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