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(b(x1)) → b(b(c(a(c(x1)))))
b(x1) → x1
c(c(x1)) → a(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(b(x1)) → b(b(c(a(c(x1)))))
b(x1) → x1
c(c(x1)) → 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:
A(b(x1)) → C(a(c(x1)))
A(b(x1)) → B(c(a(c(x1))))
C(c(x1)) → A(x1)
A(b(x1)) → A(c(x1))
A(b(x1)) → B(b(c(a(c(x1)))))
A(b(x1)) → C(x1)
The TRS R consists of the following rules:
a(x1) → x1
a(b(x1)) → b(b(c(a(c(x1)))))
b(x1) → x1
c(c(x1)) → a(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(b(x1)) → C(a(c(x1)))
A(b(x1)) → B(c(a(c(x1))))
C(c(x1)) → A(x1)
A(b(x1)) → A(c(x1))
A(b(x1)) → B(b(c(a(c(x1)))))
A(b(x1)) → C(x1)
The TRS R consists of the following rules:
a(x1) → x1
a(b(x1)) → b(b(c(a(c(x1)))))
b(x1) → x1
c(c(x1)) → a(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(b(x1)) → C(a(c(x1)))
C(c(x1)) → A(x1)
A(b(x1)) → A(c(x1))
A(b(x1)) → C(x1)
The TRS R consists of the following rules:
a(x1) → x1
a(b(x1)) → b(b(c(a(c(x1)))))
b(x1) → x1
c(c(x1)) → a(x1)
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule A(b(x1)) → C(a(c(x1))) at position [0] we obtained the following new rules:
A(b(y0)) → C(c(y0))
A(b(c(x0))) → C(a(a(x0)))
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QTRS Reverse
↳ QTRS Reverse
Q DP problem:
The TRS P consists of the following rules:
A(b(y0)) → C(c(y0))
A(b(c(x0))) → C(a(a(x0)))
C(c(x1)) → A(x1)
A(b(x1)) → A(c(x1))
A(b(x1)) → C(x1)
The TRS R consists of the following rules:
a(x1) → x1
a(b(x1)) → b(b(c(a(c(x1)))))
b(x1) → x1
c(c(x1)) → a(x1)
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule A(b(x1)) → A(c(x1)) at position [0] we obtained the following new rules:
A(b(c(x0))) → A(a(x0))
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ QDPToSRSProof
↳ QTRS Reverse
↳ QTRS Reverse
Q DP problem:
The TRS P consists of the following rules:
A(b(y0)) → C(c(y0))
A(b(c(x0))) → C(a(a(x0)))
C(c(x1)) → A(x1)
A(b(x1)) → C(x1)
A(b(c(x0))) → A(a(x0))
The TRS R consists of the following rules:
a(x1) → x1
a(b(x1)) → b(b(c(a(c(x1)))))
b(x1) → x1
c(c(x1)) → 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
↳ DependencyGraphProof
↳ 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) → x1
a(b(x1)) → b(b(c(a(c(x1)))))
b(x1) → x1
c(c(x1)) → a(x1)
A(b(y0)) → C(c(y0))
A(b(c(x0))) → C(a(a(x0)))
C(c(x1)) → A(x1)
A(b(x1)) → C(x1)
A(b(c(x0))) → A(a(x0))
Q is empty.
We have reversed the following QTRS:
The set of rules R is
a(x1) → x1
a(b(x1)) → b(b(c(a(c(x1)))))
b(x1) → x1
c(c(x1)) → a(x1)
A(b(y0)) → C(c(y0))
A(b(c(x0))) → C(a(a(x0)))
C(c(x1)) → A(x1)
A(b(x1)) → C(x1)
A(b(c(x0))) → A(a(x0))
The set Q is empty.
We have obtained the following QTRS:
a(x) → x
b(a(x)) → c(a(c(b(b(x)))))
b(x) → x
c(c(x)) → a(x)
b(A(x)) → c(C(x))
c(b(A(x))) → a(a(C(x)))
c(C(x)) → A(x)
b(A(x)) → C(x)
c(b(A(x))) → a(A(x))
The set Q is empty.
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ QDPToSRSProof
↳ QTRS
↳ QTRS Reverse
↳ QTRS
↳ DependencyPairsProof
↳ QTRS Reverse
↳ QTRS Reverse
Q restricted rewrite system:
The TRS R consists of the following rules:
a(x) → x
b(a(x)) → c(a(c(b(b(x)))))
b(x) → x
c(c(x)) → a(x)
b(A(x)) → c(C(x))
c(b(A(x))) → a(a(C(x)))
c(C(x)) → A(x)
b(A(x)) → C(x)
c(b(A(x))) → a(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:
C1(b(A(x))) → A1(A(x))
B(a(x)) → B(b(x))
B(a(x)) → A1(c(b(b(x))))
B(a(x)) → C1(a(c(b(b(x)))))
B(A(x)) → C1(C(x))
C1(b(A(x))) → A1(C(x))
C1(c(x)) → A1(x)
C1(b(A(x))) → A1(a(C(x)))
B(a(x)) → B(x)
B(a(x)) → C1(b(b(x)))
The TRS R consists of the following rules:
a(x) → x
b(a(x)) → c(a(c(b(b(x)))))
b(x) → x
c(c(x)) → a(x)
b(A(x)) → c(C(x))
c(b(A(x))) → a(a(C(x)))
c(C(x)) → A(x)
b(A(x)) → C(x)
c(b(A(x))) → a(A(x))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ QDPToSRSProof
↳ QTRS
↳ QTRS Reverse
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QTRS Reverse
↳ QTRS Reverse
Q DP problem:
The TRS P consists of the following rules:
C1(b(A(x))) → A1(A(x))
B(a(x)) → B(b(x))
B(a(x)) → A1(c(b(b(x))))
B(a(x)) → C1(a(c(b(b(x)))))
B(A(x)) → C1(C(x))
C1(b(A(x))) → A1(C(x))
C1(c(x)) → A1(x)
C1(b(A(x))) → A1(a(C(x)))
B(a(x)) → B(x)
B(a(x)) → C1(b(b(x)))
The TRS R consists of the following rules:
a(x) → x
b(a(x)) → c(a(c(b(b(x)))))
b(x) → x
c(c(x)) → a(x)
b(A(x)) → c(C(x))
c(b(A(x))) → a(a(C(x)))
c(C(x)) → A(x)
b(A(x)) → C(x)
c(b(A(x))) → a(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 8 less nodes.
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ QDPToSRSProof
↳ QTRS
↳ QTRS Reverse
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QTRS Reverse
↳ QTRS Reverse
Q DP problem:
The TRS P consists of the following rules:
B(a(x)) → B(b(x))
B(a(x)) → B(x)
The TRS R consists of the following rules:
a(x) → x
b(a(x)) → c(a(c(b(b(x)))))
b(x) → x
c(c(x)) → a(x)
b(A(x)) → c(C(x))
c(b(A(x))) → a(a(C(x)))
c(C(x)) → A(x)
b(A(x)) → C(x)
c(b(A(x))) → a(A(x))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We have reversed the following QTRS:
The set of rules R is
a(x1) → x1
a(b(x1)) → b(b(c(a(c(x1)))))
b(x1) → x1
c(c(x1)) → a(x1)
The set Q is empty.
We have obtained the following QTRS:
a(x) → x
b(a(x)) → c(a(c(b(b(x)))))
b(x) → x
c(c(x)) → a(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
b(a(x)) → c(a(c(b(b(x)))))
b(x) → x
c(c(x)) → a(x)
Q is empty.
We have reversed the following QTRS:
The set of rules R is
a(x1) → x1
a(b(x1)) → b(b(c(a(c(x1)))))
b(x1) → x1
c(c(x1)) → a(x1)
The set Q is empty.
We have obtained the following QTRS:
a(x) → x
b(a(x)) → c(a(c(b(b(x)))))
b(x) → x
c(c(x)) → a(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
b(a(x)) → c(a(c(b(b(x)))))
b(x) → x
c(c(x)) → a(x)
Q is empty.