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(a(x1)))) → b(b(c(c(x1))))
b(b(b(b(x1)))) → a(a(c(c(x1))))
c(c(c(c(x1)))) → b(a(a(a(x1))))
Q is empty.
↳ QTRS
↳ QTRS Reverse
↳ QTRS Reverse
↳ DependencyPairsProof
Q restricted rewrite system:
The TRS R consists of the following rules:
a(a(a(a(x1)))) → b(b(c(c(x1))))
b(b(b(b(x1)))) → a(a(c(c(x1))))
c(c(c(c(x1)))) → b(a(a(a(x1))))
Q is empty.
We have reversed the following QTRS:
The set of rules R is
a(a(a(a(x1)))) → b(b(c(c(x1))))
b(b(b(b(x1)))) → a(a(c(c(x1))))
c(c(c(c(x1)))) → b(a(a(a(x1))))
The set Q is empty.
We have obtained the following QTRS:
a(a(a(a(x)))) → c(c(b(b(x))))
b(b(b(b(x)))) → c(c(a(a(x))))
c(c(c(c(x)))) → a(a(a(b(x))))
The set Q is empty.
↳ QTRS
↳ QTRS Reverse
↳ QTRS
↳ QTRS Reverse
↳ DependencyPairsProof
Q restricted rewrite system:
The TRS R consists of the following rules:
a(a(a(a(x)))) → c(c(b(b(x))))
b(b(b(b(x)))) → c(c(a(a(x))))
c(c(c(c(x)))) → a(a(a(b(x))))
Q is empty.
We have reversed the following QTRS:
The set of rules R is
a(a(a(a(x1)))) → b(b(c(c(x1))))
b(b(b(b(x1)))) → a(a(c(c(x1))))
c(c(c(c(x1)))) → b(a(a(a(x1))))
The set Q is empty.
We have obtained the following QTRS:
a(a(a(a(x)))) → c(c(b(b(x))))
b(b(b(b(x)))) → c(c(a(a(x))))
c(c(c(c(x)))) → a(a(a(b(x))))
The set Q is empty.
↳ QTRS
↳ QTRS Reverse
↳ QTRS Reverse
↳ QTRS
↳ DependencyPairsProof
Q restricted rewrite system:
The TRS R consists of the following rules:
a(a(a(a(x)))) → c(c(b(b(x))))
b(b(b(b(x)))) → c(c(a(a(x))))
c(c(c(c(x)))) → a(a(a(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(c(c(c(x1)))) → A(a(x1))
A(a(a(a(x1)))) → C(c(x1))
A(a(a(a(x1)))) → B(c(c(x1)))
C(c(c(c(x1)))) → A(a(a(x1)))
B(b(b(b(x1)))) → A(a(c(c(x1))))
B(b(b(b(x1)))) → C(c(x1))
A(a(a(a(x1)))) → B(b(c(c(x1))))
A(a(a(a(x1)))) → C(x1)
C(c(c(c(x1)))) → B(a(a(a(x1))))
C(c(c(c(x1)))) → A(x1)
B(b(b(b(x1)))) → C(x1)
B(b(b(b(x1)))) → A(c(c(x1)))
The TRS R consists of the following rules:
a(a(a(a(x1)))) → b(b(c(c(x1))))
b(b(b(b(x1)))) → a(a(c(c(x1))))
c(c(c(c(x1)))) → b(a(a(a(x1))))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
↳ QTRS
↳ QTRS Reverse
↳ QTRS Reverse
↳ DependencyPairsProof
↳ QDP
↳ RuleRemovalProof
Q DP problem:
The TRS P consists of the following rules:
C(c(c(c(x1)))) → A(a(x1))
A(a(a(a(x1)))) → C(c(x1))
A(a(a(a(x1)))) → B(c(c(x1)))
C(c(c(c(x1)))) → A(a(a(x1)))
B(b(b(b(x1)))) → A(a(c(c(x1))))
B(b(b(b(x1)))) → C(c(x1))
A(a(a(a(x1)))) → B(b(c(c(x1))))
A(a(a(a(x1)))) → C(x1)
C(c(c(c(x1)))) → B(a(a(a(x1))))
C(c(c(c(x1)))) → A(x1)
B(b(b(b(x1)))) → C(x1)
B(b(b(b(x1)))) → A(c(c(x1)))
The TRS R consists of the following rules:
a(a(a(a(x1)))) → b(b(c(c(x1))))
b(b(b(b(x1)))) → a(a(c(c(x1))))
c(c(c(c(x1)))) → b(a(a(a(x1))))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By using the rule removal processor [15] with the following polynomial ordering [25], at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented.
Strictly oriented dependency pairs:
C(c(c(c(x1)))) → A(a(x1))
A(a(a(a(x1)))) → C(c(x1))
A(a(a(a(x1)))) → B(c(c(x1)))
C(c(c(c(x1)))) → A(a(a(x1)))
B(b(b(b(x1)))) → C(c(x1))
A(a(a(a(x1)))) → C(x1)
C(c(c(c(x1)))) → B(a(a(a(x1))))
C(c(c(c(x1)))) → A(x1)
B(b(b(b(x1)))) → C(x1)
B(b(b(b(x1)))) → A(c(c(x1)))
Used ordering: POLO with Polynomial interpretation [25]:
POL(A(x1)) = 2·x1
POL(B(x1)) = 2·x1
POL(C(x1)) = 2 + 2·x1
POL(a(x1)) = 2 + 2·x1
POL(b(x1)) = 2 + 2·x1
POL(c(x1)) = 2 + 2·x1
↳ QTRS
↳ QTRS Reverse
↳ QTRS Reverse
↳ DependencyPairsProof
↳ QDP
↳ RuleRemovalProof
↳ QDP
↳ Narrowing
Q DP problem:
The TRS P consists of the following rules:
B(b(b(b(x1)))) → A(a(c(c(x1))))
A(a(a(a(x1)))) → B(b(c(c(x1))))
The TRS R consists of the following rules:
a(a(a(a(x1)))) → b(b(c(c(x1))))
b(b(b(b(x1)))) → a(a(c(c(x1))))
c(c(c(c(x1)))) → b(a(a(a(x1))))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule B(b(b(b(x1)))) → A(a(c(c(x1)))) at position [0] we obtained the following new rules:
B(b(b(b(c(c(x0)))))) → A(a(b(a(a(a(x0))))))
B(b(b(b(c(c(c(x0))))))) → A(a(c(b(a(a(a(x0)))))))
↳ QTRS
↳ QTRS Reverse
↳ QTRS Reverse
↳ DependencyPairsProof
↳ QDP
↳ RuleRemovalProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
Q DP problem:
The TRS P consists of the following rules:
B(b(b(b(c(c(c(x0))))))) → A(a(c(b(a(a(a(x0)))))))
A(a(a(a(x1)))) → B(b(c(c(x1))))
B(b(b(b(c(c(x0)))))) → A(a(b(a(a(a(x0))))))
The TRS R consists of the following rules:
a(a(a(a(x1)))) → b(b(c(c(x1))))
b(b(b(b(x1)))) → a(a(c(c(x1))))
c(c(c(c(x1)))) → b(a(a(a(x1))))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule A(a(a(a(x1)))) → B(b(c(c(x1)))) at position [0] we obtained the following new rules:
A(a(a(a(c(c(x0)))))) → B(b(b(a(a(a(x0))))))
A(a(a(a(c(c(c(x0))))))) → B(b(c(b(a(a(a(x0)))))))
↳ QTRS
↳ QTRS Reverse
↳ QTRS Reverse
↳ DependencyPairsProof
↳ QDP
↳ RuleRemovalProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ QDPToSRSProof
Q DP problem:
The TRS P consists of the following rules:
B(b(b(b(c(c(c(x0))))))) → A(a(c(b(a(a(a(x0)))))))
B(b(b(b(c(c(x0)))))) → A(a(b(a(a(a(x0))))))
A(a(a(a(c(c(x0)))))) → B(b(b(a(a(a(x0))))))
A(a(a(a(c(c(c(x0))))))) → B(b(c(b(a(a(a(x0)))))))
The TRS R consists of the following rules:
a(a(a(a(x1)))) → b(b(c(c(x1))))
b(b(b(b(x1)))) → a(a(c(c(x1))))
c(c(c(c(x1)))) → b(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
↳ QTRS Reverse
↳ QTRS Reverse
↳ DependencyPairsProof
↳ QDP
↳ RuleRemovalProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ QDPToSRSProof
↳ QTRS
↳ QTRS Reverse
Q restricted rewrite system:
The TRS R consists of the following rules:
a(a(a(a(x1)))) → b(b(c(c(x1))))
b(b(b(b(x1)))) → a(a(c(c(x1))))
c(c(c(c(x1)))) → b(a(a(a(x1))))
B(b(b(b(c(c(c(x0))))))) → A(a(c(b(a(a(a(x0)))))))
B(b(b(b(c(c(x0)))))) → A(a(b(a(a(a(x0))))))
A(a(a(a(c(c(x0)))))) → B(b(b(a(a(a(x0))))))
A(a(a(a(c(c(c(x0))))))) → B(b(c(b(a(a(a(x0)))))))
Q is empty.
We have reversed the following QTRS:
The set of rules R is
a(a(a(a(x1)))) → b(b(c(c(x1))))
b(b(b(b(x1)))) → a(a(c(c(x1))))
c(c(c(c(x1)))) → b(a(a(a(x1))))
B(b(b(b(c(c(c(x0))))))) → A(a(c(b(a(a(a(x0)))))))
B(b(b(b(c(c(x0)))))) → A(a(b(a(a(a(x0))))))
A(a(a(a(c(c(x0)))))) → B(b(b(a(a(a(x0))))))
A(a(a(a(c(c(c(x0))))))) → B(b(c(b(a(a(a(x0)))))))
The set Q is empty.
We have obtained the following QTRS:
a(a(a(a(x)))) → c(c(b(b(x))))
b(b(b(b(x)))) → c(c(a(a(x))))
c(c(c(c(x)))) → a(a(a(b(x))))
c(c(c(b(b(b(B(x))))))) → a(a(a(b(c(a(A(x)))))))
c(c(b(b(b(B(x)))))) → a(a(a(b(a(A(x))))))
c(c(a(a(a(A(x)))))) → a(a(a(b(b(B(x))))))
c(c(c(a(a(a(A(x))))))) → a(a(a(b(c(b(B(x)))))))
The set Q is empty.
↳ QTRS
↳ QTRS Reverse
↳ QTRS Reverse
↳ DependencyPairsProof
↳ QDP
↳ RuleRemovalProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ QDPToSRSProof
↳ QTRS
↳ QTRS Reverse
↳ QTRS
↳ QTRS Reverse
↳ QTRS Reverse
↳ DependencyPairsProof
Q restricted rewrite system:
The TRS R consists of the following rules:
a(a(a(a(x)))) → c(c(b(b(x))))
b(b(b(b(x)))) → c(c(a(a(x))))
c(c(c(c(x)))) → a(a(a(b(x))))
c(c(c(b(b(b(B(x))))))) → a(a(a(b(c(a(A(x)))))))
c(c(b(b(b(B(x)))))) → a(a(a(b(a(A(x))))))
c(c(a(a(a(A(x)))))) → a(a(a(b(b(B(x))))))
c(c(c(a(a(a(A(x))))))) → a(a(a(b(c(b(B(x)))))))
Q is empty.
We have reversed the following QTRS:
The set of rules R is
a(a(a(a(x)))) → c(c(b(b(x))))
b(b(b(b(x)))) → c(c(a(a(x))))
c(c(c(c(x)))) → a(a(a(b(x))))
c(c(c(b(b(b(B(x))))))) → a(a(a(b(c(a(A(x)))))))
c(c(b(b(b(B(x)))))) → a(a(a(b(a(A(x))))))
c(c(a(a(a(A(x)))))) → a(a(a(b(b(B(x))))))
c(c(c(a(a(a(A(x))))))) → a(a(a(b(c(b(B(x)))))))
The set Q is empty.
We have obtained the following QTRS:
a(a(a(a(x)))) → b(b(c(c(x))))
b(b(b(b(x)))) → a(a(c(c(x))))
c(c(c(c(x)))) → b(a(a(a(x))))
B(b(b(b(c(c(c(x))))))) → A(a(c(b(a(a(a(x)))))))
B(b(b(b(c(c(x)))))) → A(a(b(a(a(a(x))))))
A(a(a(a(c(c(x)))))) → B(b(b(a(a(a(x))))))
A(a(a(a(c(c(c(x))))))) → B(b(c(b(a(a(a(x)))))))
The set Q is empty.
↳ QTRS
↳ QTRS Reverse
↳ QTRS Reverse
↳ DependencyPairsProof
↳ QDP
↳ RuleRemovalProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ QDPToSRSProof
↳ QTRS
↳ QTRS Reverse
↳ QTRS
↳ QTRS Reverse
↳ QTRS
↳ QTRS Reverse
↳ DependencyPairsProof
Q restricted rewrite system:
The TRS R consists of the following rules:
a(a(a(a(x)))) → b(b(c(c(x))))
b(b(b(b(x)))) → a(a(c(c(x))))
c(c(c(c(x)))) → b(a(a(a(x))))
B(b(b(b(c(c(c(x))))))) → A(a(c(b(a(a(a(x)))))))
B(b(b(b(c(c(x)))))) → A(a(b(a(a(a(x))))))
A(a(a(a(c(c(x)))))) → B(b(b(a(a(a(x))))))
A(a(a(a(c(c(c(x))))))) → B(b(c(b(a(a(a(x)))))))
Q is empty.
We have reversed the following QTRS:
The set of rules R is
a(a(a(a(x)))) → c(c(b(b(x))))
b(b(b(b(x)))) → c(c(a(a(x))))
c(c(c(c(x)))) → a(a(a(b(x))))
c(c(c(b(b(b(B(x))))))) → a(a(a(b(c(a(A(x)))))))
c(c(b(b(b(B(x)))))) → a(a(a(b(a(A(x))))))
c(c(a(a(a(A(x)))))) → a(a(a(b(b(B(x))))))
c(c(c(a(a(a(A(x))))))) → a(a(a(b(c(b(B(x)))))))
The set Q is empty.
We have obtained the following QTRS:
a(a(a(a(x)))) → b(b(c(c(x))))
b(b(b(b(x)))) → a(a(c(c(x))))
c(c(c(c(x)))) → b(a(a(a(x))))
B(b(b(b(c(c(c(x))))))) → A(a(c(b(a(a(a(x)))))))
B(b(b(b(c(c(x)))))) → A(a(b(a(a(a(x))))))
A(a(a(a(c(c(x)))))) → B(b(b(a(a(a(x))))))
A(a(a(a(c(c(c(x))))))) → B(b(c(b(a(a(a(x)))))))
The set Q is empty.
↳ QTRS
↳ QTRS Reverse
↳ QTRS Reverse
↳ DependencyPairsProof
↳ QDP
↳ RuleRemovalProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ QDPToSRSProof
↳ QTRS
↳ QTRS Reverse
↳ QTRS
↳ QTRS Reverse
↳ QTRS Reverse
↳ QTRS
↳ DependencyPairsProof
Q restricted rewrite system:
The TRS R consists of the following rules:
a(a(a(a(x)))) → b(b(c(c(x))))
b(b(b(b(x)))) → a(a(c(c(x))))
c(c(c(c(x)))) → b(a(a(a(x))))
B(b(b(b(c(c(c(x))))))) → A(a(c(b(a(a(a(x)))))))
B(b(b(b(c(c(x)))))) → A(a(b(a(a(a(x))))))
A(a(a(a(c(c(x)))))) → B(b(b(a(a(a(x))))))
A(a(a(a(c(c(c(x))))))) → B(b(c(b(a(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:
C(c(c(a(a(a(A(x))))))) → A1(a(a(b(c(b(B(x)))))))
C(c(a(a(a(A(x)))))) → A1(b(b(B(x))))
A1(a(a(a(x)))) → B1(b(x))
A1(a(a(a(x)))) → C(b(b(x)))
C(c(c(b(b(b(B(x))))))) → A1(a(a(b(c(a(A(x)))))))
C(c(c(c(x)))) → A1(b(x))
C(c(b(b(b(B(x)))))) → A1(a(b(a(A(x)))))
C(c(c(c(x)))) → A1(a(a(b(x))))
C(c(c(b(b(b(B(x))))))) → A1(b(c(a(A(x)))))
C(c(c(a(a(a(A(x))))))) → A1(a(b(c(b(B(x))))))
C(c(b(b(b(B(x)))))) → A1(a(a(b(a(A(x))))))
C(c(c(a(a(a(A(x))))))) → C(b(B(x)))
C(c(a(a(a(A(x)))))) → A1(a(a(b(b(B(x))))))
B1(b(b(b(x)))) → C(a(a(x)))
C(c(c(a(a(a(A(x))))))) → A1(b(c(b(B(x)))))
C(c(b(b(b(B(x)))))) → A1(A(x))
C(c(b(b(b(B(x)))))) → A1(b(a(A(x))))
B1(b(b(b(x)))) → A1(x)
B1(b(b(b(x)))) → A1(a(x))
C(c(a(a(a(A(x)))))) → A1(a(b(b(B(x)))))
C(c(c(c(x)))) → B1(x)
C(c(c(a(a(a(A(x))))))) → B1(c(b(B(x))))
C(c(b(b(b(B(x)))))) → B1(a(A(x)))
C(c(c(c(x)))) → A1(a(b(x)))
B1(b(b(b(x)))) → C(c(a(a(x))))
C(c(a(a(a(A(x)))))) → B1(B(x))
C(c(c(b(b(b(B(x))))))) → C(a(A(x)))
A1(a(a(a(x)))) → C(c(b(b(x))))
C(c(c(b(b(b(B(x))))))) → A1(A(x))
C(c(c(b(b(b(B(x))))))) → B1(c(a(A(x))))
C(c(a(a(a(A(x)))))) → B1(b(B(x)))
C(c(c(b(b(b(B(x))))))) → A1(a(b(c(a(A(x))))))
A1(a(a(a(x)))) → B1(x)
C(c(c(a(a(a(A(x))))))) → B1(B(x))
The TRS R consists of the following rules:
a(a(a(a(x)))) → c(c(b(b(x))))
b(b(b(b(x)))) → c(c(a(a(x))))
c(c(c(c(x)))) → a(a(a(b(x))))
c(c(c(b(b(b(B(x))))))) → a(a(a(b(c(a(A(x)))))))
c(c(b(b(b(B(x)))))) → a(a(a(b(a(A(x))))))
c(c(a(a(a(A(x)))))) → a(a(a(b(b(B(x))))))
c(c(c(a(a(a(A(x))))))) → a(a(a(b(c(b(B(x)))))))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
↳ QTRS
↳ QTRS Reverse
↳ QTRS Reverse
↳ DependencyPairsProof
↳ QDP
↳ RuleRemovalProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ QDPToSRSProof
↳ QTRS
↳ QTRS Reverse
↳ QTRS
↳ QTRS Reverse
↳ QTRS Reverse
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
Q DP problem:
The TRS P consists of the following rules:
C(c(c(a(a(a(A(x))))))) → A1(a(a(b(c(b(B(x)))))))
C(c(a(a(a(A(x)))))) → A1(b(b(B(x))))
A1(a(a(a(x)))) → B1(b(x))
A1(a(a(a(x)))) → C(b(b(x)))
C(c(c(b(b(b(B(x))))))) → A1(a(a(b(c(a(A(x)))))))
C(c(c(c(x)))) → A1(b(x))
C(c(b(b(b(B(x)))))) → A1(a(b(a(A(x)))))
C(c(c(c(x)))) → A1(a(a(b(x))))
C(c(c(b(b(b(B(x))))))) → A1(b(c(a(A(x)))))
C(c(c(a(a(a(A(x))))))) → A1(a(b(c(b(B(x))))))
C(c(b(b(b(B(x)))))) → A1(a(a(b(a(A(x))))))
C(c(c(a(a(a(A(x))))))) → C(b(B(x)))
C(c(a(a(a(A(x)))))) → A1(a(a(b(b(B(x))))))
B1(b(b(b(x)))) → C(a(a(x)))
C(c(c(a(a(a(A(x))))))) → A1(b(c(b(B(x)))))
C(c(b(b(b(B(x)))))) → A1(A(x))
C(c(b(b(b(B(x)))))) → A1(b(a(A(x))))
B1(b(b(b(x)))) → A1(x)
B1(b(b(b(x)))) → A1(a(x))
C(c(a(a(a(A(x)))))) → A1(a(b(b(B(x)))))
C(c(c(c(x)))) → B1(x)
C(c(c(a(a(a(A(x))))))) → B1(c(b(B(x))))
C(c(b(b(b(B(x)))))) → B1(a(A(x)))
C(c(c(c(x)))) → A1(a(b(x)))
B1(b(b(b(x)))) → C(c(a(a(x))))
C(c(a(a(a(A(x)))))) → B1(B(x))
C(c(c(b(b(b(B(x))))))) → C(a(A(x)))
A1(a(a(a(x)))) → C(c(b(b(x))))
C(c(c(b(b(b(B(x))))))) → A1(A(x))
C(c(c(b(b(b(B(x))))))) → B1(c(a(A(x))))
C(c(a(a(a(A(x)))))) → B1(b(B(x)))
C(c(c(b(b(b(B(x))))))) → A1(a(b(c(a(A(x))))))
A1(a(a(a(x)))) → B1(x)
C(c(c(a(a(a(A(x))))))) → B1(B(x))
The TRS R consists of the following rules:
a(a(a(a(x)))) → c(c(b(b(x))))
b(b(b(b(x)))) → c(c(a(a(x))))
c(c(c(c(x)))) → a(a(a(b(x))))
c(c(c(b(b(b(B(x))))))) → a(a(a(b(c(a(A(x)))))))
c(c(b(b(b(B(x)))))) → a(a(a(b(a(A(x))))))
c(c(a(a(a(A(x)))))) → a(a(a(b(b(B(x))))))
c(c(c(a(a(a(A(x))))))) → a(a(a(b(c(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 22 less nodes.
↳ QTRS
↳ QTRS Reverse
↳ QTRS Reverse
↳ DependencyPairsProof
↳ QDP
↳ RuleRemovalProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ QDPToSRSProof
↳ QTRS
↳ QTRS Reverse
↳ QTRS
↳ QTRS Reverse
↳ QTRS Reverse
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ RuleRemovalProof
Q DP problem:
The TRS P consists of the following rules:
A1(a(a(a(x)))) → B1(b(x))
A1(a(a(a(x)))) → C(b(b(x)))
C(c(c(c(x)))) → A1(a(b(x)))
A1(a(a(a(x)))) → C(c(b(b(x))))
C(c(c(c(x)))) → A1(b(x))
C(c(c(c(x)))) → A1(a(a(b(x))))
B1(b(b(b(x)))) → A1(x)
B1(b(b(b(x)))) → A1(a(x))
B1(b(b(b(x)))) → C(c(a(a(x))))
C(c(c(c(x)))) → B1(x)
A1(a(a(a(x)))) → B1(x)
B1(b(b(b(x)))) → C(a(a(x)))
The TRS R consists of the following rules:
a(a(a(a(x)))) → c(c(b(b(x))))
b(b(b(b(x)))) → c(c(a(a(x))))
c(c(c(c(x)))) → a(a(a(b(x))))
c(c(c(b(b(b(B(x))))))) → a(a(a(b(c(a(A(x)))))))
c(c(b(b(b(B(x)))))) → a(a(a(b(a(A(x))))))
c(c(a(a(a(A(x)))))) → a(a(a(b(b(B(x))))))
c(c(c(a(a(a(A(x))))))) → a(a(a(b(c(b(B(x)))))))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By using the rule removal processor [15] with the following polynomial ordering [25], at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented.
Strictly oriented dependency pairs:
A1(a(a(a(x)))) → B1(b(x))
A1(a(a(a(x)))) → C(b(b(x)))
C(c(c(c(x)))) → A1(a(b(x)))
C(c(c(c(x)))) → A1(b(x))
B1(b(b(b(x)))) → A1(x)
B1(b(b(b(x)))) → A1(a(x))
B1(b(b(b(x)))) → C(c(a(a(x))))
C(c(c(c(x)))) → B1(x)
A1(a(a(a(x)))) → B1(x)
B1(b(b(b(x)))) → C(a(a(x)))
Used ordering: POLO with Polynomial interpretation [25]:
POL(A(x1)) = x1
POL(A1(x1)) = 1 + 2·x1
POL(B(x1)) = x1
POL(B1(x1)) = 2 + 2·x1
POL(C(x1)) = 1 + 2·x1
POL(a(x1)) = 2 + 2·x1
POL(b(x1)) = 2 + 2·x1
POL(c(x1)) = 2 + 2·x1
↳ QTRS
↳ QTRS Reverse
↳ QTRS Reverse
↳ DependencyPairsProof
↳ QDP
↳ RuleRemovalProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ QDPToSRSProof
↳ QTRS
↳ QTRS Reverse
↳ QTRS
↳ QTRS Reverse
↳ QTRS Reverse
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ RuleRemovalProof
↳ QDP
↳ Narrowing
Q DP problem:
The TRS P consists of the following rules:
A1(a(a(a(x)))) → C(c(b(b(x))))
C(c(c(c(x)))) → A1(a(a(b(x))))
The TRS R consists of the following rules:
a(a(a(a(x)))) → c(c(b(b(x))))
b(b(b(b(x)))) → c(c(a(a(x))))
c(c(c(c(x)))) → a(a(a(b(x))))
c(c(c(b(b(b(B(x))))))) → a(a(a(b(c(a(A(x)))))))
c(c(b(b(b(B(x)))))) → a(a(a(b(a(A(x))))))
c(c(a(a(a(A(x)))))) → a(a(a(b(b(B(x))))))
c(c(c(a(a(a(A(x))))))) → a(a(a(b(c(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(x)))) → C(c(b(b(x)))) at position [0] we obtained the following new rules:
A1(a(a(a(b(b(b(x0))))))) → C(c(b(c(c(a(a(x0)))))))
A1(a(a(a(b(b(x0)))))) → C(c(c(c(a(a(x0))))))
↳ QTRS
↳ QTRS Reverse
↳ QTRS Reverse
↳ DependencyPairsProof
↳ QDP
↳ RuleRemovalProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ QDPToSRSProof
↳ QTRS
↳ QTRS Reverse
↳ QTRS
↳ QTRS Reverse
↳ QTRS Reverse
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ RuleRemovalProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
Q DP problem:
The TRS P consists of the following rules:
A1(a(a(a(b(b(b(x0))))))) → C(c(b(c(c(a(a(x0)))))))
C(c(c(c(x)))) → A1(a(a(b(x))))
A1(a(a(a(b(b(x0)))))) → C(c(c(c(a(a(x0))))))
The TRS R consists of the following rules:
a(a(a(a(x)))) → c(c(b(b(x))))
b(b(b(b(x)))) → c(c(a(a(x))))
c(c(c(c(x)))) → a(a(a(b(x))))
c(c(c(b(b(b(B(x))))))) → a(a(a(b(c(a(A(x)))))))
c(c(b(b(b(B(x)))))) → a(a(a(b(a(A(x))))))
c(c(a(a(a(A(x)))))) → a(a(a(b(b(B(x))))))
c(c(c(a(a(a(A(x))))))) → a(a(a(b(c(b(B(x)))))))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule C(c(c(c(x)))) → A1(a(a(b(x)))) at position [0] we obtained the following new rules:
C(c(c(c(b(b(b(x0))))))) → A1(a(a(c(c(a(a(x0)))))))
↳ QTRS
↳ QTRS Reverse
↳ QTRS Reverse
↳ DependencyPairsProof
↳ QDP
↳ RuleRemovalProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ QDPToSRSProof
↳ QTRS
↳ QTRS Reverse
↳ QTRS
↳ QTRS Reverse
↳ QTRS Reverse
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ RuleRemovalProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
C(c(c(c(b(b(b(x0))))))) → A1(a(a(c(c(a(a(x0)))))))
A1(a(a(a(b(b(b(x0))))))) → C(c(b(c(c(a(a(x0)))))))
A1(a(a(a(b(b(x0)))))) → C(c(c(c(a(a(x0))))))
The TRS R consists of the following rules:
a(a(a(a(x)))) → c(c(b(b(x))))
b(b(b(b(x)))) → c(c(a(a(x))))
c(c(c(c(x)))) → a(a(a(b(x))))
c(c(c(b(b(b(B(x))))))) → a(a(a(b(c(a(A(x)))))))
c(c(b(b(b(B(x)))))) → a(a(a(b(a(A(x))))))
c(c(a(a(a(A(x)))))) → a(a(a(b(b(B(x))))))
c(c(c(a(a(a(A(x))))))) → a(a(a(b(c(b(B(x)))))))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.