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(x1)) → b(x1)
b(a(c(x1))) → c(c(a(a(a(a(x1))))))
Q is empty.
↳ QTRS
↳ DependencyPairsProof
↳ QTRS Reverse
↳ QTRS Reverse
Q restricted rewrite system:
The TRS R consists of the following rules:
a(a(x1)) → b(x1)
b(a(c(x1))) → c(c(a(a(a(a(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:
B(a(c(x1))) → A(a(a(x1)))
B(a(c(x1))) → A(a(a(a(x1))))
B(a(c(x1))) → A(a(x1))
B(a(c(x1))) → A(x1)
A(a(x1)) → B(x1)
The TRS R consists of the following rules:
a(a(x1)) → b(x1)
b(a(c(x1))) → c(c(a(a(a(a(x1))))))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ ForwardInstantiation
↳ QTRS Reverse
↳ QTRS Reverse
Q DP problem:
The TRS P consists of the following rules:
B(a(c(x1))) → A(a(a(x1)))
B(a(c(x1))) → A(a(a(a(x1))))
B(a(c(x1))) → A(a(x1))
B(a(c(x1))) → A(x1)
A(a(x1)) → B(x1)
The TRS R consists of the following rules:
a(a(x1)) → b(x1)
b(a(c(x1))) → c(c(a(a(a(a(x1))))))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By forward instantiating [14] the rule A(a(x1)) → B(x1) we obtained the following new rules:
A(a(a(y_1))) → B(a(y_1))
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ ForwardInstantiation
↳ QTRS Reverse
↳ QTRS Reverse
Q DP problem:
The TRS P consists of the following rules:
B(a(c(x1))) → A(a(a(x1)))
B(a(c(x1))) → A(a(a(a(x1))))
B(a(c(x1))) → A(a(x1))
B(a(c(x1))) → A(x1)
A(a(a(y_1))) → B(a(y_1))
The TRS R consists of the following rules:
a(a(x1)) → b(x1)
b(a(c(x1))) → c(c(a(a(a(a(x1))))))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By forward instantiating [14] the rule B(a(c(x1))) → A(a(x1)) we obtained the following new rules:
B(a(c(a(y_0)))) → A(a(a(y_0)))
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ ForwardInstantiation
↳ QTRS Reverse
↳ QTRS Reverse
Q DP problem:
The TRS P consists of the following rules:
B(a(c(x1))) → A(a(a(x1)))
B(a(c(a(y_0)))) → A(a(a(y_0)))
B(a(c(x1))) → A(a(a(a(x1))))
B(a(c(x1))) → A(x1)
A(a(a(y_1))) → B(a(y_1))
The TRS R consists of the following rules:
a(a(x1)) → b(x1)
b(a(c(x1))) → c(c(a(a(a(a(x1))))))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By forward instantiating [14] the rule B(a(c(x1))) → A(x1) we obtained the following new rules:
B(a(c(a(a(y_0))))) → A(a(a(y_0)))
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ QDPToSRSProof
↳ QTRS Reverse
↳ QTRS Reverse
Q DP problem:
The TRS P consists of the following rules:
B(a(c(x1))) → A(a(a(x1)))
B(a(c(a(y_0)))) → A(a(a(y_0)))
B(a(c(x1))) → A(a(a(a(x1))))
B(a(c(a(a(y_0))))) → A(a(a(y_0)))
A(a(a(y_1))) → B(a(y_1))
The TRS R consists of the following rules:
a(a(x1)) → b(x1)
b(a(c(x1))) → c(c(a(a(a(a(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
↳ ForwardInstantiation
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ QDPToSRSProof
↳ QTRS
↳ QTRS Reverse
↳ QTRS Reverse
↳ QTRS Reverse
Q restricted rewrite system:
The TRS R consists of the following rules:
a(a(x1)) → b(x1)
b(a(c(x1))) → c(c(a(a(a(a(x1))))))
B(a(c(x1))) → A(a(a(x1)))
B(a(c(a(y_0)))) → A(a(a(y_0)))
B(a(c(x1))) → A(a(a(a(x1))))
B(a(c(a(a(y_0))))) → A(a(a(y_0)))
A(a(a(y_1))) → B(a(y_1))
Q is empty.
We have reversed the following QTRS:
The set of rules R is
a(a(x1)) → b(x1)
b(a(c(x1))) → c(c(a(a(a(a(x1))))))
B(a(c(x1))) → A(a(a(x1)))
B(a(c(a(y_0)))) → A(a(a(y_0)))
B(a(c(x1))) → A(a(a(a(x1))))
B(a(c(a(a(y_0))))) → A(a(a(y_0)))
A(a(a(y_1))) → B(a(y_1))
The set Q is empty.
We have obtained the following QTRS:
a(a(x)) → b(x)
c(a(b(x))) → a(a(a(a(c(c(x))))))
c(a(B(x))) → a(a(A(x)))
a(c(a(B(x)))) → a(a(A(x)))
c(a(B(x))) → a(a(a(A(x))))
a(a(c(a(B(x))))) → a(a(A(x)))
a(a(A(x))) → a(B(x))
The set Q is empty.
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ QDPToSRSProof
↳ QTRS
↳ QTRS Reverse
↳ QTRS
↳ QTRS Reverse
↳ DependencyPairsProof
↳ QTRS Reverse
↳ QTRS Reverse
↳ QTRS Reverse
Q restricted rewrite system:
The TRS R consists of the following rules:
a(a(x)) → b(x)
c(a(b(x))) → a(a(a(a(c(c(x))))))
c(a(B(x))) → a(a(A(x)))
a(c(a(B(x)))) → a(a(A(x)))
c(a(B(x))) → a(a(a(A(x))))
a(a(c(a(B(x))))) → a(a(A(x)))
a(a(A(x))) → a(B(x))
Q is empty.
We have reversed the following QTRS:
The set of rules R is
a(a(x)) → b(x)
c(a(b(x))) → a(a(a(a(c(c(x))))))
c(a(B(x))) → a(a(A(x)))
a(c(a(B(x)))) → a(a(A(x)))
c(a(B(x))) → a(a(a(A(x))))
a(a(c(a(B(x))))) → a(a(A(x)))
a(a(A(x))) → a(B(x))
The set Q is empty.
We have obtained the following QTRS:
a(a(x)) → b(x)
b(a(c(x))) → c(c(a(a(a(a(x))))))
B(a(c(x))) → A(a(a(x)))
B(a(c(a(x)))) → A(a(a(x)))
B(a(c(x))) → A(a(a(a(x))))
B(a(c(a(a(x))))) → A(a(a(x)))
A(a(a(x))) → B(a(x))
The set Q is empty.
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ QDPToSRSProof
↳ QTRS
↳ QTRS Reverse
↳ QTRS
↳ QTRS Reverse
↳ QTRS
↳ DependencyPairsProof
↳ QTRS Reverse
↳ QTRS Reverse
↳ QTRS Reverse
Q restricted rewrite system:
The TRS R consists of the following rules:
a(a(x)) → b(x)
b(a(c(x))) → c(c(a(a(a(a(x))))))
B(a(c(x))) → A(a(a(x)))
B(a(c(a(x)))) → A(a(a(x)))
B(a(c(x))) → A(a(a(a(x))))
B(a(c(a(a(x))))) → A(a(a(x)))
A(a(a(x))) → B(a(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:
A1(c(a(B(x)))) → A1(a(A(x)))
C(a(b(x))) → C(c(x))
C(a(b(x))) → A1(a(a(a(c(c(x))))))
C(a(b(x))) → A1(c(c(x)))
C(a(B(x))) → A1(a(a(A(x))))
A1(a(c(a(B(x))))) → A1(a(A(x)))
C(a(b(x))) → A1(a(c(c(x))))
A1(a(c(a(B(x))))) → A1(A(x))
A1(c(a(B(x)))) → A1(A(x))
C(a(b(x))) → A1(a(a(c(c(x)))))
C(a(B(x))) → A1(A(x))
C(a(B(x))) → A1(a(A(x)))
A1(a(A(x))) → A1(B(x))
C(a(b(x))) → C(x)
The TRS R consists of the following rules:
a(a(x)) → b(x)
c(a(b(x))) → a(a(a(a(c(c(x))))))
c(a(B(x))) → a(a(A(x)))
a(c(a(B(x)))) → a(a(A(x)))
c(a(B(x))) → a(a(a(A(x))))
a(a(c(a(B(x))))) → a(a(A(x)))
a(a(A(x))) → a(B(x))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ QDPToSRSProof
↳ QTRS
↳ QTRS Reverse
↳ QTRS
↳ QTRS Reverse
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QTRS Reverse
↳ QTRS Reverse
↳ QTRS Reverse
Q DP problem:
The TRS P consists of the following rules:
A1(c(a(B(x)))) → A1(a(A(x)))
C(a(b(x))) → C(c(x))
C(a(b(x))) → A1(a(a(a(c(c(x))))))
C(a(b(x))) → A1(c(c(x)))
C(a(B(x))) → A1(a(a(A(x))))
A1(a(c(a(B(x))))) → A1(a(A(x)))
C(a(b(x))) → A1(a(c(c(x))))
A1(a(c(a(B(x))))) → A1(A(x))
A1(c(a(B(x)))) → A1(A(x))
C(a(b(x))) → A1(a(a(c(c(x)))))
C(a(B(x))) → A1(A(x))
C(a(B(x))) → A1(a(A(x)))
A1(a(A(x))) → A1(B(x))
C(a(b(x))) → C(x)
The TRS R consists of the following rules:
a(a(x)) → b(x)
c(a(b(x))) → a(a(a(a(c(c(x))))))
c(a(B(x))) → a(a(A(x)))
a(c(a(B(x)))) → a(a(A(x)))
c(a(B(x))) → a(a(a(A(x))))
a(a(c(a(B(x))))) → a(a(A(x)))
a(a(A(x))) → a(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
↳ ForwardInstantiation
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ QDPToSRSProof
↳ QTRS
↳ QTRS Reverse
↳ QTRS
↳ QTRS Reverse
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QTRS Reverse
↳ QTRS Reverse
↳ QTRS Reverse
Q DP problem:
The TRS P consists of the following rules:
C(a(b(x))) → C(c(x))
C(a(b(x))) → C(x)
The TRS R consists of the following rules:
a(a(x)) → b(x)
c(a(b(x))) → a(a(a(a(c(c(x))))))
c(a(B(x))) → a(a(A(x)))
a(c(a(B(x)))) → a(a(A(x)))
c(a(B(x))) → a(a(a(A(x))))
a(a(c(a(B(x))))) → a(a(A(x)))
a(a(A(x))) → a(B(x))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule C(a(b(x))) → C(c(x)) at position [0] we obtained the following new rules:
C(a(b(a(B(x0))))) → C(a(a(a(A(x0)))))
C(a(b(a(B(x0))))) → C(a(a(A(x0))))
C(a(b(a(b(x0))))) → C(a(a(a(a(c(c(x0)))))))
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ QDPToSRSProof
↳ QTRS
↳ QTRS Reverse
↳ QTRS
↳ QTRS Reverse
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QTRS Reverse
↳ QTRS Reverse
↳ QTRS Reverse
Q DP problem:
The TRS P consists of the following rules:
C(a(b(a(B(x0))))) → C(a(a(a(A(x0)))))
C(a(b(a(b(x0))))) → C(a(a(a(a(c(c(x0)))))))
C(a(b(a(B(x0))))) → C(a(a(A(x0))))
C(a(b(x))) → C(x)
The TRS R consists of the following rules:
a(a(x)) → b(x)
c(a(b(x))) → a(a(a(a(c(c(x))))))
c(a(B(x))) → a(a(A(x)))
a(c(a(B(x)))) → a(a(A(x)))
c(a(B(x))) → a(a(a(A(x))))
a(a(c(a(B(x))))) → a(a(A(x)))
a(a(A(x))) → a(B(x))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule C(a(b(a(B(x0))))) → C(a(a(A(x0)))) at position [0] we obtained the following new rules:
C(a(b(a(B(y0))))) → C(b(A(y0)))
C(a(b(a(B(x0))))) → C(a(B(x0)))
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ QDPToSRSProof
↳ QTRS
↳ QTRS Reverse
↳ QTRS
↳ QTRS Reverse
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QTRS Reverse
↳ QTRS Reverse
↳ QTRS Reverse
Q DP problem:
The TRS P consists of the following rules:
C(a(b(a(B(y0))))) → C(b(A(y0)))
C(a(b(a(B(x0))))) → C(a(B(x0)))
C(a(b(a(B(x0))))) → C(a(a(a(A(x0)))))
C(a(b(x))) → C(x)
C(a(b(a(b(x0))))) → C(a(a(a(a(c(c(x0)))))))
The TRS R consists of the following rules:
a(a(x)) → b(x)
c(a(b(x))) → a(a(a(a(c(c(x))))))
c(a(B(x))) → a(a(A(x)))
a(c(a(B(x)))) → a(a(A(x)))
c(a(B(x))) → a(a(a(A(x))))
a(a(c(a(B(x))))) → a(a(A(x)))
a(a(A(x))) → a(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 2 less nodes.
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ QDPToSRSProof
↳ QTRS
↳ QTRS Reverse
↳ QTRS
↳ QTRS Reverse
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QTRS Reverse
↳ QTRS Reverse
↳ QTRS Reverse
Q DP problem:
The TRS P consists of the following rules:
C(a(b(a(B(x0))))) → C(a(a(a(A(x0)))))
C(a(b(a(b(x0))))) → C(a(a(a(a(c(c(x0)))))))
C(a(b(x))) → C(x)
The TRS R consists of the following rules:
a(a(x)) → b(x)
c(a(b(x))) → a(a(a(a(c(c(x))))))
c(a(B(x))) → a(a(A(x)))
a(c(a(B(x)))) → a(a(A(x)))
c(a(B(x))) → a(a(a(A(x))))
a(a(c(a(B(x))))) → a(a(A(x)))
a(a(A(x))) → a(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(x)) → b(x)
c(a(b(x))) → a(a(a(a(c(c(x))))))
c(a(B(x))) → a(a(A(x)))
a(c(a(B(x)))) → a(a(A(x)))
c(a(B(x))) → a(a(a(A(x))))
a(a(c(a(B(x))))) → a(a(A(x)))
a(a(A(x))) → a(B(x))
The set Q is empty.
We have obtained the following QTRS:
a(a(x)) → b(x)
b(a(c(x))) → c(c(a(a(a(a(x))))))
B(a(c(x))) → A(a(a(x)))
B(a(c(a(x)))) → A(a(a(x)))
B(a(c(x))) → A(a(a(a(x))))
B(a(c(a(a(x))))) → A(a(a(x)))
A(a(a(x))) → B(a(x))
The set Q is empty.
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ QDPToSRSProof
↳ QTRS
↳ QTRS Reverse
↳ QTRS
↳ QTRS Reverse
↳ DependencyPairsProof
↳ QTRS Reverse
↳ QTRS
↳ QTRS Reverse
↳ QTRS Reverse
Q restricted rewrite system:
The TRS R consists of the following rules:
a(a(x)) → b(x)
b(a(c(x))) → c(c(a(a(a(a(x))))))
B(a(c(x))) → A(a(a(x)))
B(a(c(a(x)))) → A(a(a(x)))
B(a(c(x))) → A(a(a(a(x))))
B(a(c(a(a(x))))) → A(a(a(x)))
A(a(a(x))) → B(a(x))
Q is empty.
We have reversed the following QTRS:
The set of rules R is
a(a(x1)) → b(x1)
b(a(c(x1))) → c(c(a(a(a(a(x1))))))
The set Q is empty.
We have obtained the following QTRS:
a(a(x)) → b(x)
c(a(b(x))) → a(a(a(a(c(c(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(x)) → b(x)
c(a(b(x))) → a(a(a(a(c(c(x))))))
Q is empty.
We have reversed the following QTRS:
The set of rules R is
a(a(x1)) → b(x1)
b(a(c(x1))) → c(c(a(a(a(a(x1))))))
The set Q is empty.
We have obtained the following QTRS:
a(a(x)) → b(x)
c(a(b(x))) → a(a(a(a(c(c(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(x)) → b(x)
c(a(b(x))) → a(a(a(a(c(c(x))))))
Q is empty.