(a(a(a(A(x)))))) → A1(a(a(b(b(B(x))))))
A1(a(a(a(a(b(x0)))))) → B1(b(b(a(a(a(b(x0)))))))
A1(a(a(a(a(b(a(a(B(x0))))))))) → B1(b(b(a(a(a(b(a(a(a(B(x0)))))))))))
B1(a(a(b(a(a(a(a(A(x0))))))))) → B1(a(a(a(b(a(a(a(b(b(B(x0)))))))))))
B1(a(a(b(a(a(a(a(A(x0))))))))) → B1(a(a(a(B(x0)))))
B1(a(a(b(a(a(B(x))))))) → A1(a(a(B(x))))
B1(a(a(b(a(a(B(y0))))))) → B1(b(b(b(b(a(a(a(B(y0)))))))))
B1(a(a(b(a(a(a(a(A(x0))))))))) → B1(a(a(a(b(B(x0))))))
A1(a(a(a(a(b(x0)))))) → B1(b(a(a(a(b(x0))))))
B1(a(a(a(a(A(x)))))) → A1(a(a(b(B(x)))))
A1(a(a(a(a(b(x0)))))) → B1(b(x0))
B1(a(a(b(x)))) → A1(a(a(b(x))))
A1(a(a(a(a(a(a(A(x0)))))))) → B1(b(b(a(a(a(b(b(B(x0)))))))))
A1(a(a(x))) → B1(x)
B1(a(a(b(a(a(a(a(A(x0))))))))) → B1(a(a(a(b(a(a(a(b(B(x0))))))))))
B1(a(a(b(a(a(B(x))))))) → A1(a(a(b(a(a(a(B(x))))))))
A1(a(a(a(a(b(x0)))))) → B1(x0)
A1(a(a(a(a(b(a(a(B(x0))))))))) → B1(a(a(a(B(x0)))))
B1(a(a(b(a(a(B(x0))))))) → B1(a(a(a(b(b(b(b(B(x0)))))))))
A1(a(a(a(a(a(a(A(x0)))))))) → B1(b(a(a(a(b(b(B(x0))))))))
B1(a(a(b(a(a(b(a(a(B(x0)))))))))) → B1(a(a(a(a(a(a(B(x0))))))))
A1(a(a(a(a(a(a(A(x0)))))))) → B1(b(b(a(a(a(b(B(x0))))))))
B1(a(a(b(y0)))) → B1(b(b(b(b(y0)))))
B1(a(a(b(a(a(b(a(a(B(x0)))))))))) → B1(a(a(a(b(a(a(a(b(a(a(a(B(x0)))))))))))))
A1(a(a(a(a(B(x0)))))) → B1(b(a(a(A(x0)))))
B1(a(a(b(a(a(B(x0))))))) → B1(a(a(a(b(a(a(A(x0))))))))
The TRS R consists of the following rules:
a(a(a(x))) → b(b(b(x)))
b(a(a(b(x)))) → x
b(a(a(b(x)))) → b(a(a(a(b(x)))))
a(a(A(x))) → B(x)
b(a(a(a(a(A(x)))))) → b(a(a(a(b(B(x))))))
b(a(a(b(a(a(B(x))))))) → b(a(a(a(b(a(a(a(B(x)))))))))
b(a(a(B(x)))) → b(b(b(b(B(x)))))
b(a(a(b(a(a(B(x))))))) → a(a(a(B(x))))
b(a(a(a(a(A(x)))))) → b(a(a(a(b(b(B(x)))))))
b(a(a(a(a(A(x)))))) → B(x)
b(a(a(B(x)))) → b(a(a(A(x))))
b(a(a(a(a(A(x)))))) → b(B(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 1 less node.
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ QDPToSRSProof
↳ QTRS
↳ QTRS Reverse
↳ QTRS
↳ QTRS Reverse
↳ QTRS Reverse
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QTRS Reverse
↳ QTRS Reverse
Q DP problem:
The TRS P consists of the following rules:
B1(a(a(b(a(a(b(x0))))))) → B1(a(a(a(x0))))
A1(a(a(a(a(b(a(a(B(x0))))))))) → B1(b(a(a(a(b(a(a(a(B(x0))))))))))
A1(a(a(a(a(a(a(A(x0)))))))) → B1(b(a(a(a(b(B(x0)))))))
B1(a(a(b(a(a(b(x0))))))) → B1(a(a(a(b(a(a(a(b(x0)))))))))
B1(a(a(a(a(A(x)))))) → A1(a(a(b(b(B(x))))))
A1(a(a(a(a(b(x0)))))) → B1(b(b(a(a(a(b(x0)))))))
A1(a(a(a(a(b(a(a(B(x0))))))))) → B1(b(b(a(a(a(b(a(a(a(B(x0)))))))))))
B1(a(a(b(a(a(a(a(A(x0))))))))) → B1(a(a(a(b(a(a(a(b(b(B(x0)))))))))))
B1(a(a(b(a(a(a(a(A(x0))))))))) → B1(a(a(a(B(x0)))))
B1(a(a(b(a(a(B(x))))))) → A1(a(a(B(x))))
B1(a(a(b(a(a(B(y0))))))) → B1(b(b(b(b(a(a(a(B(y0)))))))))
B1(a(a(b(a(a(a(a(A(x0))))))))) → B1(a(a(a(b(B(x0))))))
A1(a(a(a(a(b(x0)))))) → B1(b(a(a(a(b(x0))))))
B1(a(a(a(a(A(x)))))) → A1(a(a(b(B(x)))))
A1(a(a(a(a(b(x0)))))) → B1(b(x0))
B1(a(a(b(x)))) → A1(a(a(b(x))))
A1(a(a(a(a(a(a(A(x0)))))))) → B1(b(b(a(a(a(b(b(B(x0)))))))))
A1(a(a(x))) → B1(x)
B1(a(a(b(a(a(a(a(A(x0))))))))) → B1(a(a(a(b(a(a(a(b(B(x0))))))))))
B1(a(a(b(a(a(B(x))))))) → A1(a(a(b(a(a(a(B(x))))))))
A1(a(a(a(a(b(x0)))))) → B1(x0)
A1(a(a(a(a(b(a(a(B(x0))))))))) → B1(a(a(a(B(x0)))))
B1(a(a(b(a(a(B(x0))))))) → B1(a(a(a(b(b(b(b(B(x0)))))))))
A1(a(a(a(a(a(a(A(x0)))))))) → B1(b(a(a(a(b(b(B(x0))))))))
B1(a(a(b(a(a(b(a(a(B(x0)))))))))) → B1(a(a(a(a(a(a(B(x0))))))))
A1(a(a(a(a(a(a(A(x0)))))))) → B1(b(b(a(a(a(b(B(x0))))))))
B1(a(a(b(y0)))) → B1(b(b(b(b(y0)))))
B1(a(a(b(a(a(b(a(a(B(x0)))))))))) → B1(a(a(a(b(a(a(a(b(a(a(a(B(x0)))))))))))))
A1(a(a(a(a(B(x0)))))) → B1(b(a(a(A(x0)))))
B1(a(a(b(a(a(B(x0))))))) → B1(a(a(a(b(a(a(A(x0))))))))
The TRS R consists of the following rules:
a(a(a(x))) → b(b(b(x)))
b(a(a(b(x)))) → x
b(a(a(b(x)))) → b(a(a(a(b(x)))))
a(a(A(x))) → B(x)
b(a(a(a(a(A(x)))))) → b(a(a(a(b(B(x))))))
b(a(a(b(a(a(B(x))))))) → b(a(a(a(b(a(a(a(B(x)))))))))
b(a(a(B(x)))) → b(b(b(b(B(x)))))
b(a(a(b(a(a(B(x))))))) → a(a(a(B(x))))
b(a(a(a(a(A(x)))))) → b(a(a(a(b(b(B(x)))))))
b(a(a(a(a(A(x)))))) → B(x)
b(a(a(B(x)))) → b(a(a(A(x))))
b(a(a(a(a(A(x)))))) → b(B(x))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule A1(a(a(a(a(a(a(A(x0)))))))) → B1(b(b(a(a(a(b(B(x0)))))))) at position [0] we obtained the following new rules:
A1(a(a(a(a(a(a(A(y0)))))))) → B1(b(b(b(b(b(b(B(y0))))))))
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ QDPToSRSProof
↳ QTRS
↳ QTRS Reverse
↳ QTRS
↳ QTRS Reverse
↳ QTRS Reverse
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QTRS Reverse
↳ QTRS Reverse
Q DP problem:
The TRS P consists of the following rules:
B1(a(a(b(a(a(b(x0))))))) → B1(a(a(a(x0))))
A1(a(a(a(a(b(a(a(B(x0))))))))) → B1(b(a(a(a(b(a(a(a(B(x0))))))))))
A1(a(a(a(a(a(a(A(x0)))))))) → B1(b(a(a(a(b(B(x0)))))))
B1(a(a(b(a(a(b(x0))))))) → B1(a(a(a(b(a(a(a(b(x0)))))))))
A1(a(a(a(a(b(a(a(B(x0))))))))) → B1(b(b(a(a(a(b(a(a(a(B(x0)))))))))))
A1(a(a(a(a(b(x0)))))) → B1(b(b(a(a(a(b(x0)))))))
B1(a(a(a(a(A(x)))))) → A1(a(a(b(b(B(x))))))
B1(a(a(b(a(a(a(a(A(x0))))))))) → B1(a(a(a(B(x0)))))
B1(a(a(b(a(a(a(a(A(x0))))))))) → B1(a(a(a(b(a(a(a(b(b(B(x0)))))))))))
B1(a(a(b(a(a(B(x))))))) → A1(a(a(B(x))))
B1(a(a(b(a(a(B(y0))))))) → B1(b(b(b(b(a(a(a(B(y0)))))))))
B1(a(a(b(a(a(a(a(A(x0))))))))) → B1(a(a(a(b(B(x0))))))
A1(a(a(a(a(b(x0)))))) → B1(b(a(a(a(b(x0))))))
B1(a(a(a(a(A(x)))))) → A1(a(a(b(B(x)))))
A1(a(a(a(a(b(x0)))))) → B1(b(x0))
A1(a(a(a(a(a(a(A(x0)))))))) → B1(b(b(a(a(a(b(b(B(x0)))))))))
B1(a(a(b(x)))) → A1(a(a(b(x))))
A1(a(a(x))) → B1(x)
B1(a(a(b(a(a(B(x))))))) → A1(a(a(b(a(a(a(B(x))))))))
B1(a(a(b(a(a(a(a(A(x0))))))))) → B1(a(a(a(b(a(a(a(b(B(x0))))))))))
A1(a(a(a(a(b(x0)))))) → B1(x0)
B1(a(a(b(a(a(B(x0))))))) → B1(a(a(a(b(b(b(b(B(x0)))))))))
A1(a(a(a(a(b(a(a(B(x0))))))))) → B1(a(a(a(B(x0)))))
B1(a(a(b(a(a(b(a(a(B(x0)))))))))) → B1(a(a(a(a(a(a(B(x0))))))))
A1(a(a(a(a(a(a(A(x0)))))))) → B1(b(a(a(a(b(b(B(x0))))))))
A1(a(a(a(a(a(a(A(y0)))))))) → B1(b(b(b(b(b(b(B(y0))))))))
B1(a(a(b(y0)))) → B1(b(b(b(b(y0)))))
B1(a(a(b(a(a(b(a(a(B(x0)))))))))) → B1(a(a(a(b(a(a(a(b(a(a(a(B(x0)))))))))))))
B1(a(a(b(a(a(B(x0))))))) → B1(a(a(a(b(a(a(A(x0))))))))
A1(a(a(a(a(B(x0)))))) → B1(b(a(a(A(x0)))))
The TRS R consists of the following rules:
a(a(a(x))) → b(b(b(x)))
b(a(a(b(x)))) → x
b(a(a(b(x)))) → b(a(a(a(b(x)))))
a(a(A(x))) → B(x)
b(a(a(a(a(A(x)))))) → b(a(a(a(b(B(x))))))
b(a(a(b(a(a(B(x))))))) → b(a(a(a(b(a(a(a(B(x)))))))))
b(a(a(B(x)))) → b(b(b(b(B(x)))))
b(a(a(b(a(a(B(x))))))) → a(a(a(B(x))))
b(a(a(a(a(A(x)))))) → b(a(a(a(b(b(B(x)))))))
b(a(a(a(a(A(x)))))) → B(x)
b(a(a(B(x)))) → b(a(a(A(x))))
b(a(a(a(a(A(x)))))) → b(B(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 1 less node.
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ QDPToSRSProof
↳ QTRS
↳ QTRS Reverse
↳ QTRS
↳ QTRS Reverse
↳ QTRS Reverse
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QTRS Reverse
↳ QTRS Reverse
Q DP problem:
The TRS P consists of the following rules:
B1(a(a(b(a(a(b(x0))))))) → B1(a(a(a(x0))))
A1(a(a(a(a(b(a(a(B(x0))))))))) → B1(b(a(a(a(b(a(a(a(B(x0))))))))))
A1(a(a(a(a(a(a(A(x0)))))))) → B1(b(a(a(a(b(B(x0)))))))
B1(a(a(b(a(a(b(x0))))))) → B1(a(a(a(b(a(a(a(b(x0)))))))))
B1(a(a(a(a(A(x)))))) → A1(a(a(b(b(B(x))))))
A1(a(a(a(a(b(x0)))))) → B1(b(b(a(a(a(b(x0)))))))
A1(a(a(a(a(b(a(a(B(x0))))))))) → B1(b(b(a(a(a(b(a(a(a(B(x0)))))))))))
B1(a(a(b(a(a(a(a(A(x0))))))))) → B1(a(a(a(b(a(a(a(b(b(B(x0)))))))))))
B1(a(a(b(a(a(a(a(A(x0))))))))) → B1(a(a(a(B(x0)))))
B1(a(a(b(a(a(B(x))))))) → A1(a(a(B(x))))
B1(a(a(b(a(a(B(y0))))))) → B1(b(b(b(b(a(a(a(B(y0)))))))))
B1(a(a(b(a(a(a(a(A(x0))))))))) → B1(a(a(a(b(B(x0))))))
A1(a(a(a(a(b(x0)))))) → B1(b(a(a(a(b(x0))))))
B1(a(a(a(a(A(x)))))) → A1(a(a(b(B(x)))))
A1(a(a(a(a(b(x0)))))) → B1(b(x0))
B1(a(a(b(x)))) → A1(a(a(b(x))))
A1(a(a(a(a(a(a(A(x0)))))))) → B1(b(b(a(a(a(b(b(B(x0)))))))))
A1(a(a(x))) → B1(x)
B1(a(a(b(a(a(a(a(A(x0))))))))) → B1(a(a(a(b(a(a(a(b(B(x0))))))))))
B1(a(a(b(a(a(B(x))))))) → A1(a(a(b(a(a(a(B(x))))))))
A1(a(a(a(a(b(x0)))))) → B1(x0)
A1(a(a(a(a(b(a(a(B(x0))))))))) → B1(a(a(a(B(x0)))))
B1(a(a(b(a(a(B(x0))))))) → B1(a(a(a(b(b(b(b(B(x0)))))))))
A1(a(a(a(a(a(a(A(x0)))))))) → B1(b(a(a(a(b(b(B(x0))))))))
B1(a(a(b(a(a(b(a(a(B(x0)))))))))) → B1(a(a(a(a(a(a(B(x0))))))))
B1(a(a(b(y0)))) → B1(b(b(b(b(y0)))))
B1(a(a(b(a(a(b(a(a(B(x0)))))))))) → B1(a(a(a(b(a(a(a(b(a(a(a(B(x0)))))))))))))
A1(a(a(a(a(B(x0)))))) → B1(b(a(a(A(x0)))))
B1(a(a(b(a(a(B(x0))))))) → B1(a(a(a(b(a(a(A(x0))))))))
The TRS R consists of the following rules:
a(a(a(x))) → b(b(b(x)))
b(a(a(b(x)))) → x
b(a(a(b(x)))) → b(a(a(a(b(x)))))
a(a(A(x))) → B(x)
b(a(a(a(a(A(x)))))) → b(a(a(a(b(B(x))))))
b(a(a(b(a(a(B(x))))))) → b(a(a(a(b(a(a(a(B(x)))))))))
b(a(a(B(x)))) → b(b(b(b(B(x)))))
b(a(a(b(a(a(B(x))))))) → a(a(a(B(x))))
b(a(a(a(a(A(x)))))) → b(a(a(a(b(b(B(x)))))))
b(a(a(a(a(A(x)))))) → B(x)
b(a(a(B(x)))) → b(a(a(A(x))))
b(a(a(a(a(A(x)))))) → b(B(x))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule B1(a(a(b(a(a(a(a(A(x0))))))))) → B1(a(a(a(B(x0))))) at position [0] we obtained the following new rules:
B1(a(a(b(a(a(a(a(A(y0))))))))) → B1(b(b(b(B(y0)))))
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ QDPToSRSProof
↳ QTRS
↳ QTRS Reverse
↳ QTRS
↳ QTRS Reverse
↳ QTRS Reverse
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QTRS Reverse
↳ QTRS Reverse
Q DP problem:
The TRS P consists of the following rules:
B1(a(a(b(a(a(a(a(A(y0))))))))) → B1(b(b(b(B(y0)))))
B1(a(a(b(a(a(b(x0))))))) → B1(a(a(a(x0))))
A1(a(a(a(a(b(a(a(B(x0))))))))) → B1(b(a(a(a(b(a(a(a(B(x0))))))))))
A1(a(a(a(a(a(a(A(x0)))))))) → B1(b(a(a(a(b(B(x0)))))))
B1(a(a(b(a(a(b(x0))))))) → B1(a(a(a(b(a(a(a(b(x0)))))))))
A1(a(a(a(a(b(a(a(B(x0))))))))) → B1(b(b(a(a(a(b(a(a(a(B(x0)))))))))))
A1(a(a(a(a(b(x0)))))) → B1(b(b(a(a(a(b(x0)))))))
B1(a(a(a(a(A(x)))))) → A1(a(a(b(b(B(x))))))
B1(a(a(b(a(a(a(a(A(x0))))))))) → B1(a(a(a(b(a(a(a(b(b(B(x0)))))))))))
B1(a(a(b(a(a(B(x))))))) → A1(a(a(B(x))))
B1(a(a(b(a(a(B(y0))))))) → B1(b(b(b(b(a(a(a(B(y0)))))))))
B1(a(a(b(a(a(a(a(A(x0))))))))) → B1(a(a(a(b(B(x0))))))
A1(a(a(a(a(b(x0)))))) → B1(b(a(a(a(b(x0))))))
B1(a(a(a(a(A(x)))))) → A1(a(a(b(B(x)))))
A1(a(a(a(a(b(x0)))))) → B1(b(x0))
A1(a(a(a(a(a(a(A(x0)))))))) → B1(b(b(a(a(a(b(b(B(x0)))))))))
B1(a(a(b(x)))) → A1(a(a(b(x))))
A1(a(a(x))) → B1(x)
B1(a(a(b(a(a(B(x))))))) → A1(a(a(b(a(a(a(B(x))))))))
B1(a(a(b(a(a(a(a(A(x0))))))))) → B1(a(a(a(b(a(a(a(b(B(x0))))))))))
A1(a(a(a(a(b(x0)))))) → B1(x0)
B1(a(a(b(a(a(B(x0))))))) → B1(a(a(a(b(b(b(b(B(x0)))))))))
A1(a(a(a(a(b(a(a(B(x0))))))))) → B1(a(a(a(B(x0)))))
B1(a(a(b(a(a(b(a(a(B(x0)))))))))) → B1(a(a(a(a(a(a(B(x0))))))))
A1(a(a(a(a(a(a(A(x0)))))))) → B1(b(a(a(a(b(b(B(x0))))))))
B1(a(a(b(y0)))) → B1(b(b(b(b(y0)))))
B1(a(a(b(a(a(b(a(a(B(x0)))))))))) → B1(a(a(a(b(a(a(a(b(a(a(a(B(x0)))))))))))))
B1(a(a(b(a(a(B(x0))))))) → B1(a(a(a(b(a(a(A(x0))))))))
A1(a(a(a(a(B(x0)))))) → B1(b(a(a(A(x0)))))
The TRS R consists of the following rules:
a(a(a(x))) → b(b(b(x)))
b(a(a(b(x)))) → x
b(a(a(b(x)))) → b(a(a(a(b(x)))))
a(a(A(x))) → B(x)
b(a(a(a(a(A(x)))))) → b(a(a(a(b(B(x))))))
b(a(a(b(a(a(B(x))))))) → b(a(a(a(b(a(a(a(B(x)))))))))
b(a(a(B(x)))) → b(b(b(b(B(x)))))
b(a(a(b(a(a(B(x))))))) → a(a(a(B(x))))
b(a(a(a(a(A(x)))))) → b(a(a(a(b(b(B(x)))))))
b(a(a(a(a(A(x)))))) → B(x)
b(a(a(B(x)))) → b(a(a(A(x))))
b(a(a(a(a(A(x)))))) → b(B(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 1 less node.
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ QDPToSRSProof
↳ QTRS
↳ QTRS Reverse
↳ QTRS
↳ QTRS Reverse
↳ QTRS Reverse
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QTRS Reverse
↳ QTRS Reverse
Q DP problem:
The TRS P consists of the following rules:
B1(a(a(b(a(a(b(x0))))))) → B1(a(a(a(x0))))
A1(a(a(a(a(b(a(a(B(x0))))))))) → B1(b(a(a(a(b(a(a(a(B(x0))))))))))
A1(a(a(a(a(a(a(A(x0)))))))) → B1(b(a(a(a(b(B(x0)))))))
B1(a(a(b(a(a(b(x0))))))) → B1(a(a(a(b(a(a(a(b(x0)))))))))
B1(a(a(a(a(A(x)))))) → A1(a(a(b(b(B(x))))))
A1(a(a(a(a(b(x0)))))) → B1(b(b(a(a(a(b(x0)))))))
A1(a(a(a(a(b(a(a(B(x0))))))))) → B1(b(b(a(a(a(b(a(a(a(B(x0)))))))))))
B1(a(a(b(a(a(a(a(A(x0))))))))) → B1(a(a(a(b(a(a(a(b(b(B(x0)))))))))))
B1(a(a(b(a(a(B(x))))))) → A1(a(a(B(x))))
B1(a(a(b(a(a(B(y0))))))) → B1(b(b(b(b(a(a(a(B(y0)))))))))
B1(a(a(b(a(a(a(a(A(x0))))))))) → B1(a(a(a(b(B(x0))))))
A1(a(a(a(a(b(x0)))))) → B1(b(a(a(a(b(x0))))))
B1(a(a(a(a(A(x)))))) → A1(a(a(b(B(x)))))
A1(a(a(a(a(b(x0)))))) → B1(b(x0))
A1(a(a(a(a(a(a(A(x0)))))))) → B1(b(b(a(a(a(b(b(B(x0)))))))))
B1(a(a(b(x)))) → A1(a(a(b(x))))
A1(a(a(x))) → B1(x)
B1(a(a(b(a(a(a(a(A(x0))))))))) → B1(a(a(a(b(a(a(a(b(B(x0))))))))))
B1(a(a(b(a(a(B(x))))))) → A1(a(a(b(a(a(a(B(x))))))))
A1(a(a(a(a(b(x0)))))) → B1(x0)
A1(a(a(a(a(b(a(a(B(x0))))))))) → B1(a(a(a(B(x0)))))
B1(a(a(b(a(a(B(x0))))))) → B1(a(a(a(b(b(b(b(B(x0)))))))))
A1(a(a(a(a(a(a(A(x0)))))))) → B1(b(a(a(a(b(b(B(x0))))))))
B1(a(a(b(a(a(b(a(a(B(x0)))))))))) → B1(a(a(a(a(a(a(B(x0))))))))
B1(a(a(b(y0)))) → B1(b(b(b(b(y0)))))
B1(a(a(b(a(a(b(a(a(B(x0)))))))))) → B1(a(a(a(b(a(a(a(b(a(a(a(B(x0)))))))))))))
B1(a(a(b(a(a(B(x0))))))) → B1(a(a(a(b(a(a(A(x0))))))))
A1(a(a(a(a(B(x0)))))) → B1(b(a(a(A(x0)))))
The TRS R consists of the following rules:
a(a(a(x))) → b(b(b(x)))
b(a(a(b(x)))) → x
b(a(a(b(x)))) → b(a(a(a(b(x)))))
a(a(A(x))) → B(x)
b(a(a(a(a(A(x)))))) → b(a(a(a(b(B(x))))))
b(a(a(b(a(a(B(x))))))) → b(a(a(a(b(a(a(a(B(x)))))))))
b(a(a(B(x)))) → b(b(b(b(B(x)))))
b(a(a(b(a(a(B(x))))))) → a(a(a(B(x))))
b(a(a(a(a(A(x)))))) → b(a(a(a(b(b(B(x)))))))
b(a(a(a(a(A(x)))))) → B(x)
b(a(a(B(x)))) → b(a(a(A(x))))
b(a(a(a(a(A(x)))))) → b(B(x))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule A1(a(a(a(a(a(a(A(x0)))))))) → B1(b(b(a(a(a(b(b(B(x0))))))))) at position [0] we obtained the following new rules:
A1(a(a(a(a(a(a(A(y0)))))))) → B1(b(b(b(b(b(b(b(B(y0)))))))))
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ QDPToSRSProof
↳ QTRS
↳ QTRS Reverse
↳ QTRS
↳ QTRS Reverse
↳ QTRS Reverse
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QTRS Reverse
↳ QTRS Reverse
Q DP problem:
The TRS P consists of the following rules:
B1(a(a(b(a(a(b(x0))))))) → B1(a(a(a(x0))))
A1(a(a(a(a(b(a(a(B(x0))))))))) → B1(b(a(a(a(b(a(a(a(B(x0))))))))))
A1(a(a(a(a(a(a(A(x0)))))))) → B1(b(a(a(a(b(B(x0)))))))
B1(a(a(b(a(a(b(x0))))))) → B1(a(a(a(b(a(a(a(b(x0)))))))))
A1(a(a(a(a(b(a(a(B(x0))))))))) → B1(b(b(a(a(a(b(a(a(a(B(x0)))))))))))
A1(a(a(a(a(b(x0)))))) → B1(b(b(a(a(a(b(x0)))))))
B1(a(a(a(a(A(x)))))) → A1(a(a(b(b(B(x))))))
B1(a(a(b(a(a(a(a(A(x0))))))))) → B1(a(a(a(b(a(a(a(b(b(B(x0)))))))))))
B1(a(a(b(a(a(B(x))))))) → A1(a(a(B(x))))
B1(a(a(b(a(a(B(y0))))))) → B1(b(b(b(b(a(a(a(B(y0)))))))))
B1(a(a(b(a(a(a(a(A(x0))))))))) → B1(a(a(a(b(B(x0))))))
A1(a(a(a(a(b(x0)))))) → B1(b(a(a(a(b(x0))))))
A1(a(a(a(a(a(a(A(y0)))))))) → B1(b(b(b(b(b(b(b(B(y0)))))))))
B1(a(a(a(a(A(x)))))) → A1(a(a(b(B(x)))))
A1(a(a(a(a(b(x0)))))) → B1(b(x0))
B1(a(a(b(x)))) → A1(a(a(b(x))))
A1(a(a(x))) → B1(x)
B1(a(a(b(a(a(B(x))))))) → A1(a(a(b(a(a(a(B(x))))))))
B1(a(a(b(a(a(a(a(A(x0))))))))) → B1(a(a(a(b(a(a(a(b(B(x0))))))))))
A1(a(a(a(a(b(x0)))))) → B1(x0)
B1(a(a(b(a(a(B(x0))))))) → B1(a(a(a(b(b(b(b(B(x0)))))))))
A1(a(a(a(a(b(a(a(B(x0))))))))) → B1(a(a(a(B(x0)))))
B1(a(a(b(a(a(b(a(a(B(x0)))))))))) → B1(a(a(a(a(a(a(B(x0))))))))
A1(a(a(a(a(a(a(A(x0)))))))) → B1(b(a(a(a(b(b(B(x0))))))))
B1(a(a(b(y0)))) → B1(b(b(b(b(y0)))))
B1(a(a(b(a(a(b(a(a(B(x0)))))))))) → B1(a(a(a(b(a(a(a(b(a(a(a(B(x0)))))))))))))
A1(a(a(a(a(B(x0)))))) → B1(b(a(a(A(x0)))))
B1(a(a(b(a(a(B(x0))))))) → B1(a(a(a(b(a(a(A(x0))))))))
The TRS R consists of the following rules:
a(a(a(x))) → b(b(b(x)))
b(a(a(b(x)))) → x
b(a(a(b(x)))) → b(a(a(a(b(x)))))
a(a(A(x))) → B(x)
b(a(a(a(a(A(x)))))) → b(a(a(a(b(B(x))))))
b(a(a(b(a(a(B(x))))))) → b(a(a(a(b(a(a(a(B(x)))))))))
b(a(a(B(x)))) → b(b(b(b(B(x)))))
b(a(a(b(a(a(B(x))))))) → a(a(a(B(x))))
b(a(a(a(a(A(x)))))) → b(a(a(a(b(b(B(x)))))))
b(a(a(a(a(A(x)))))) → B(x)
b(a(a(B(x)))) → b(a(a(A(x))))
b(a(a(a(a(A(x)))))) → b(B(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 1 less node.
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ QDPToSRSProof
↳ QTRS
↳ QTRS Reverse
↳ QTRS
↳ QTRS Reverse
↳ QTRS Reverse
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QTRS Reverse
↳ QTRS Reverse
Q DP problem:
The TRS P consists of the following rules:
B1(a(a(b(a(a(b(x0))))))) → B1(a(a(a(x0))))
A1(a(a(a(a(b(a(a(B(x0))))))))) → B1(b(a(a(a(b(a(a(a(B(x0))))))))))
A1(a(a(a(a(a(a(A(x0)))))))) → B1(b(a(a(a(b(B(x0)))))))
B1(a(a(b(a(a(b(x0))))))) → B1(a(a(a(b(a(a(a(b(x0)))))))))
B1(a(a(a(a(A(x)))))) → A1(a(a(b(b(B(x))))))
A1(a(a(a(a(b(x0)))))) → B1(b(b(a(a(a(b(x0)))))))
A1(a(a(a(a(b(a(a(B(x0))))))))) → B1(b(b(a(a(a(b(a(a(a(B(x0)))))))))))
B1(a(a(b(a(a(a(a(A(x0))))))))) → B1(a(a(a(b(a(a(a(b(b(B(x0)))))))))))
B1(a(a(b(a(a(B(x))))))) → A1(a(a(B(x))))
B1(a(a(b(a(a(B(y0))))))) → B1(b(b(b(b(a(a(a(B(y0)))))))))
B1(a(a(b(a(a(a(a(A(x0))))))))) → B1(a(a(a(b(B(x0))))))
A1(a(a(a(a(b(x0)))))) → B1(b(a(a(a(b(x0))))))
B1(a(a(a(a(A(x)))))) → A1(a(a(b(B(x)))))
A1(a(a(a(a(b(x0)))))) → B1(b(x0))
B1(a(a(b(x)))) → A1(a(a(b(x))))
A1(a(a(x))) → B1(x)
B1(a(a(b(a(a(a(a(A(x0))))))))) → B1(a(a(a(b(a(a(a(b(B(x0))))))))))
B1(a(a(b(a(a(B(x))))))) → A1(a(a(b(a(a(a(B(x))))))))
A1(a(a(a(a(b(x0)))))) → B1(x0)
B1(a(a(b(a(a(B(x0))))))) → B1(a(a(a(b(b(b(b(B(x0)))))))))
A1(a(a(a(a(b(a(a(B(x0))))))))) → B1(a(a(a(B(x0)))))
A1(a(a(a(a(a(a(A(x0)))))))) → B1(b(a(a(a(b(b(B(x0))))))))
B1(a(a(b(a(a(b(a(a(B(x0)))))))))) → B1(a(a(a(a(a(a(B(x0))))))))
B1(a(a(b(y0)))) → B1(b(b(b(b(y0)))))
B1(a(a(b(a(a(b(a(a(B(x0)))))))))) → B1(a(a(a(b(a(a(a(b(a(a(a(B(x0)))))))))))))
B1(a(a(b(a(a(B(x0))))))) → B1(a(a(a(b(a(a(A(x0))))))))
A1(a(a(a(a(B(x0)))))) → B1(b(a(a(A(x0)))))
The TRS R consists of the following rules:
a(a(a(x))) → b(b(b(x)))
b(a(a(b(x)))) → x
b(a(a(b(x)))) → b(a(a(a(b(x)))))
a(a(A(x))) → B(x)
b(a(a(a(a(A(x)))))) → b(a(a(a(b(B(x))))))
b(a(a(b(a(a(B(x))))))) → b(a(a(a(b(a(a(a(B(x)))))))))
b(a(a(B(x)))) → b(b(b(b(B(x)))))
b(a(a(b(a(a(B(x))))))) → a(a(a(B(x))))
b(a(a(a(a(A(x)))))) → b(a(a(a(b(b(B(x)))))))
b(a(a(a(a(A(x)))))) → B(x)
b(a(a(B(x)))) → b(a(a(A(x))))
b(a(a(a(a(A(x)))))) → b(B(x))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule B1(a(a(b(a(a(a(a(A(x0))))))))) → B1(a(a(a(b(B(x0)))))) at position [0] we obtained the following new rules:
B1(a(a(b(a(a(a(a(A(y0))))))))) → B1(b(b(b(b(B(y0))))))
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ QDPToSRSProof
↳ QTRS
↳ QTRS Reverse
↳ QTRS
↳ QTRS Reverse
↳ QTRS Reverse
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QTRS Reverse
↳ QTRS Reverse
Q DP problem:
The TRS P consists of the following rules:
B1(a(a(b(a(a(b(x0))))))) → B1(a(a(a(x0))))
A1(a(a(a(a(b(a(a(B(x0))))))))) → B1(b(a(a(a(b(a(a(a(B(x0))))))))))
A1(a(a(a(a(a(a(A(x0)))))))) → B1(b(a(a(a(b(B(x0)))))))
B1(a(a(b(a(a(b(x0))))))) → B1(a(a(a(b(a(a(a(b(x0)))))))))
A1(a(a(a(a(b(a(a(B(x0))))))))) → B1(b(b(a(a(a(b(a(a(a(B(x0)))))))))))
A1(a(a(a(a(b(x0)))))) → B1(b(b(a(a(a(b(x0)))))))
B1(a(a(a(a(A(x)))))) → A1(a(a(b(b(B(x))))))
B1(a(a(b(a(a(a(a(A(x0))))))))) → B1(a(a(a(b(a(a(a(b(b(B(x0)))))))))))
B1(a(a(b(a(a(B(x))))))) → A1(a(a(B(x))))
B1(a(a(b(a(a(B(y0))))))) → B1(b(b(b(b(a(a(a(B(y0)))))))))
A1(a(a(a(a(b(x0)))))) → B1(b(a(a(a(b(x0))))))
B1(a(a(a(a(A(x)))))) → A1(a(a(b(B(x)))))
B1(a(a(b(a(a(a(a(A(y0))))))))) → B1(b(b(b(b(B(y0))))))
A1(a(a(a(a(b(x0)))))) → B1(b(x0))
B1(a(a(b(x)))) → A1(a(a(b(x))))
A1(a(a(x))) → B1(x)
B1(a(a(b(a(a(B(x))))))) → A1(a(a(b(a(a(a(B(x))))))))
B1(a(a(b(a(a(a(a(A(x0))))))))) → B1(a(a(a(b(a(a(a(b(B(x0))))))))))
A1(a(a(a(a(b(x0)))))) → B1(x0)
A1(a(a(a(a(b(a(a(B(x0))))))))) → B1(a(a(a(B(x0)))))
B1(a(a(b(a(a(B(x0))))))) → B1(a(a(a(b(b(b(b(B(x0)))))))))
B1(a(a(b(a(a(b(a(a(B(x0)))))))))) → B1(a(a(a(a(a(a(B(x0))))))))
A1(a(a(a(a(a(a(A(x0)))))))) → B1(b(a(a(a(b(b(B(x0))))))))
B1(a(a(b(y0)))) → B1(b(b(b(b(y0)))))
B1(a(a(b(a(a(b(a(a(B(x0)))))))))) → B1(a(a(a(b(a(a(a(b(a(a(a(B(x0)))))))))))))
A1(a(a(a(a(B(x0)))))) → B1(b(a(a(A(x0)))))
B1(a(a(b(a(a(B(x0))))))) → B1(a(a(a(b(a(a(A(x0))))))))
The TRS R consists of the following rules:
a(a(a(x))) → b(b(b(x)))
b(a(a(b(x)))) → x
b(a(a(b(x)))) → b(a(a(a(b(x)))))
a(a(A(x))) → B(x)
b(a(a(a(a(A(x)))))) → b(a(a(a(b(B(x))))))
b(a(a(b(a(a(B(x))))))) → b(a(a(a(b(a(a(a(B(x)))))))))
b(a(a(B(x)))) → b(b(b(b(B(x)))))
b(a(a(b(a(a(B(x))))))) → a(a(a(B(x))))
b(a(a(a(a(A(x)))))) → b(a(a(a(b(b(B(x)))))))
b(a(a(a(a(A(x)))))) → B(x)
b(a(a(B(x)))) → b(a(a(A(x))))
b(a(a(a(a(A(x)))))) → b(B(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 1 less node.
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ QDPToSRSProof
↳ QTRS
↳ QTRS Reverse
↳ QTRS
↳ QTRS Reverse
↳ QTRS Reverse
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QTRS Reverse
↳ QTRS Reverse
Q DP problem:
The TRS P consists of the following rules:
B1(a(a(b(a(a(b(x0))))))) → B1(a(a(a(x0))))
A1(a(a(a(a(b(a(a(B(x0))))))))) → B1(b(a(a(a(b(a(a(a(B(x0))))))))))
A1(a(a(a(a(a(a(A(x0)))))))) → B1(b(a(a(a(b(B(x0)))))))
B1(a(a(b(a(a(b(x0))))))) → B1(a(a(a(b(a(a(a(b(x0)))))))))
B1(a(a(a(a(A(x)))))) → A1(a(a(b(b(B(x))))))
A1(a(a(a(a(b(x0)))))) → B1(b(b(a(a(a(b(x0)))))))
A1(a(a(a(a(b(a(a(B(x0))))))))) → B1(b(b(a(a(a(b(a(a(a(B(x0)))))))))))
B1(a(a(b(a(a(a(a(A(x0))))))))) → B1(a(a(a(b(a(a(a(b(b(B(x0)))))))))))
B1(a(a(b(a(a(B(x))))))) → A1(a(a(B(x))))
B1(a(a(b(a(a(B(y0))))))) → B1(b(b(b(b(a(a(a(B(y0)))))))))
A1(a(a(a(a(b(x0)))))) → B1(b(a(a(a(b(x0))))))
B1(a(a(a(a(A(x)))))) → A1(a(a(b(B(x)))))
A1(a(a(a(a(b(x0)))))) → B1(b(x0))
B1(a(a(b(x)))) → A1(a(a(b(x))))
A1(a(a(x))) → B1(x)
B1(a(a(b(a(a(a(a(A(x0))))))))) → B1(a(a(a(b(a(a(a(b(B(x0))))))))))
B1(a(a(b(a(a(B(x))))))) → A1(a(a(b(a(a(a(B(x))))))))
A1(a(a(a(a(b(x0)))))) → B1(x0)
B1(a(a(b(a(a(B(x0))))))) → B1(a(a(a(b(b(b(b(B(x0)))))))))
A1(a(a(a(a(b(a(a(B(x0))))))))) → B1(a(a(a(B(x0)))))
B1(a(a(b(a(a(b(a(a(B(x0)))))))))) → B1(a(a(a(a(a(a(B(x0))))))))
A1(a(a(a(a(a(a(A(x0)))))))) → B1(b(a(a(a(b(b(B(x0))))))))
B1(a(a(b(y0)))) → B1(b(b(b(b(y0)))))
B1(a(a(b(a(a(b(a(a(B(x0)))))))))) → B1(a(a(a(b(a(a(a(b(a(a(a(B(x0)))))))))))))
B1(a(a(b(a(a(B(x0))))))) → B1(a(a(a(b(a(a(A(x0))))))))
A1(a(a(a(a(B(x0)))))) → B1(b(a(a(A(x0)))))
The TRS R consists of the following rules:
a(a(a(x))) → b(b(b(x)))
b(a(a(b(x)))) → x
b(a(a(b(x)))) → b(a(a(a(b(x)))))
a(a(A(x))) → B(x)
b(a(a(a(a(A(x)))))) → b(a(a(a(b(B(x))))))
b(a(a(b(a(a(B(x))))))) → b(a(a(a(b(a(a(a(B(x)))))))))
b(a(a(B(x)))) → b(b(b(b(B(x)))))
b(a(a(b(a(a(B(x))))))) → a(a(a(B(x))))
b(a(a(a(a(A(x)))))) → b(a(a(a(b(b(B(x)))))))
b(a(a(a(a(A(x)))))) → B(x)
b(a(a(B(x)))) → b(a(a(A(x))))
b(a(a(a(a(A(x)))))) → b(B(x))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule B1(a(a(b(a(a(B(x0))))))) → B1(a(a(a(b(b(b(b(B(x0))))))))) at position [0] we obtained the following new rules:
B1(a(a(b(a(a(B(y0))))))) → B1(b(b(b(b(b(b(b(B(y0)))))))))
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ QDPToSRSProof
↳ QTRS
↳ QTRS Reverse
↳ QTRS
↳ QTRS Reverse
↳ QTRS Reverse
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QTRS Reverse
↳ QTRS Reverse
Q DP problem:
The TRS P consists of the following rules:
B1(a(a(b(a(a(b(x0))))))) → B1(a(a(a(x0))))
A1(a(a(a(a(b(a(a(B(x0))))))))) → B1(b(a(a(a(b(a(a(a(B(x0))))))))))
B1(a(a(b(a(a(B(y0))))))) → B1(b(b(b(b(b(b(b(B(y0)))))))))
A1(a(a(a(a(a(a(A(x0)))))))) → B1(b(a(a(a(b(B(x0)))))))
B1(a(a(b(a(a(b(x0))))))) → B1(a(a(a(b(a(a(a(b(x0)))))))))
A1(a(a(a(a(b(a(a(B(x0))))))))) → B1(b(b(a(a(a(b(a(a(a(B(x0)))))))))))
A1(a(a(a(a(b(x0)))))) → B1(b(b(a(a(a(b(x0)))))))
B1(a(a(a(a(A(x)))))) → A1(a(a(b(b(B(x))))))
B1(a(a(b(a(a(a(a(A(x0))))))))) → B1(a(a(a(b(a(a(a(b(b(B(x0)))))))))))
B1(a(a(b(a(a(B(x))))))) → A1(a(a(B(x))))
B1(a(a(b(a(a(B(y0))))))) → B1(b(b(b(b(a(a(a(B(y0)))))))))
A1(a(a(a(a(b(x0)))))) → B1(b(a(a(a(b(x0))))))
B1(a(a(a(a(A(x)))))) → A1(a(a(b(B(x)))))
A1(a(a(a(a(b(x0)))))) → B1(b(x0))
B1(a(a(b(x)))) → A1(a(a(b(x))))
A1(a(a(x))) → B1(x)
B1(a(a(b(a(a(B(x))))))) → A1(a(a(b(a(a(a(B(x))))))))
B1(a(a(b(a(a(a(a(A(x0))))))))) → B1(a(a(a(b(a(a(a(b(B(x0))))))))))
A1(a(a(a(a(b(x0)))))) → B1(x0)
A1(a(a(a(a(b(a(a(B(x0))))))))) → B1(a(a(a(B(x0)))))
A1(a(a(a(a(a(a(A(x0)))))))) → B1(b(a(a(a(b(b(B(x0))))))))
B1(a(a(b(a(a(b(a(a(B(x0)))))))))) → B1(a(a(a(a(a(a(B(x0))))))))
B1(a(a(b(y0)))) → B1(b(b(b(b(y0)))))
B1(a(a(b(a(a(b(a(a(B(x0)))))))))) → B1(a(a(a(b(a(a(a(b(a(a(a(B(x0)))))))))))))
A1(a(a(a(a(B(x0)))))) → B1(b(a(a(A(x0)))))
B1(a(a(b(a(a(B(x0))))))) → B1(a(a(a(b(a(a(A(x0))))))))
The TRS R consists of the following rules:
a(a(a(x))) → b(b(b(x)))
b(a(a(b(x)))) → x
b(a(a(b(x)))) → b(a(a(a(b(x)))))
a(a(A(x))) → B(x)
b(a(a(a(a(A(x)))))) → b(a(a(a(b(B(x))))))
b(a(a(b(a(a(B(x))))))) → b(a(a(a(b(a(a(a(B(x)))))))))
b(a(a(B(x)))) → b(b(b(b(B(x)))))
b(a(a(b(a(a(B(x))))))) → a(a(a(B(x))))
b(a(a(a(a(A(x)))))) → b(a(a(a(b(b(B(x)))))))
b(a(a(a(a(A(x)))))) → B(x)
b(a(a(B(x)))) → b(a(a(A(x))))
b(a(a(a(a(A(x)))))) → b(B(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 1 less node.
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ QDPToSRSProof
↳ QTRS
↳ QTRS Reverse
↳ QTRS
↳ QTRS Reverse
↳ QTRS Reverse
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QTRS Reverse
↳ QTRS Reverse
Q DP problem:
The TRS P consists of the following rules:
B1(a(a(b(a(a(b(x0))))))) → B1(a(a(a(x0))))
A1(a(a(a(a(b(a(a(B(x0))))))))) → B1(b(a(a(a(b(a(a(a(B(x0))))))))))
A1(a(a(a(a(a(a(A(x0)))))))) → B1(b(a(a(a(b(B(x0)))))))
B1(a(a(b(a(a(b(x0))))))) → B1(a(a(a(b(a(a(a(b(x0)))))))))
B1(a(a(a(a(A(x)))))) → A1(a(a(b(b(B(x))))))
A1(a(a(a(a(b(x0)))))) → B1(b(b(a(a(a(b(x0)))))))
A1(a(a(a(a(b(a(a(B(x0))))))))) → B1(b(b(a(a(a(b(a(a(a(B(x0)))))))))))
B1(a(a(b(a(a(a(a(A(x0))))))))) → B1(a(a(a(b(a(a(a(b(b(B(x0)))))))))))
B1(a(a(b(a(a(B(x))))))) → A1(a(a(B(x))))
B1(a(a(b(a(a(B(y0))))))) → B1(b(b(b(b(a(a(a(B(y0)))))))))
A1(a(a(a(a(b(x0)))))) → B1(b(a(a(a(b(x0))))))
B1(a(a(a(a(A(x)))))) → A1(a(a(b(B(x)))))
A1(a(a(a(a(b(x0)))))) → B1(b(x0))
B1(a(a(b(x)))) → A1(a(a(b(x))))
A1(a(a(x))) → B1(x)
B1(a(a(b(a(a(a(a(A(x0))))))))) → B1(a(a(a(b(a(a(a(b(B(x0))))))))))
B1(a(a(b(a(a(B(x))))))) → A1(a(a(b(a(a(a(B(x))))))))
A1(a(a(a(a(b(x0)))))) → B1(x0)
A1(a(a(a(a(b(a(a(B(x0))))))))) → B1(a(a(a(B(x0)))))
B1(a(a(b(a(a(b(a(a(B(x0)))))))))) → B1(a(a(a(a(a(a(B(x0))))))))
A1(a(a(a(a(a(a(A(x0)))))))) → B1(b(a(a(a(b(b(B(x0))))))))
B1(a(a(b(y0)))) → B1(b(b(b(b(y0)))))
B1(a(a(b(a(a(b(a(a(B(x0)))))))))) → B1(a(a(a(b(a(a(a(b(a(a(a(B(x0)))))))))))))
B1(a(a(b(a(a(B(x0))))))) → B1(a(a(a(b(a(a(A(x0))))))))
A1(a(a(a(a(B(x0)))))) → B1(b(a(a(A(x0)))))
The TRS R consists of the following rules:
a(a(a(x))) → b(b(b(x)))
b(a(a(b(x)))) → x
b(a(a(b(x)))) → b(a(a(a(b(x)))))
a(a(A(x))) → B(x)
b(a(a(a(a(A(x)))))) → b(a(a(a(b(B(x))))))
b(a(a(b(a(a(B(x))))))) → b(a(a(a(b(a(a(a(B(x)))))))))
b(a(a(B(x)))) → b(b(b(b(B(x)))))
b(a(a(b(a(a(B(x))))))) → a(a(a(B(x))))
b(a(a(a(a(A(x)))))) → b(a(a(a(b(b(B(x)))))))
b(a(a(a(a(A(x)))))) → B(x)
b(a(a(B(x)))) → b(a(a(A(x))))
b(a(a(a(a(A(x)))))) → b(B(x))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule B1(a(a(b(a(a(B(y0))))))) → B1(b(b(b(b(a(a(a(B(y0))))))))) at position [0] we obtained the following new rules:
B1(a(a(b(a(a(B(y0))))))) → B1(b(b(b(b(b(b(b(B(y0)))))))))
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ QDPToSRSProof
↳ QTRS
↳ QTRS Reverse
↳ QTRS
↳ QTRS Reverse
↳ QTRS Reverse
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QTRS Reverse
↳ QTRS Reverse
Q DP problem:
The TRS P consists of the following rules:
B1(a(a(b(a(a(b(x0))))))) → B1(a(a(a(x0))))
A1(a(a(a(a(b(a(a(B(x0))))))))) → B1(b(a(a(a(b(a(a(a(B(x0))))))))))
B1(a(a(b(a(a(B(y0))))))) → B1(b(b(b(b(b(b(b(B(y0)))))))))
A1(a(a(a(a(a(a(A(x0)))))))) → B1(b(a(a(a(b(B(x0)))))))
B1(a(a(b(a(a(b(x0))))))) → B1(a(a(a(b(a(a(a(b(x0)))))))))
A1(a(a(a(a(b(a(a(B(x0))))))))) → B1(b(b(a(a(a(b(a(a(a(B(x0)))))))))))
A1(a(a(a(a(b(x0)))))) → B1(b(b(a(a(a(b(x0)))))))
B1(a(a(a(a(A(x)))))) → A1(a(a(b(b(B(x))))))
B1(a(a(b(a(a(a(a(A(x0))))))))) → B1(a(a(a(b(a(a(a(b(b(B(x0)))))))))))
B1(a(a(b(a(a(B(x))))))) → A1(a(a(B(x))))
A1(a(a(a(a(b(x0)))))) → B1(b(a(a(a(b(x0))))))
B1(a(a(a(a(A(x)))))) → A1(a(a(b(B(x)))))
A1(a(a(a(a(b(x0)))))) → B1(b(x0))
B1(a(a(b(x)))) → A1(a(a(b(x))))
A1(a(a(x))) → B1(x)
B1(a(a(b(a(a(B(x))))))) → A1(a(a(b(a(a(a(B(x))))))))
B1(a(a(b(a(a(a(a(A(x0))))))))) → B1(a(a(a(b(a(a(a(b(B(x0))))))))))
A1(a(a(a(a(b(x0)))))) → B1(x0)
A1(a(a(a(a(b(a(a(B(x0))))))))) → B1(a(a(a(B(x0)))))
A1(a(a(a(a(a(a(A(x0)))))))) → B1(b(a(a(a(b(b(B(x0))))))))
B1(a(a(b(a(a(b(a(a(B(x0)))))))))) → B1(a(a(a(a(a(a(B(x0))))))))
B1(a(a(b(y0)))) → B1(b(b(b(b(y0)))))
B1(a(a(b(a(a(b(a(a(B(x0)))))))))) → B1(a(a(a(b(a(a(a(b(a(a(a(B(x0)))))))))))))
A1(a(a(a(a(B(x0)))))) → B1(b(a(a(A(x0)))))
B1(a(a(b(a(a(B(x0))))))) → B1(a(a(a(b(a(a(A(x0))))))))
The TRS R consists of the following rules:
a(a(a(x))) → b(b(b(x)))
b(a(a(b(x)))) → x
b(a(a(b(x)))) → b(a(a(a(b(x)))))
a(a(A(x))) → B(x)
b(a(a(a(a(A(x)))))) → b(a(a(a(b(B(x))))))
b(a(a(b(a(a(B(x))))))) → b(a(a(a(b(a(a(a(B(x)))))))))
b(a(a(B(x)))) → b(b(b(b(B(x)))))
b(a(a(b(a(a(B(x))))))) → a(a(a(B(x))))
b(a(a(a(a(A(x)))))) → b(a(a(a(b(b(B(x)))))))
b(a(a(a(a(A(x)))))) → B(x)
b(a(a(B(x)))) → b(a(a(A(x))))
b(a(a(a(a(A(x)))))) → b(B(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 1 less node.
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ QDPToSRSProof
↳ QTRS
↳ QTRS Reverse
↳ QTRS
↳ QTRS Reverse
↳ QTRS Reverse
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QTRS Reverse
↳ QTRS Reverse
Q DP problem:
The TRS P consists of the following rules:
B1(a(a(b(a(a(B(x))))))) → A1(a(a(b(a(a(a(B(x))))))))
B1(a(a(b(a(a(a(a(A(x0))))))))) → B1(a(a(a(b(a(a(a(b(B(x0))))))))))
A1(a(a(a(a(b(x0)))))) → B1(x0)
B1(a(a(b(a(a(b(x0))))))) → B1(a(a(a(x0))))
A1(a(a(a(a(b(a(a(B(x0))))))))) → B1(b(a(a(a(b(a(a(a(B(x0))))))))))
A1(a(a(a(a(b(a(a(B(x0))))))))) → B1(a(a(a(B(x0)))))
A1(a(a(a(a(a(a(A(x0)))))))) → B1(b(a(a(a(b(b(B(x0))))))))
B1(a(a(b(a(a(b(a(a(B(x0)))))))))) → B1(a(a(a(a(a(a(B(x0))))))))
A1(a(a(a(a(a(a(A(x0)))))))) → B1(b(a(a(a(b(B(x0)))))))
B1(a(a(b(a(a(b(x0))))))) → B1(a(a(a(b(a(a(a(b(x0)))))))))
A1(a(a(a(a(b(a(a(B(x0))))))))) → B1(b(b(a(a(a(b(a(a(a(B(x0)))))))))))
B1(a(a(b(y0)))) → B1(b(b(b(b(y0)))))
A1(a(a(a(a(b(x0)))))) → B1(b(b(a(a(a(b(x0)))))))
B1(a(a(a(a(A(x)))))) → A1(a(a(b(b(B(x))))))
B1(a(a(b(a(a(a(a(A(x0))))))))) → B1(a(a(a(b(a(a(a(b(b(B(x0)))))))))))
B1(a(a(b(a(a(B(x))))))) → A1(a(a(B(x))))
A1(a(a(a(a(b(x0)))))) → B1(b(a(a(a(b(x0))))))
B1(a(a(b(a(a(b(a(a(B(x0)))))))))) → B1(a(a(a(b(a(a(a(b(a(a(a(B(x0)))))))))))))
B1(a(a(a(a(A(x)))))) → A1(a(a(b(B(x)))))
A1(a(a(a(a(B(x0)))))) → B1(b(a(a(A(x0)))))
B1(a(a(b(a(a(B(x0))))))) → B1(a(a(a(b(a(a(A(x0))))))))
A1(a(a(a(a(b(x0)))))) → B1(b(x0))
B1(a(a(b(x)))) → A1(a(a(b(x))))
A1(a(a(x))) → B1(x)
The TRS R consists of the following rules:
a(a(a(x))) → b(b(b(x)))
b(a(a(b(x)))) → x
b(a(a(b(x)))) → b(a(a(a(b(x)))))
a(a(A(x))) → B(x)
b(a(a(a(a(A(x)))))) → b(a(a(a(b(B(x))))))
b(a(a(b(a(a(B(x))))))) → b(a(a(a(b(a(a(a(B(x)))))))))
b(a(a(B(x)))) → b(b(b(b(B(x)))))
b(a(a(b(a(a(B(x))))))) → a(a(a(B(x))))
b(a(a(a(a(A(x)))))) → b(a(a(a(b(b(B(x)))))))
b(a(a(a(a(A(x)))))) → B(x)
b(a(a(B(x)))) → b(a(a(A(x))))
b(a(a(a(a(A(x)))))) → b(B(x))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule A1(a(a(a(a(B(x0)))))) → B1(b(a(a(A(x0))))) at position [0] we obtained the following new rules:
A1(a(a(a(a(B(x0)))))) → B1(b(B(x0)))
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ QDPToSRSProof
↳ QTRS
↳ QTRS Reverse
↳ QTRS
↳ QTRS Reverse
↳ QTRS Reverse
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QTRS Reverse
↳ QTRS Reverse
Q DP problem:
The TRS P consists of the following rules:
B1(a(a(b(a(a(b(x0))))))) → B1(a(a(a(x0))))
A1(a(a(a(a(b(a(a(B(x0))))))))) → B1(b(a(a(a(b(a(a(a(B(x0))))))))))
A1(a(a(a(a(a(a(A(x0)))))))) → B1(b(a(a(a(b(B(x0)))))))
B1(a(a(b(a(a(b(x0))))))) → B1(a(a(a(b(a(a(a(b(x0)))))))))
B1(a(a(a(a(A(x)))))) → A1(a(a(b(b(B(x))))))
A1(a(a(a(a(b(x0)))))) → B1(b(b(a(a(a(b(x0)))))))
A1(a(a(a(a(b(a(a(B(x0))))))))) → B1(b(b(a(a(a(b(a(a(a(B(x0)))))))))))
B1(a(a(b(a(a(a(a(A(x0))))))))) → B1(a(a(a(b(a(a(a(b(b(B(x0)))))))))))
B1(a(a(b(a(a(B(x))))))) → A1(a(a(B(x))))
A1(a(a(a(a(b(x0)))))) → B1(b(a(a(a(b(x0))))))
B1(a(a(a(a(A(x)))))) → A1(a(a(b(B(x)))))
A1(a(a(a(a(B(x0)))))) → B1(b(B(x0)))
A1(a(a(a(a(b(x0)))))) → B1(b(x0))
B1(a(a(b(x)))) → A1(a(a(b(x))))
A1(a(a(x))) → B1(x)
B1(a(a(b(a(a(a(a(A(x0))))))))) → B1(a(a(a(b(a(a(a(b(B(x0))))))))))
B1(a(a(b(a(a(B(x))))))) → A1(a(a(b(a(a(a(B(x))))))))
A1(a(a(a(a(b(x0)))))) → B1(x0)
A1(a(a(a(a(b(a(a(B(x0))))))))) → B1(a(a(a(B(x0)))))
B1(a(a(b(a(a(b(a(a(B(x0)))))))))) → B1(a(a(a(a(a(a(B(x0))))))))
A1(a(a(a(a(a(a(A(x0)))))))) → B1(b(a(a(a(b(b(B(x0))))))))
B1(a(a(b(y0)))) → B1(b(b(b(b(y0)))))
B1(a(a(b(a(a(b(a(a(B(x0)))))))))) → B1(a(a(a(b(a(a(a(b(a(a(a(B(x0)))))))))))))
B1(a(a(b(a(a(B(x0))))))) → B1(a(a(a(b(a(a(A(x0))))))))
The TRS R consists of the following rules:
a(a(a(x))) → b(b(b(x)))
b(a(a(b(x)))) → x
b(a(a(b(x)))) → b(a(a(a(b(x)))))
a(a(A(x))) → B(x)
b(a(a(a(a(A(x)))))) → b(a(a(a(b(B(x))))))
b(a(a(b(a(a(B(x))))))) → b(a(a(a(b(a(a(a(B(x)))))))))
b(a(a(B(x)))) → b(b(b(b(B(x)))))
b(a(a(b(a(a(B(x))))))) → a(a(a(B(x))))
b(a(a(a(a(A(x)))))) → b(a(a(a(b(b(B(x)))))))
b(a(a(a(a(A(x)))))) → B(x)
b(a(a(B(x)))) → b(a(a(A(x))))
b(a(a(a(a(A(x)))))) → b(B(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 1 less node.
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ QDPToSRSProof
↳ QTRS
↳ QTRS Reverse
↳ QTRS
↳ QTRS Reverse
↳ QTRS Reverse
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QTRS Reverse
↳ QTRS Reverse
Q DP problem:
The TRS P consists of the following rules:
B1(a(a(b(a(a(B(x))))))) → A1(a(a(b(a(a(a(B(x))))))))
B1(a(a(b(a(a(a(a(A(x0))))))))) → B1(a(a(a(b(a(a(a(b(B(x0))))))))))
A1(a(a(a(a(b(x0)))))) → B1(x0)
B1(a(a(b(a(a(b(x0))))))) → B1(a(a(a(x0))))
A1(a(a(a(a(b(a(a(B(x0))))))))) → B1(b(a(a(a(b(a(a(a(B(x0))))))))))
A1(a(a(a(a(b(a(a(B(x0))))))))) → B1(a(a(a(B(x0)))))
A1(a(a(a(a(a(a(A(x0)))))))) → B1(b(a(a(a(b(b(B(x0))))))))
B1(a(a(b(a(a(b(a(a(B(x0)))))))))) → B1(a(a(a(a(a(a(B(x0))))))))
A1(a(a(a(a(a(a(A(x0)))))))) → B1(b(a(a(a(b(B(x0)))))))
B1(a(a(b(a(a(b(x0))))))) → B1(a(a(a(b(a(a(a(b(x0)))))))))
A1(a(a(a(a(b(a(a(B(x0))))))))) → B1(b(b(a(a(a(b(a(a(a(B(x0)))))))))))
A1(a(a(a(a(b(x0)))))) → B1(b(b(a(a(a(b(x0)))))))
B1(a(a(b(y0)))) → B1(b(b(b(b(y0)))))
B1(a(a(a(a(A(x)))))) → A1(a(a(b(b(B(x))))))
B1(a(a(b(a(a(a(a(A(x0))))))))) → B1(a(a(a(b(a(a(a(b(b(B(x0)))))))))))
B1(a(a(b(a(a(B(x))))))) → A1(a(a(B(x))))
A1(a(a(a(a(b(x0)))))) → B1(b(a(a(a(b(x0))))))
B1(a(a(b(a(a(b(a(a(B(x0)))))))))) → B1(a(a(a(b(a(a(a(b(a(a(a(B(x0)))))))))))))
B1(a(a(a(a(A(x)))))) → A1(a(a(b(B(x)))))
B1(a(a(b(a(a(B(x0))))))) → B1(a(a(a(b(a(a(A(x0))))))))
A1(a(a(a(a(b(x0)))))) → B1(b(x0))
B1(a(a(b(x)))) → A1(a(a(b(x))))
A1(a(a(x))) → B1(x)
The TRS R consists of the following rules:
a(a(a(x))) → b(b(b(x)))
b(a(a(b(x)))) → x
b(a(a(b(x)))) → b(a(a(a(b(x)))))
a(a(A(x))) → B(x)
b(a(a(a(a(A(x)))))) → b(a(a(a(b(B(x))))))
b(a(a(b(a(a(B(x))))))) → b(a(a(a(b(a(a(a(B(x)))))))))
b(a(a(B(x)))) → b(b(b(b(B(x)))))
b(a(a(b(a(a(B(x))))))) → a(a(a(B(x))))
b(a(a(a(a(A(x)))))) → b(a(a(a(b(b(B(x)))))))
b(a(a(a(a(A(x)))))) → B(x)
b(a(a(B(x)))) → b(a(a(A(x))))
b(a(a(a(a(A(x)))))) → b(B(x))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule A1(a(a(a(a(b(a(a(B(x0))))))))) → B1(a(a(a(B(x0))))) at position [0] we obtained the following new rules:
A1(a(a(a(a(b(a(a(B(y0))))))))) → B1(b(b(b(B(y0)))))
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ QDPToSRSProof
↳ QTRS
↳ QTRS Reverse
↳ QTRS
↳ QTRS Reverse
↳ QTRS Reverse
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QTRS Reverse
↳ QTRS Reverse
Q DP problem:
The TRS P consists of the following rules:
B1(a(a(b(a(a(a(a(A(x0))))))))) → B1(a(a(a(b(a(a(a(b(B(x0))))))))))
B1(a(a(b(a(a(B(x))))))) → A1(a(a(b(a(a(a(B(x))))))))
A1(a(a(a(a(b(x0)))))) → B1(x0)
B1(a(a(b(a(a(b(x0))))))) → B1(a(a(a(x0))))
A1(a(a(a(a(b(a(a(B(x0))))))))) → B1(b(a(a(a(b(a(a(a(B(x0))))))))))
B1(a(a(b(a(a(b(a(a(B(x0)))))))))) → B1(a(a(a(a(a(a(B(x0))))))))
A1(a(a(a(a(a(a(A(x0)))))))) → B1(b(a(a(a(b(b(B(x0))))))))
A1(a(a(a(a(a(a(A(x0)))))))) → B1(b(a(a(a(b(B(x0)))))))
B1(a(a(b(a(a(b(x0))))))) → B1(a(a(a(b(a(a(a(b(x0)))))))))
B1(a(a(a(a(A(x)))))) → A1(a(a(b(b(B(x))))))
B1(a(a(b(y0)))) → B1(b(b(b(b(y0)))))
A1(a(a(a(a(b(x0)))))) → B1(b(b(a(a(a(b(x0)))))))
A1(a(a(a(a(b(a(a(B(x0))))))))) → B1(b(b(a(a(a(b(a(a(a(B(x0)))))))))))
B1(a(a(b(a(a(a(a(A(x0))))))))) → B1(a(a(a(b(a(a(a(b(b(B(x0)))))))))))
B1(a(a(b(a(a(B(x))))))) → A1(a(a(B(x))))
A1(a(a(a(a(b(a(a(B(y0))))))))) → B1(b(b(b(B(y0)))))
A1(a(a(a(a(b(x0)))))) → B1(b(a(a(a(b(x0))))))
B1(a(a(b(a(a(b(a(a(B(x0)))))))))) → B1(a(a(a(b(a(a(a(b(a(a(a(B(x0)))))))))))))
B1(a(a(a(a(A(x)))))) → A1(a(a(b(B(x)))))
B1(a(a(b(a(a(B(x0))))))) → B1(a(a(a(b(a(a(A(x0))))))))
A1(a(a(a(a(b(x0)))))) → B1(b(x0))
B1(a(a(b(x)))) → A1(a(a(b(x))))
A1(a(a(x))) → B1(x)
The TRS R consists of the following rules:
a(a(a(x))) → b(b(b(x)))
b(a(a(b(x)))) → x
b(a(a(b(x)))) → b(a(a(a(b(x)))))
a(a(A(x))) → B(x)
b(a(a(a(a(A(x)))))) → b(a(a(a(b(B(x))))))
b(a(a(b(a(a(B(x))))))) → b(a(a(a(b(a(a(a(B(x)))))))))
b(a(a(B(x)))) → b(b(b(b(B(x)))))
b(a(a(b(a(a(B(x))))))) → a(a(a(B(x))))
b(a(a(a(a(A(x)))))) → b(a(a(a(b(b(B(x)))))))
b(a(a(a(a(A(x)))))) → B(x)
b(a(a(B(x)))) → b(a(a(A(x))))
b(a(a(a(a(A(x)))))) → b(B(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 1 less node.
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ QDPToSRSProof
↳ QTRS
↳ QTRS Reverse
↳ QTRS
↳ QTRS Reverse
↳ QTRS Reverse
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QTRS Reverse
↳ QTRS Reverse
Q DP problem:
The TRS P consists of the following rules:
B1(a(a(b(a(a(B(x))))))) → A1(a(a(b(a(a(a(B(x))))))))
B1(a(a(b(a(a(a(a(A(x0))))))))) → B1(a(a(a(b(a(a(a(b(B(x0))))))))))
A1(a(a(a(a(b(x0)))))) → B1(x0)
B1(a(a(b(a(a(b(x0))))))) → B1(a(a(a(x0))))
A1(a(a(a(a(b(a(a(B(x0))))))))) → B1(b(a(a(a(b(a(a(a(B(x0))))))))))
A1(a(a(a(a(a(a(A(x0)))))))) → B1(b(a(a(a(b(b(B(x0))))))))
B1(a(a(b(a(a(b(a(a(B(x0)))))))))) → B1(a(a(a(a(a(a(B(x0))))))))
A1(a(a(a(a(a(a(A(x0)))))))) → B1(b(a(a(a(b(B(x0)))))))
B1(a(a(b(a(a(b(x0))))))) → B1(a(a(a(b(a(a(a(b(x0)))))))))
A1(a(a(a(a(b(a(a(B(x0))))))))) → B1(b(b(a(a(a(b(a(a(a(B(x0)))))))))))
B1(a(a(b(y0)))) → B1(b(b(b(b(y0)))))
A1(a(a(a(a(b(x0)))))) → B1(b(b(a(a(a(b(x0)))))))
B1(a(a(a(a(A(x)))))) → A1(a(a(b(b(B(x))))))
B1(a(a(b(a(a(a(a(A(x0))))))))) → B1(a(a(a(b(a(a(a(b(b(B(x0)))))))))))
B1(a(a(b(a(a(B(x))))))) → A1(a(a(B(x))))
A1(a(a(a(a(b(x0)))))) → B1(b(a(a(a(b(x0))))))
B1(a(a(b(a(a(b(a(a(B(x0)))))))))) → B1(a(a(a(b(a(a(a(b(a(a(a(B(x0)))))))))))))
B1(a(a(a(a(A(x)))))) → A1(a(a(b(B(x)))))
B1(a(a(b(a(a(B(x0))))))) → B1(a(a(a(b(a(a(A(x0))))))))
A1(a(a(a(a(b(x0)))))) → B1(b(x0))
B1(a(a(b(x)))) → A1(a(a(b(x))))
A1(a(a(x))) → B1(x)
The TRS R consists of the following rules:
a(a(a(x))) → b(b(b(x)))
b(a(a(b(x)))) → x
b(a(a(b(x)))) → b(a(a(a(b(x)))))
a(a(A(x))) → B(x)
b(a(a(a(a(A(x)))))) → b(a(a(a(b(B(x))))))
b(a(a(b(a(a(B(x))))))) → b(a(a(a(b(a(a(a(B(x)))))))))
b(a(a(B(x)))) → b(b(b(b(B(x)))))
b(a(a(b(a(a(B(x))))))) → a(a(a(B(x))))
b(a(a(a(a(A(x)))))) → b(a(a(a(b(b(B(x)))))))
b(a(a(a(a(A(x)))))) → B(x)
b(a(a(B(x)))) → b(a(a(A(x))))
b(a(a(a(a(A(x)))))) → b(B(x))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule A1(a(a(a(a(a(a(A(x0)))))))) → B1(b(a(a(a(b(b(B(x0)))))))) at position [0] we obtained the following new rules:
A1(a(a(a(a(a(a(A(y0)))))))) → B1(b(b(b(b(b(b(B(y0))))))))
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ QDPToSRSProof
↳ QTRS
↳ QTRS Reverse
↳ QTRS
↳ QTRS Reverse
↳ QTRS Reverse
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QTRS Reverse
↳ QTRS Reverse
Q DP problem:
The TRS P consists of the following rules:
B1(a(a(b(a(a(a(a(A(x0))))))))) → B1(a(a(a(b(a(a(a(b(B(x0))))))))))
B1(a(a(b(a(a(B(x))))))) → A1(a(a(b(a(a(a(B(x))))))))
A1(a(a(a(a(b(x0)))))) → B1(x0)
B1(a(a(b(a(a(b(x0))))))) → B1(a(a(a(x0))))
A1(a(a(a(a(b(a(a(B(x0))))))))) → B1(b(a(a(a(b(a(a(a(B(x0))))))))))
B1(a(a(b(a(a(b(a(a(B(x0)))))))))) → B1(a(a(a(a(a(a(B(x0))))))))
A1(a(a(a(a(a(a(A(x0)))))))) → B1(b(a(a(a(b(B(x0)))))))
A1(a(a(a(a(a(a(A(y0)))))))) → B1(b(b(b(b(b(b(B(y0))))))))
B1(a(a(b(a(a(b(x0))))))) → B1(a(a(a(b(a(a(a(b(x0)))))))))
B1(a(a(a(a(A(x)))))) → A1(a(a(b(b(B(x))))))
A1(a(a(a(a(b(x0)))))) → B1(b(b(a(a(a(b(x0)))))))
B1(a(a(b(y0)))) → B1(b(b(b(b(y0)))))
A1(a(a(a(a(b(a(a(B(x0))))))))) → B1(b(b(a(a(a(b(a(a(a(B(x0)))))))))))
B1(a(a(b(a(a(a(a(A(x0))))))))) → B1(a(a(a(b(a(a(a(b(b(B(x0)))))))))))
B1(a(a(b(a(a(B(x))))))) → A1(a(a(B(x))))
A1(a(a(a(a(b(x0)))))) → B1(b(a(a(a(b(x0))))))
B1(a(a(b(a(a(b(a(a(B(x0)))))))))) → B1(a(a(a(b(a(a(a(b(a(a(a(B(x0)))))))))))))
B1(a(a(a(a(A(x)))))) → A1(a(a(b(B(x)))))
B1(a(a(b(a(a(B(x0))))))) → B1(a(a(a(b(a(a(A(x0))))))))
A1(a(a(a(a(b(x0)))))) → B1(b(x0))
B1(a(a(b(x)))) → A1(a(a(b(x))))
A1(a(a(x))) → B1(x)
The TRS R consists of the following rules:
a(a(a(x))) → b(b(b(x)))
b(a(a(b(x)))) → x
b(a(a(b(x)))) → b(a(a(a(b(x)))))
a(a(A(x))) → B(x)
b(a(a(a(a(A(x)))))) → b(a(a(a(b(B(x))))))
b(a(a(b(a(a(B(x))))))) → b(a(a(a(b(a(a(a(B(x)))))))))
b(a(a(B(x)))) → b(b(b(b(B(x)))))
b(a(a(b(a(a(B(x))))))) → a(a(a(B(x))))
b(a(a(a(a(A(x)))))) → b(a(a(a(b(b(B(x)))))))
b(a(a(a(a(A(x)))))) → B(x)
b(a(a(B(x)))) → b(a(a(A(x))))
b(a(a(a(a(A(x)))))) → b(B(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 1 less node.
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ QDPToSRSProof
↳ QTRS
↳ QTRS Reverse
↳ QTRS
↳ QTRS Reverse
↳ QTRS Reverse
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QTRS Reverse
↳ QTRS Reverse
Q DP problem:
The TRS P consists of the following rules:
B1(a(a(b(a(a(B(x))))))) → A1(a(a(b(a(a(a(B(x))))))))
B1(a(a(b(a(a(a(a(A(x0))))))))) → B1(a(a(a(b(a(a(a(b(B(x0))))))))))
A1(a(a(a(a(b(x0)))))) → B1(x0)
B1(a(a(b(a(a(b(x0))))))) → B1(a(a(a(x0))))
A1(a(a(a(a(b(a(a(B(x0))))))))) → B1(b(a(a(a(b(a(a(a(B(x0))))))))))
B1(a(a(b(a(a(b(a(a(B(x0)))))))))) → B1(a(a(a(a(a(a(B(x0))))))))
A1(a(a(a(a(a(a(A(x0)))))))) → B1(b(a(a(a(b(B(x0)))))))
B1(a(a(b(a(a(b(x0))))))) → B1(a(a(a(b(a(a(a(b(x0)))))))))
A1(a(a(a(a(b(a(a(B(x0))))))))) → B1(b(b(a(a(a(b(a(a(a(B(x0)))))))))))
A1(a(a(a(a(b(x0)))))) → B1(b(b(a(a(a(b(x0)))))))
B1(a(a(b(y0)))) → B1(b(b(b(b(y0)))))
B1(a(a(a(a(A(x)))))) → A1(a(a(b(b(B(x))))))
B1(a(a(b(a(a(a(a(A(x0))))))))) → B1(a(a(a(b(a(a(a(b(b(B(x0)))))))))))
B1(a(a(b(a(a(B(x))))))) → A1(a(a(B(x))))
A1(a(a(a(a(b(x0)))))) → B1(b(a(a(a(b(x0))))))
B1(a(a(b(a(a(b(a(a(B(x0)))))))))) → B1(a(a(a(b(a(a(a(b(a(a(a(B(x0)))))))))))))
B1(a(a(a(a(A(x)))))) → A1(a(a(b(B(x)))))
B1(a(a(b(a(a(B(x0))))))) → B1(a(a(a(b(a(a(A(x0))))))))
A1(a(a(a(a(b(x0)))))) → B1(b(x0))
B1(a(a(b(x)))) → A1(a(a(b(x))))
A1(a(a(x))) → B1(x)
The TRS R consists of the following rules:
a(a(a(x))) → b(b(b(x)))
b(a(a(b(x)))) → x
b(a(a(b(x)))) → b(a(a(a(b(x)))))
a(a(A(x))) → B(x)
b(a(a(a(a(A(x)))))) → b(a(a(a(b(B(x))))))
b(a(a(b(a(a(B(x))))))) → b(a(a(a(b(a(a(a(B(x)))))))))
b(a(a(B(x)))) → b(b(b(b(B(x)))))
b(a(a(b(a(a(B(x))))))) → a(a(a(B(x))))
b(a(a(a(a(A(x)))))) → b(a(a(a(b(b(B(x)))))))
b(a(a(a(a(A(x)))))) → B(x)
b(a(a(B(x)))) → b(a(a(A(x))))
b(a(a(a(a(A(x)))))) → b(B(x))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule A1(a(a(a(a(a(a(A(x0)))))))) → B1(b(a(a(a(b(B(x0))))))) at position [0] we obtained the following new rules:
A1(a(a(a(a(a(a(A(y0)))))))) → B1(b(b(b(b(b(B(y0)))))))
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ QDPToSRSProof
↳ QTRS
↳ QTRS Reverse
↳ QTRS
↳ QTRS Reverse
↳ QTRS Reverse
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QTRS Reverse
↳ QTRS Reverse
Q DP problem:
The TRS P consists of the following rules:
B1(a(a(b(a(a(a(a(A(x0))))))))) → B1(a(a(a(b(a(a(a(b(B(x0))))))))))
B1(a(a(b(a(a(B(x))))))) → A1(a(a(b(a(a(a(B(x))))))))
A1(a(a(a(a(b(x0)))))) → B1(x0)
B1(a(a(b(a(a(b(x0))))))) → B1(a(a(a(x0))))
A1(a(a(a(a(b(a(a(B(x0))))))))) → B1(b(a(a(a(b(a(a(a(B(x0))))))))))
A1(a(a(a(a(a(a(A(y0)))))))) → B1(b(b(b(b(b(B(y0)))))))
B1(a(a(b(a(a(b(a(a(B(x0)))))))))) → B1(a(a(a(a(a(a(B(x0))))))))
B1(a(a(b(a(a(b(x0))))))) → B1(a(a(a(b(a(a(a(b(x0)))))))))
B1(a(a(a(a(A(x)))))) → A1(a(a(b(b(B(x))))))
B1(a(a(b(y0)))) → B1(b(b(b(b(y0)))))
A1(a(a(a(a(b(x0)))))) → B1(b(b(a(a(a(b(x0)))))))
A1(a(a(a(a(b(a(a(B(x0))))))))) → B1(b(b(a(a(a(b(a(a(a(B(x0)))))))))))
B1(a(a(b(a(a(a(a(A(x0))))))))) → B1(a(a(a(b(a(a(a(b(b(B(x0)))))))))))
B1(a(a(b(a(a(B(x))))))) → A1(a(a(B(x))))
A1(a(a(a(a(b(x0)))))) → B1(b(a(a(a(b(x0))))))
B1(a(a(b(a(a(b(a(a(B(x0)))))))))) → B1(a(a(a(b(a(a(a(b(a(a(a(B(x0)))))))))))))
B1(a(a(a(a(A(x)))))) → A1(a(a(b(B(x)))))
B1(a(a(b(a(a(B(x0))))))) → B1(a(a(a(b(a(a(A(x0))))))))
A1(a(a(a(a(b(x0)))))) → B1(b(x0))
B1(a(a(b(x)))) → A1(a(a(b(x))))
A1(a(a(x))) → B1(x)
The TRS R consists of the following rules:
a(a(a(x))) → b(b(b(x)))
b(a(a(b(x)))) → x
b(a(a(b(x)))) → b(a(a(a(b(x)))))
a(a(A(x))) → B(x)
b(a(a(a(a(A(x)))))) → b(a(a(a(b(B(x))))))
b(a(a(b(a(a(B(x))))))) → b(a(a(a(b(a(a(a(B(x)))))))))
b(a(a(B(x)))) → b(b(b(b(B(x)))))
b(a(a(b(a(a(B(x))))))) → a(a(a(B(x))))
b(a(a(a(a(A(x)))))) → b(a(a(a(b(b(B(x)))))))
b(a(a(a(a(A(x)))))) → B(x)
b(a(a(B(x)))) → b(a(a(A(x))))
b(a(a(a(a(A(x)))))) → b(B(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 1 less node.
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ QDPToSRSProof
↳ QTRS
↳ QTRS Reverse
↳ QTRS
↳ QTRS Reverse
↳ QTRS Reverse
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QTRS Reverse
↳ QTRS Reverse
Q DP problem:
The TRS P consists of the following rules:
B1(a(a(b(a(a(B(x))))))) → A1(a(a(b(a(a(a(B(x))))))))
B1(a(a(b(a(a(a(a(A(x0))))))))) → B1(a(a(a(b(a(a(a(b(B(x0))))))))))
A1(a(a(a(a(b(x0)))))) → B1(x0)
B1(a(a(b(a(a(b(x0))))))) → B1(a(a(a(x0))))
A1(a(a(a(a(b(a(a(B(x0))))))))) → B1(b(a(a(a(b(a(a(a(B(x0))))))))))
B1(a(a(b(a(a(b(a(a(B(x0)))))))))) → B1(a(a(a(a(a(a(B(x0))))))))
B1(a(a(b(a(a(b(x0))))))) → B1(a(a(a(b(a(a(a(b(x0)))))))))
A1(a(a(a(a(b(a(a(B(x0))))))))) → B1(b(b(a(a(a(b(a(a(a(B(x0)))))))))))
B1(a(a(b(y0)))) → B1(b(b(b(b(y0)))))
A1(a(a(a(a(b(x0)))))) → B1(b(b(a(a(a(b(x0)))))))
B1(a(a(a(a(A(x)))))) → A1(a(a(b(b(B(x))))))
B1(a(a(b(a(a(a(a(A(x0))))))))) → B1(a(a(a(b(a(a(a(b(b(B(x0)))))))))))
B1(a(a(b(a(a(B(x))))))) → A1(a(a(B(x))))
A1(a(a(a(a(b(x0)))))) → B1(b(a(a(a(b(x0))))))
B1(a(a(b(a(a(b(a(a(B(x0)))))))))) → B1(a(a(a(b(a(a(a(b(a(a(a(B(x0)))))))))))))
B1(a(a(a(a(A(x)))))) → A1(a(a(b(B(x)))))
B1(a(a(b(a(a(B(x0))))))) → B1(a(a(a(b(a(a(A(x0))))))))
A1(a(a(a(a(b(x0)))))) → B1(b(x0))
B1(a(a(b(x)))) → A1(a(a(b(x))))
A1(a(a(x))) → B1(x)
The TRS R consists of the following rules:
a(a(a(x))) → b(b(b(x)))
b(a(a(b(x)))) → x
b(a(a(b(x)))) → b(a(a(a(b(x)))))
a(a(A(x))) → B(x)
b(a(a(a(a(A(x)))))) → b(a(a(a(b(B(x))))))
b(a(a(b(a(a(B(x))))))) → b(a(a(a(b(a(a(a(B(x)))))))))
b(a(a(B(x)))) → b(b(b(b(B(x)))))
b(a(a(b(a(a(B(x))))))) → a(a(a(B(x))))
b(a(a(a(a(A(x)))))) → b(a(a(a(b(b(B(x)))))))
b(a(a(a(a(A(x)))))) → B(x)
b(a(a(B(x)))) → b(a(a(A(x))))
b(a(a(a(a(A(x)))))) → b(B(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
a(a(a(x1))) → b(b(b(x1)))
b(a(a(b(x1)))) → x1
b(a(a(b(x1)))) → b(a(a(a(b(x1)))))
The set Q is empty.
We have obtained the following QTRS:
a(a(a(x))) → b(b(b(x)))
b(a(a(b(x)))) → x
b(a(a(b(x)))) → b(a(a(a(b(x)))))
The set Q is empty.
↳ QTRS
↳ DependencyPairsProof
↳ QTRS Reverse
↳ QTRS
↳ QTRS Reverse
Q restricted rewrite system:
The TRS R consists of the following rules:
a(a(a(x))) → b(b(b(x)))
b(a(a(b(x)))) → x
b(a(a(b(x)))) → b(a(a(a(b(x)))))
Q is empty.
We have reversed the following QTRS:
The set of rules R is
a(a(a(x1))) → b(b(b(x1)))
b(a(a(b(x1)))) → x1
b(a(a(b(x1)))) → b(a(a(a(b(x1)))))
The set Q is empty.
We have obtained the following QTRS:
a(a(a(x))) → b(b(b(x)))
b(a(a(b(x)))) → x
b(a(a(b(x)))) → b(a(a(a(b(x)))))
The set Q is empty.
↳ QTRS
↳ DependencyPairsProof
↳ QTRS Reverse
↳ QTRS Reverse
↳ QTRS
Q restricted rewrite system:
The TRS R consists of the following rules:
a(a(a(x))) → b(b(b(x)))
b(a(a(b(x)))) → x
b(a(a(b(x)))) → b(a(a(a(b(x)))))
Q is empty.