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(x1) → b(x1)
a(c(b(x1))) → c(a(b(a(c(x1)))))
b(c(x1)) → x1
Q is empty.
↳ QTRS
↳ DependencyPairsProof
↳ QTRS Reverse
↳ QTRS Reverse
Q restricted rewrite system:
The TRS R consists of the following rules:
a(x1) → b(x1)
a(c(b(x1))) → c(a(b(a(c(x1)))))
b(c(x1)) → 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(c(b(x1))) → B(a(c(x1)))
A(c(b(x1))) → A(b(a(c(x1))))
A(x1) → B(x1)
A(c(b(x1))) → A(c(x1))
The TRS R consists of the following rules:
a(x1) → b(x1)
a(c(b(x1))) → c(a(b(a(c(x1)))))
b(c(x1)) → x1
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QTRS Reverse
↳ QTRS Reverse
Q DP problem:
The TRS P consists of the following rules:
A(c(b(x1))) → B(a(c(x1)))
A(c(b(x1))) → A(b(a(c(x1))))
A(x1) → B(x1)
A(c(b(x1))) → A(c(x1))
The TRS R consists of the following rules:
a(x1) → b(x1)
a(c(b(x1))) → c(a(b(a(c(x1)))))
b(c(x1)) → x1
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
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QTRS Reverse
↳ QTRS Reverse
Q DP problem:
The TRS P consists of the following rules:
A(c(b(x1))) → A(b(a(c(x1))))
A(c(b(x1))) → A(c(x1))
The TRS R consists of the following rules:
a(x1) → b(x1)
a(c(b(x1))) → c(a(b(a(c(x1)))))
b(c(x1)) → x1
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule A(c(b(x1))) → A(b(a(c(x1)))) at position [0] we obtained the following new rules:
A(c(b(b(x0)))) → A(b(c(a(b(a(c(x0)))))))
A(c(b(y0))) → A(b(b(c(y0))))
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QTRS Reverse
↳ QTRS Reverse
Q DP problem:
The TRS P consists of the following rules:
A(c(b(b(x0)))) → A(b(c(a(b(a(c(x0)))))))
A(c(b(y0))) → A(b(b(c(y0))))
A(c(b(x1))) → A(c(x1))
The TRS R consists of the following rules:
a(x1) → b(x1)
a(c(b(x1))) → c(a(b(a(c(x1)))))
b(c(x1)) → x1
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule A(c(b(y0))) → A(b(b(c(y0)))) at position [0] we obtained the following new rules:
A(c(b(x0))) → A(b(x0))
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QTRS Reverse
↳ QTRS Reverse
Q DP problem:
The TRS P consists of the following rules:
A(c(b(x0))) → A(b(x0))
A(c(b(b(x0)))) → A(b(c(a(b(a(c(x0)))))))
A(c(b(x1))) → A(c(x1))
The TRS R consists of the following rules:
a(x1) → b(x1)
a(c(b(x1))) → c(a(b(a(c(x1)))))
b(c(x1)) → x1
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule A(c(b(x0))) → A(b(x0)) at position [0] we obtained the following new rules:
A(c(b(c(x0)))) → A(x0)
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ QDPToSRSProof
↳ QTRS Reverse
↳ QTRS Reverse
Q DP problem:
The TRS P consists of the following rules:
A(c(b(c(x0)))) → A(x0)
A(c(b(b(x0)))) → A(b(c(a(b(a(c(x0)))))))
A(c(b(x1))) → A(c(x1))
The TRS R consists of the following rules:
a(x1) → b(x1)
a(c(b(x1))) → c(a(b(a(c(x1)))))
b(c(x1)) → 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
↳ DependencyGraphProof
↳ 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(x1) → b(x1)
a(c(b(x1))) → c(a(b(a(c(x1)))))
b(c(x1)) → x1
A(c(b(c(x0)))) → A(x0)
A(c(b(b(x0)))) → A(b(c(a(b(a(c(x0)))))))
A(c(b(x1))) → A(c(x1))
Q is empty.
We have reversed the following QTRS:
The set of rules R is
a(x1) → b(x1)
a(c(b(x1))) → c(a(b(a(c(x1)))))
b(c(x1)) → x1
A(c(b(c(x0)))) → A(x0)
A(c(b(b(x0)))) → A(b(c(a(b(a(c(x0)))))))
A(c(b(x1))) → A(c(x1))
The set Q is empty.
We have obtained the following QTRS:
a(x) → b(x)
b(c(a(x))) → c(a(b(a(c(x)))))
c(b(x)) → x
c(b(c(A(x)))) → A(x)
b(b(c(A(x)))) → c(a(b(a(c(b(A(x)))))))
b(c(A(x))) → c(A(x))
The set Q is empty.
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ QDPToSRSProof
↳ QTRS
↳ QTRS Reverse
↳ QTRS
↳ DependencyPairsProof
↳ QTRS Reverse
↳ QTRS Reverse
↳ QTRS Reverse
↳ QTRS Reverse
Q restricted rewrite system:
The TRS R consists of the following rules:
a(x) → b(x)
b(c(a(x))) → c(a(b(a(c(x)))))
c(b(x)) → x
c(b(c(A(x)))) → A(x)
b(b(c(A(x)))) → c(a(b(a(c(b(A(x)))))))
b(c(A(x))) → c(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:
B(b(c(A(x)))) → B(A(x))
B(b(c(A(x)))) → B(a(c(b(A(x)))))
A1(x) → B(x)
B(b(c(A(x)))) → A1(c(b(A(x))))
B(b(c(A(x)))) → A1(b(a(c(b(A(x))))))
B(c(a(x))) → A1(c(x))
B(b(c(A(x)))) → C(b(A(x)))
B(c(a(x))) → C(a(b(a(c(x)))))
B(c(a(x))) → C(x)
B(c(a(x))) → A1(b(a(c(x))))
B(b(c(A(x)))) → C(a(b(a(c(b(A(x)))))))
B(c(a(x))) → B(a(c(x)))
The TRS R consists of the following rules:
a(x) → b(x)
b(c(a(x))) → c(a(b(a(c(x)))))
c(b(x)) → x
c(b(c(A(x)))) → A(x)
b(b(c(A(x)))) → c(a(b(a(c(b(A(x)))))))
b(c(A(x))) → c(A(x))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ QDPToSRSProof
↳ QTRS
↳ QTRS Reverse
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QTRS Reverse
↳ QTRS Reverse
↳ QTRS Reverse
↳ QTRS Reverse
Q DP problem:
The TRS P consists of the following rules:
B(b(c(A(x)))) → B(A(x))
B(b(c(A(x)))) → B(a(c(b(A(x)))))
A1(x) → B(x)
B(b(c(A(x)))) → A1(c(b(A(x))))
B(b(c(A(x)))) → A1(b(a(c(b(A(x))))))
B(c(a(x))) → A1(c(x))
B(b(c(A(x)))) → C(b(A(x)))
B(c(a(x))) → C(a(b(a(c(x)))))
B(c(a(x))) → C(x)
B(c(a(x))) → A1(b(a(c(x))))
B(b(c(A(x)))) → C(a(b(a(c(b(A(x)))))))
B(c(a(x))) → B(a(c(x)))
The TRS R consists of the following rules:
a(x) → b(x)
b(c(a(x))) → c(a(b(a(c(x)))))
c(b(x)) → x
c(b(c(A(x)))) → A(x)
b(b(c(A(x)))) → c(a(b(a(c(b(A(x)))))))
b(c(A(x))) → c(A(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 5 less nodes.
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ QDPToSRSProof
↳ QTRS
↳ QTRS Reverse
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QTRS Reverse
↳ QTRS Reverse
↳ QTRS Reverse
↳ QTRS Reverse
Q DP problem:
The TRS P consists of the following rules:
B(b(c(A(x)))) → B(a(c(b(A(x)))))
A1(x) → B(x)
B(b(c(A(x)))) → A1(c(b(A(x))))
B(b(c(A(x)))) → A1(b(a(c(b(A(x))))))
B(c(a(x))) → A1(c(x))
B(c(a(x))) → A1(b(a(c(x))))
B(c(a(x))) → B(a(c(x)))
The TRS R consists of the following rules:
a(x) → b(x)
b(c(a(x))) → c(a(b(a(c(x)))))
c(b(x)) → x
c(b(c(A(x)))) → A(x)
b(b(c(A(x)))) → c(a(b(a(c(b(A(x)))))))
b(c(A(x))) → c(A(x))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule B(c(a(x))) → B(a(c(x))) at position [0] we obtained the following new rules:
B(c(a(b(x0)))) → B(a(x0))
B(c(a(b(c(A(x0)))))) → B(a(A(x0)))
B(c(a(y0))) → B(b(c(y0)))
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ QDPToSRSProof
↳ QTRS
↳ QTRS Reverse
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QTRS Reverse
↳ QTRS Reverse
↳ QTRS Reverse
↳ QTRS Reverse
Q DP problem:
The TRS P consists of the following rules:
B(b(c(A(x)))) → B(a(c(b(A(x)))))
A1(x) → B(x)
B(c(a(b(x0)))) → B(a(x0))
B(c(a(b(c(A(x0)))))) → B(a(A(x0)))
B(c(a(y0))) → B(b(c(y0)))
B(b(c(A(x)))) → A1(c(b(A(x))))
B(c(a(x))) → A1(c(x))
B(b(c(A(x)))) → A1(b(a(c(b(A(x))))))
B(c(a(x))) → A1(b(a(c(x))))
The TRS R consists of the following rules:
a(x) → b(x)
b(c(a(x))) → c(a(b(a(c(x)))))
c(b(x)) → x
c(b(c(A(x)))) → A(x)
b(b(c(A(x)))) → c(a(b(a(c(b(A(x)))))))
b(c(A(x))) → c(A(x))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule B(b(c(A(x)))) → B(a(c(b(A(x))))) at position [0] we obtained the following new rules:
B(b(c(A(y0)))) → B(b(c(b(A(y0)))))
B(b(c(A(y0)))) → B(a(A(y0)))
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ QDPToSRSProof
↳ QTRS
↳ QTRS Reverse
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QTRS Reverse
↳ QTRS Reverse
↳ QTRS Reverse
↳ QTRS Reverse
Q DP problem:
The TRS P consists of the following rules:
B(b(c(A(y0)))) → B(b(c(b(A(y0)))))
A1(x) → B(x)
B(c(a(b(x0)))) → B(a(x0))
B(c(a(b(c(A(x0)))))) → B(a(A(x0)))
B(b(c(A(y0)))) → B(a(A(y0)))
B(b(c(A(x)))) → A1(c(b(A(x))))
B(c(a(y0))) → B(b(c(y0)))
B(b(c(A(x)))) → A1(b(a(c(b(A(x))))))
B(c(a(x))) → A1(c(x))
B(c(a(x))) → A1(b(a(c(x))))
The TRS R consists of the following rules:
a(x) → b(x)
b(c(a(x))) → c(a(b(a(c(x)))))
c(b(x)) → x
c(b(c(A(x)))) → A(x)
b(b(c(A(x)))) → c(a(b(a(c(b(A(x)))))))
b(c(A(x))) → c(A(x))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule B(c(a(b(c(A(x0)))))) → B(a(A(x0))) at position [0] we obtained the following new rules:
B(c(a(b(c(A(y0)))))) → B(b(A(y0)))
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ QDPToSRSProof
↳ QTRS
↳ QTRS Reverse
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QTRS Reverse
↳ QTRS Reverse
↳ QTRS Reverse
↳ QTRS Reverse
Q DP problem:
The TRS P consists of the following rules:
B(c(a(b(c(A(y0)))))) → B(b(A(y0)))
A1(x) → B(x)
B(b(c(A(y0)))) → B(b(c(b(A(y0)))))
B(c(a(b(x0)))) → B(a(x0))
B(c(a(y0))) → B(b(c(y0)))
B(b(c(A(x)))) → A1(c(b(A(x))))
B(b(c(A(y0)))) → B(a(A(y0)))
B(c(a(x))) → A1(c(x))
B(b(c(A(x)))) → A1(b(a(c(b(A(x))))))
B(c(a(x))) → A1(b(a(c(x))))
The TRS R consists of the following rules:
a(x) → b(x)
b(c(a(x))) → c(a(b(a(c(x)))))
c(b(x)) → x
c(b(c(A(x)))) → A(x)
b(b(c(A(x)))) → c(a(b(a(c(b(A(x)))))))
b(c(A(x))) → c(A(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
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ QDPToSRSProof
↳ QTRS
↳ QTRS Reverse
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QTRS Reverse
↳ QTRS Reverse
↳ QTRS Reverse
↳ QTRS Reverse
Q DP problem:
The TRS P consists of the following rules:
B(b(c(A(y0)))) → B(b(c(b(A(y0)))))
A1(x) → B(x)
B(c(a(b(x0)))) → B(a(x0))
B(b(c(A(y0)))) → B(a(A(y0)))
B(c(a(y0))) → B(b(c(y0)))
B(b(c(A(x)))) → A1(c(b(A(x))))
B(b(c(A(x)))) → A1(b(a(c(b(A(x))))))
B(c(a(x))) → A1(c(x))
B(c(a(x))) → A1(b(a(c(x))))
The TRS R consists of the following rules:
a(x) → b(x)
b(c(a(x))) → c(a(b(a(c(x)))))
c(b(x)) → x
c(b(c(A(x)))) → A(x)
b(b(c(A(x)))) → c(a(b(a(c(b(A(x)))))))
b(c(A(x))) → c(A(x))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule B(b(c(A(y0)))) → B(b(c(b(A(y0))))) at position [0] we obtained the following new rules:
B(b(c(A(y0)))) → B(b(A(y0)))
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ QDPToSRSProof
↳ QTRS
↳ QTRS Reverse
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QTRS Reverse
↳ QTRS Reverse
↳ QTRS Reverse
↳ QTRS Reverse
Q DP problem:
The TRS P consists of the following rules:
B(b(c(A(y0)))) → B(b(A(y0)))
A1(x) → B(x)
B(c(a(b(x0)))) → B(a(x0))
B(b(c(A(x)))) → A1(c(b(A(x))))
B(c(a(y0))) → B(b(c(y0)))
B(b(c(A(y0)))) → B(a(A(y0)))
B(c(a(x))) → A1(c(x))
B(b(c(A(x)))) → A1(b(a(c(b(A(x))))))
B(c(a(x))) → A1(b(a(c(x))))
The TRS R consists of the following rules:
a(x) → b(x)
b(c(a(x))) → c(a(b(a(c(x)))))
c(b(x)) → x
c(b(c(A(x)))) → A(x)
b(b(c(A(x)))) → c(a(b(a(c(b(A(x)))))))
b(c(A(x))) → c(A(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
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ QDPToSRSProof
↳ QTRS
↳ QTRS Reverse
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QTRS Reverse
↳ QTRS Reverse
↳ QTRS Reverse
↳ QTRS Reverse
Q DP problem:
The TRS P consists of the following rules:
A1(x) → B(x)
B(c(a(b(x0)))) → B(a(x0))
B(b(c(A(y0)))) → B(a(A(y0)))
B(c(a(y0))) → B(b(c(y0)))
B(b(c(A(x)))) → A1(c(b(A(x))))
B(b(c(A(x)))) → A1(b(a(c(b(A(x))))))
B(c(a(x))) → A1(c(x))
B(c(a(x))) → A1(b(a(c(x))))
The TRS R consists of the following rules:
a(x) → b(x)
b(c(a(x))) → c(a(b(a(c(x)))))
c(b(x)) → x
c(b(c(A(x)))) → A(x)
b(b(c(A(x)))) → c(a(b(a(c(b(A(x)))))))
b(c(A(x))) → c(A(x))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule B(b(c(A(y0)))) → B(a(A(y0))) at position [0] we obtained the following new rules:
B(b(c(A(y0)))) → B(b(A(y0)))
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ QDPToSRSProof
↳ QTRS
↳ QTRS Reverse
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QTRS Reverse
↳ QTRS Reverse
↳ QTRS Reverse
↳ QTRS Reverse
Q DP problem:
The TRS P consists of the following rules:
B(b(c(A(y0)))) → B(b(A(y0)))
A1(x) → B(x)
B(c(a(b(x0)))) → B(a(x0))
B(b(c(A(x)))) → A1(c(b(A(x))))
B(c(a(y0))) → B(b(c(y0)))
B(c(a(x))) → A1(c(x))
B(b(c(A(x)))) → A1(b(a(c(b(A(x))))))
B(c(a(x))) → A1(b(a(c(x))))
The TRS R consists of the following rules:
a(x) → b(x)
b(c(a(x))) → c(a(b(a(c(x)))))
c(b(x)) → x
c(b(c(A(x)))) → A(x)
b(b(c(A(x)))) → c(a(b(a(c(b(A(x)))))))
b(c(A(x))) → c(A(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
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ QDPToSRSProof
↳ QTRS
↳ QTRS Reverse
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QTRS Reverse
↳ QTRS Reverse
↳ QTRS Reverse
↳ QTRS Reverse
Q DP problem:
The TRS P consists of the following rules:
A1(x) → B(x)
B(c(a(b(x0)))) → B(a(x0))
B(c(a(y0))) → B(b(c(y0)))
B(b(c(A(x)))) → A1(c(b(A(x))))
B(b(c(A(x)))) → A1(b(a(c(b(A(x))))))
B(c(a(x))) → A1(c(x))
B(c(a(x))) → A1(b(a(c(x))))
The TRS R consists of the following rules:
a(x) → b(x)
b(c(a(x))) → c(a(b(a(c(x)))))
c(b(x)) → x
c(b(c(A(x)))) → A(x)
b(b(c(A(x)))) → c(a(b(a(c(b(A(x)))))))
b(c(A(x))) → c(A(x))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We have reversed the following QTRS:
The set of rules R is
a(x) → b(x)
b(c(a(x))) → c(a(b(a(c(x)))))
c(b(x)) → x
c(b(c(A(x)))) → A(x)
b(b(c(A(x)))) → c(a(b(a(c(b(A(x)))))))
b(c(A(x))) → c(A(x))
The set Q is empty.
We have obtained the following QTRS:
a(x) → b(x)
a(c(b(x))) → c(a(b(a(c(x)))))
b(c(x)) → x
A(c(b(c(x)))) → A(x)
A(c(b(b(x)))) → A(b(c(a(b(a(c(x)))))))
A(c(b(x))) → A(c(x))
The set Q is empty.
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ QDPToSRSProof
↳ QTRS
↳ QTRS Reverse
↳ QTRS
↳ DependencyPairsProof
↳ QTRS Reverse
↳ QTRS
↳ QTRS Reverse
↳ QTRS Reverse
↳ QTRS Reverse
Q restricted rewrite system:
The TRS R consists of the following rules:
a(x) → b(x)
a(c(b(x))) → c(a(b(a(c(x)))))
b(c(x)) → x
A(c(b(c(x)))) → A(x)
A(c(b(b(x)))) → A(b(c(a(b(a(c(x)))))))
A(c(b(x))) → A(c(x))
Q is empty.
We have reversed the following QTRS:
The set of rules R is
a(x) → b(x)
b(c(a(x))) → c(a(b(a(c(x)))))
c(b(x)) → x
c(b(c(A(x)))) → A(x)
b(b(c(A(x)))) → c(a(b(a(c(b(A(x)))))))
b(c(A(x))) → c(A(x))
The set Q is empty.
We have obtained the following QTRS:
a(x) → b(x)
a(c(b(x))) → c(a(b(a(c(x)))))
b(c(x)) → x
A(c(b(c(x)))) → A(x)
A(c(b(b(x)))) → A(b(c(a(b(a(c(x)))))))
A(c(b(x))) → A(c(x))
The set Q is empty.
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ QDPToSRSProof
↳ QTRS
↳ QTRS Reverse
↳ QTRS
↳ DependencyPairsProof
↳ QTRS Reverse
↳ QTRS Reverse
↳ QTRS
↳ QTRS Reverse
↳ QTRS Reverse
Q restricted rewrite system:
The TRS R consists of the following rules:
a(x) → b(x)
a(c(b(x))) → c(a(b(a(c(x)))))
b(c(x)) → x
A(c(b(c(x)))) → A(x)
A(c(b(b(x)))) → A(b(c(a(b(a(c(x)))))))
A(c(b(x))) → A(c(x))
Q is empty.
We have reversed the following QTRS:
The set of rules R is
a(x1) → b(x1)
a(c(b(x1))) → c(a(b(a(c(x1)))))
b(c(x1)) → x1
The set Q is empty.
We have obtained the following QTRS:
a(x) → b(x)
b(c(a(x))) → c(a(b(a(c(x)))))
c(b(x)) → 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(x) → b(x)
b(c(a(x))) → c(a(b(a(c(x)))))
c(b(x)) → x
Q is empty.
We have reversed the following QTRS:
The set of rules R is
a(x1) → b(x1)
a(c(b(x1))) → c(a(b(a(c(x1)))))
b(c(x1)) → x1
The set Q is empty.
We have obtained the following QTRS:
a(x) → b(x)
b(c(a(x))) → c(a(b(a(c(x)))))
c(b(x)) → 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(x) → b(x)
b(c(a(x))) → c(a(b(a(c(x)))))
c(b(x)) → x
Q is empty.