Termination w.r.t. Q of the following Term Rewriting System could be proven:
Q restricted rewrite system:
The TRS R consists of the following rules:
c(c(x1)) → a(a(a(b(b(b(x1))))))
b(b(b(a(x1)))) → b(b(b(b(b(b(b(b(x1))))))))
b(b(c(c(x1)))) → c(c(c(a(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:
c(c(x1)) → a(a(a(b(b(b(x1))))))
b(b(b(a(x1)))) → b(b(b(b(b(b(b(b(x1))))))))
b(b(c(c(x1)))) → c(c(c(a(a(a(a(x1)))))))
Q is empty.
We have reversed the following QTRS:
The set of rules R is
c(c(x1)) → a(a(a(b(b(b(x1))))))
b(b(b(a(x1)))) → b(b(b(b(b(b(b(b(x1))))))))
b(b(c(c(x1)))) → c(c(c(a(a(a(a(x1)))))))
The set Q is empty.
We have obtained the following QTRS:
c(c(x)) → b(b(b(a(a(a(x))))))
a(b(b(b(x)))) → b(b(b(b(b(b(b(b(x))))))))
c(c(b(b(x)))) → a(a(a(a(c(c(c(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:
c(c(x)) → b(b(b(a(a(a(x))))))
a(b(b(b(x)))) → b(b(b(b(b(b(b(b(x))))))))
c(c(b(b(x)))) → a(a(a(a(c(c(c(x)))))))
Q is empty.
We have reversed the following QTRS:
The set of rules R is
c(c(x1)) → a(a(a(b(b(b(x1))))))
b(b(b(a(x1)))) → b(b(b(b(b(b(b(b(x1))))))))
b(b(c(c(x1)))) → c(c(c(a(a(a(a(x1)))))))
The set Q is empty.
We have obtained the following QTRS:
c(c(x)) → b(b(b(a(a(a(x))))))
a(b(b(b(x)))) → b(b(b(b(b(b(b(b(x))))))))
c(c(b(b(x)))) → a(a(a(a(c(c(c(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:
c(c(x)) → b(b(b(a(a(a(x))))))
a(b(b(b(x)))) → b(b(b(b(b(b(b(b(x))))))))
c(c(b(b(x)))) → a(a(a(a(c(c(c(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(b(a(x1)))) → B(b(b(b(b(x1)))))
B(b(b(a(x1)))) → B(b(b(b(x1))))
C(c(x1)) → B(x1)
B(b(c(c(x1)))) → C(a(a(a(a(x1)))))
B(b(b(a(x1)))) → B(b(b(b(b(b(b(x1)))))))
B(b(b(a(x1)))) → B(b(b(b(b(b(b(b(x1))))))))
B(b(b(a(x1)))) → B(b(b(x1)))
B(b(b(a(x1)))) → B(x1)
B(b(b(a(x1)))) → B(b(b(b(b(b(x1))))))
B(b(c(c(x1)))) → C(c(a(a(a(a(x1))))))
C(c(x1)) → B(b(x1))
B(b(b(a(x1)))) → B(b(x1))
B(b(c(c(x1)))) → C(c(c(a(a(a(a(x1)))))))
C(c(x1)) → B(b(b(x1)))
The TRS R consists of the following rules:
c(c(x1)) → a(a(a(b(b(b(x1))))))
b(b(b(a(x1)))) → b(b(b(b(b(b(b(b(x1))))))))
b(b(c(c(x1)))) → c(c(c(a(a(a(a(x1)))))))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
↳ QTRS
↳ QTRS Reverse
↳ QTRS Reverse
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
Q DP problem:
The TRS P consists of the following rules:
B(b(b(a(x1)))) → B(b(b(b(b(x1)))))
B(b(b(a(x1)))) → B(b(b(b(x1))))
C(c(x1)) → B(x1)
B(b(c(c(x1)))) → C(a(a(a(a(x1)))))
B(b(b(a(x1)))) → B(b(b(b(b(b(b(x1)))))))
B(b(b(a(x1)))) → B(b(b(b(b(b(b(b(x1))))))))
B(b(b(a(x1)))) → B(b(b(x1)))
B(b(b(a(x1)))) → B(x1)
B(b(b(a(x1)))) → B(b(b(b(b(b(x1))))))
B(b(c(c(x1)))) → C(c(a(a(a(a(x1))))))
C(c(x1)) → B(b(x1))
B(b(b(a(x1)))) → B(b(x1))
B(b(c(c(x1)))) → C(c(c(a(a(a(a(x1)))))))
C(c(x1)) → B(b(b(x1)))
The TRS R consists of the following rules:
c(c(x1)) → a(a(a(b(b(b(x1))))))
b(b(b(a(x1)))) → b(b(b(b(b(b(b(b(x1))))))))
b(b(c(c(x1)))) → c(c(c(a(a(a(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 1 less node.
↳ QTRS
↳ QTRS Reverse
↳ QTRS Reverse
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
Q DP problem:
The TRS P consists of the following rules:
B(b(b(a(x1)))) → B(b(b(b(b(x1)))))
B(b(b(a(x1)))) → B(b(b(b(x1))))
C(c(x1)) → B(x1)
B(b(b(a(x1)))) → B(b(b(b(b(b(b(x1)))))))
B(b(b(a(x1)))) → B(b(b(b(b(b(b(b(x1))))))))
B(b(b(a(x1)))) → B(b(b(x1)))
B(b(b(a(x1)))) → B(x1)
B(b(b(a(x1)))) → B(b(b(b(b(b(x1))))))
B(b(c(c(x1)))) → C(c(a(a(a(a(x1))))))
C(c(x1)) → B(b(x1))
B(b(b(a(x1)))) → B(b(x1))
B(b(c(c(x1)))) → C(c(c(a(a(a(a(x1)))))))
C(c(x1)) → B(b(b(x1)))
The TRS R consists of the following rules:
c(c(x1)) → a(a(a(b(b(b(x1))))))
b(b(b(a(x1)))) → b(b(b(b(b(b(b(b(x1))))))))
b(b(c(c(x1)))) → c(c(c(a(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(a(x1)))) → B(b(b(b(b(b(b(b(x1)))))))) at position [0] we obtained the following new rules:
B(b(b(a(b(c(c(x0))))))) → B(b(b(b(b(b(b(c(c(c(a(a(a(a(x0))))))))))))))
B(b(b(a(c(c(x0)))))) → B(b(b(b(b(b(c(c(c(a(a(a(a(x0)))))))))))))
B(b(b(a(b(a(x0)))))) → B(b(b(b(b(b(b(b(b(b(b(b(b(b(x0))))))))))))))
B(b(b(a(a(x0))))) → B(b(b(b(b(b(b(b(b(b(b(b(b(x0)))))))))))))
B(b(b(a(b(b(a(x0))))))) → B(b(b(b(b(b(b(b(b(b(b(b(b(b(b(x0)))))))))))))))
↳ QTRS
↳ QTRS Reverse
↳ QTRS Reverse
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
Q DP problem:
The TRS P consists of the following rules:
B(b(b(a(b(c(c(x0))))))) → B(b(b(b(b(b(b(c(c(c(a(a(a(a(x0))))))))))))))
B(b(b(a(x1)))) → B(b(b(b(b(x1)))))
B(b(b(a(x1)))) → B(b(b(b(x1))))
C(c(x1)) → B(x1)
B(b(b(a(c(c(x0)))))) → B(b(b(b(b(b(c(c(c(a(a(a(a(x0)))))))))))))
B(b(b(a(a(x0))))) → B(b(b(b(b(b(b(b(b(b(b(b(b(x0)))))))))))))
B(b(b(a(b(b(a(x0))))))) → B(b(b(b(b(b(b(b(b(b(b(b(b(b(b(x0)))))))))))))))
B(b(b(a(x1)))) → B(b(b(b(b(b(b(x1)))))))
B(b(b(a(x1)))) → B(b(b(x1)))
B(b(b(a(b(a(x0)))))) → B(b(b(b(b(b(b(b(b(b(b(b(b(b(x0))))))))))))))
B(b(b(a(x1)))) → B(x1)
B(b(b(a(x1)))) → B(b(b(b(b(b(x1))))))
B(b(c(c(x1)))) → C(c(a(a(a(a(x1))))))
C(c(x1)) → B(b(x1))
B(b(c(c(x1)))) → C(c(c(a(a(a(a(x1)))))))
B(b(b(a(x1)))) → B(b(x1))
C(c(x1)) → B(b(b(x1)))
The TRS R consists of the following rules:
c(c(x1)) → a(a(a(b(b(b(x1))))))
b(b(b(a(x1)))) → b(b(b(b(b(b(b(b(x1))))))))
b(b(c(c(x1)))) → c(c(c(a(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(a(x1)))) → B(b(b(b(b(b(b(x1))))))) at position [0] we obtained the following new rules:
B(b(b(a(b(c(c(x0))))))) → B(b(b(b(b(b(c(c(c(a(a(a(a(x0)))))))))))))
B(b(b(a(c(c(x0)))))) → B(b(b(b(b(c(c(c(a(a(a(a(x0))))))))))))
B(b(b(a(b(b(a(x0))))))) → B(b(b(b(b(b(b(b(b(b(b(b(b(b(x0))))))))))))))
B(b(b(a(b(a(x0)))))) → B(b(b(b(b(b(b(b(b(b(b(b(b(x0)))))))))))))
B(b(b(a(a(x0))))) → B(b(b(b(b(b(b(b(b(b(b(b(x0))))))))))))
↳ QTRS
↳ QTRS Reverse
↳ QTRS Reverse
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
Q DP problem:
The TRS P consists of the following rules:
B(b(b(a(b(c(c(x0))))))) → B(b(b(b(b(b(b(c(c(c(a(a(a(a(x0))))))))))))))
B(b(b(a(x1)))) → B(b(b(b(b(x1)))))
B(b(b(a(x1)))) → B(b(b(b(x1))))
B(b(b(a(a(x0))))) → B(b(b(b(b(b(b(b(b(b(b(b(x0))))))))))))
B(b(b(a(b(a(x0)))))) → B(b(b(b(b(b(b(b(b(b(b(b(b(x0)))))))))))))
C(c(x1)) → B(x1)
B(b(b(a(c(c(x0)))))) → B(b(b(b(b(b(c(c(c(a(a(a(a(x0)))))))))))))
B(b(b(a(a(x0))))) → B(b(b(b(b(b(b(b(b(b(b(b(b(x0)))))))))))))
B(b(b(a(b(b(a(x0))))))) → B(b(b(b(b(b(b(b(b(b(b(b(b(b(b(x0)))))))))))))))
B(b(b(a(x1)))) → B(b(b(x1)))
B(b(b(a(c(c(x0)))))) → B(b(b(b(b(c(c(c(a(a(a(a(x0))))))))))))
B(b(b(a(x1)))) → B(x1)
B(b(b(a(b(a(x0)))))) → B(b(b(b(b(b(b(b(b(b(b(b(b(b(x0))))))))))))))
B(b(b(a(b(c(c(x0))))))) → B(b(b(b(b(b(c(c(c(a(a(a(a(x0)))))))))))))
B(b(b(a(x1)))) → B(b(b(b(b(b(x1))))))
B(b(c(c(x1)))) → C(c(a(a(a(a(x1))))))
C(c(x1)) → B(b(x1))
B(b(b(a(b(b(a(x0))))))) → B(b(b(b(b(b(b(b(b(b(b(b(b(b(x0))))))))))))))
B(b(b(a(x1)))) → B(b(x1))
B(b(c(c(x1)))) → C(c(c(a(a(a(a(x1)))))))
C(c(x1)) → B(b(b(x1)))
The TRS R consists of the following rules:
c(c(x1)) → a(a(a(b(b(b(x1))))))
b(b(b(a(x1)))) → b(b(b(b(b(b(b(b(x1))))))))
b(b(c(c(x1)))) → c(c(c(a(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(a(x1)))) → B(b(b(b(b(b(x1)))))) at position [0] we obtained the following new rules:
B(b(b(a(b(b(a(x0))))))) → B(b(b(b(b(b(b(b(b(b(b(b(b(x0)))))))))))))
B(b(b(a(c(c(x0)))))) → B(b(b(b(c(c(c(a(a(a(a(x0)))))))))))
B(b(b(a(b(a(x0)))))) → B(b(b(b(b(b(b(b(b(b(b(b(x0))))))))))))
B(b(b(a(a(x0))))) → B(b(b(b(b(b(b(b(b(b(b(x0)))))))))))
B(b(b(a(b(c(c(x0))))))) → B(b(b(b(b(c(c(c(a(a(a(a(x0))))))))))))
↳ QTRS
↳ QTRS Reverse
↳ QTRS Reverse
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
Q DP problem:
The TRS P consists of the following rules:
B(b(b(a(b(c(c(x0))))))) → B(b(b(b(b(b(b(c(c(c(a(a(a(a(x0))))))))))))))
B(b(b(a(x1)))) → B(b(b(b(x1))))
B(b(b(a(a(x0))))) → B(b(b(b(b(b(b(b(b(b(b(b(x0))))))))))))
B(b(b(a(b(a(x0)))))) → B(b(b(b(b(b(b(b(b(b(b(b(x0))))))))))))
B(b(b(a(a(x0))))) → B(b(b(b(b(b(b(b(b(b(b(b(b(x0)))))))))))))
B(b(b(a(c(c(x0)))))) → B(b(b(b(c(c(c(a(a(a(a(x0)))))))))))
B(b(b(a(x1)))) → B(b(b(x1)))
B(b(b(a(x1)))) → B(x1)
B(b(b(a(b(c(c(x0))))))) → B(b(b(b(b(b(c(c(c(a(a(a(a(x0)))))))))))))
B(b(b(a(b(b(a(x0))))))) → B(b(b(b(b(b(b(b(b(b(b(b(b(b(x0))))))))))))))
B(b(b(a(b(c(c(x0))))))) → B(b(b(b(b(c(c(c(a(a(a(a(x0))))))))))))
B(b(b(a(x1)))) → B(b(b(b(b(x1)))))
B(b(b(a(b(a(x0)))))) → B(b(b(b(b(b(b(b(b(b(b(b(b(x0)))))))))))))
B(b(b(a(b(b(a(x0))))))) → B(b(b(b(b(b(b(b(b(b(b(b(b(x0)))))))))))))
C(c(x1)) → B(x1)
B(b(b(a(c(c(x0)))))) → B(b(b(b(b(b(c(c(c(a(a(a(a(x0)))))))))))))
B(b(b(a(b(b(a(x0))))))) → B(b(b(b(b(b(b(b(b(b(b(b(b(b(b(x0)))))))))))))))
B(b(b(a(c(c(x0)))))) → B(b(b(b(b(c(c(c(a(a(a(a(x0))))))))))))
B(b(b(a(b(a(x0)))))) → B(b(b(b(b(b(b(b(b(b(b(b(b(b(x0))))))))))))))
B(b(c(c(x1)))) → C(c(a(a(a(a(x1))))))
B(b(b(a(a(x0))))) → B(b(b(b(b(b(b(b(b(b(b(x0)))))))))))
C(c(x1)) → B(b(x1))
B(b(b(a(x1)))) → B(b(x1))
B(b(c(c(x1)))) → C(c(c(a(a(a(a(x1)))))))
C(c(x1)) → B(b(b(x1)))
The TRS R consists of the following rules:
c(c(x1)) → a(a(a(b(b(b(x1))))))
b(b(b(a(x1)))) → b(b(b(b(b(b(b(b(x1))))))))
b(b(c(c(x1)))) → c(c(c(a(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(a(x1)))) → B(b(b(b(b(x1))))) at position [0] we obtained the following new rules:
B(b(b(a(b(b(a(x0))))))) → B(b(b(b(b(b(b(b(b(b(b(b(x0))))))))))))
B(b(b(a(c(c(x0)))))) → B(b(b(c(c(c(a(a(a(a(x0))))))))))
B(b(b(a(a(x0))))) → B(b(b(b(b(b(b(b(b(b(x0))))))))))
B(b(b(a(b(c(c(x0))))))) → B(b(b(b(c(c(c(a(a(a(a(x0)))))))))))
B(b(b(a(b(a(x0)))))) → B(b(b(b(b(b(b(b(b(b(b(x0)))))))))))
↳ QTRS
↳ QTRS Reverse
↳ QTRS Reverse
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
Q DP problem:
The TRS P consists of the following rules:
B(b(b(a(b(c(c(x0))))))) → B(b(b(b(b(b(b(c(c(c(a(a(a(a(x0))))))))))))))
B(b(b(a(a(x0))))) → B(b(b(b(b(b(b(b(b(b(x0))))))))))
B(b(b(a(x1)))) → B(b(b(b(x1))))
B(b(b(a(a(x0))))) → B(b(b(b(b(b(b(b(b(b(b(b(x0))))))))))))
B(b(b(a(b(a(x0)))))) → B(b(b(b(b(b(b(b(b(b(b(b(x0))))))))))))
B(b(b(a(b(b(a(x0))))))) → B(b(b(b(b(b(b(b(b(b(b(b(x0))))))))))))
B(b(b(a(a(x0))))) → B(b(b(b(b(b(b(b(b(b(b(b(b(x0)))))))))))))
B(b(b(a(c(c(x0)))))) → B(b(b(b(c(c(c(a(a(a(a(x0)))))))))))
B(b(b(a(x1)))) → B(b(b(x1)))
B(b(b(a(x1)))) → B(x1)
B(b(b(a(b(c(c(x0))))))) → B(b(b(b(b(b(c(c(c(a(a(a(a(x0)))))))))))))
B(b(b(a(b(b(a(x0))))))) → B(b(b(b(b(b(b(b(b(b(b(b(b(b(x0))))))))))))))
B(b(b(a(b(c(c(x0))))))) → B(b(b(b(b(c(c(c(a(a(a(a(x0))))))))))))
B(b(b(a(b(a(x0)))))) → B(b(b(b(b(b(b(b(b(b(b(b(b(x0)))))))))))))
B(b(b(a(b(a(x0)))))) → B(b(b(b(b(b(b(b(b(b(b(x0)))))))))))
B(b(b(a(b(b(a(x0))))))) → B(b(b(b(b(b(b(b(b(b(b(b(b(x0)))))))))))))
C(c(x1)) → B(x1)
B(b(b(a(c(c(x0)))))) → B(b(b(b(b(b(c(c(c(a(a(a(a(x0)))))))))))))
B(b(b(a(b(b(a(x0))))))) → B(b(b(b(b(b(b(b(b(b(b(b(b(b(b(x0)))))))))))))))
B(b(b(a(c(c(x0)))))) → B(b(b(b(b(c(c(c(a(a(a(a(x0))))))))))))
B(b(b(a(b(a(x0)))))) → B(b(b(b(b(b(b(b(b(b(b(b(b(b(x0))))))))))))))
B(b(b(a(c(c(x0)))))) → B(b(b(c(c(c(a(a(a(a(x0))))))))))
B(b(c(c(x1)))) → C(c(a(a(a(a(x1))))))
B(b(b(a(a(x0))))) → B(b(b(b(b(b(b(b(b(b(b(x0)))))))))))
C(c(x1)) → B(b(x1))
B(b(b(a(b(c(c(x0))))))) → B(b(b(b(c(c(c(a(a(a(a(x0)))))))))))
B(b(c(c(x1)))) → C(c(c(a(a(a(a(x1)))))))
B(b(b(a(x1)))) → B(b(x1))
C(c(x1)) → B(b(b(x1)))
The TRS R consists of the following rules:
c(c(x1)) → a(a(a(b(b(b(x1))))))
b(b(b(a(x1)))) → b(b(b(b(b(b(b(b(x1))))))))
b(b(c(c(x1)))) → c(c(c(a(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(a(x1)))) → B(b(b(b(x1)))) at position [0] we obtained the following new rules:
B(b(b(a(c(c(x0)))))) → B(b(c(c(c(a(a(a(a(x0)))))))))
B(b(b(a(b(c(c(x0))))))) → B(b(b(c(c(c(a(a(a(a(x0))))))))))
B(b(b(a(b(b(a(x0))))))) → B(b(b(b(b(b(b(b(b(b(b(x0)))))))))))
B(b(b(a(b(a(x0)))))) → B(b(b(b(b(b(b(b(b(b(x0))))))))))
B(b(b(a(a(x0))))) → B(b(b(b(b(b(b(b(b(x0)))))))))
↳ QTRS
↳ QTRS Reverse
↳ QTRS Reverse
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ SemLabProof
↳ SemLabProof2
Q DP problem:
The TRS P consists of the following rules:
B(b(b(a(b(c(c(x0))))))) → B(b(b(b(b(b(b(c(c(c(a(a(a(a(x0))))))))))))))
B(b(b(a(a(x0))))) → B(b(b(b(b(b(b(b(b(b(x0))))))))))
B(b(b(a(b(a(x0)))))) → B(b(b(b(b(b(b(b(b(b(x0))))))))))
B(b(b(a(a(x0))))) → B(b(b(b(b(b(b(b(b(b(b(b(x0))))))))))))
B(b(b(a(b(a(x0)))))) → B(b(b(b(b(b(b(b(b(b(b(b(x0))))))))))))
B(b(b(a(b(b(a(x0))))))) → B(b(b(b(b(b(b(b(b(b(b(b(x0))))))))))))
B(b(b(a(b(b(a(x0))))))) → B(b(b(b(b(b(b(b(b(b(b(x0)))))))))))
B(b(b(a(a(x0))))) → B(b(b(b(b(b(b(b(b(b(b(b(b(x0)))))))))))))
B(b(b(a(c(c(x0)))))) → B(b(b(b(c(c(c(a(a(a(a(x0)))))))))))
B(b(b(a(x1)))) → B(b(b(x1)))
B(b(b(a(x1)))) → B(x1)
B(b(b(a(b(c(c(x0))))))) → B(b(b(b(b(b(c(c(c(a(a(a(a(x0)))))))))))))
B(b(b(a(b(b(a(x0))))))) → B(b(b(b(b(b(b(b(b(b(b(b(b(b(x0))))))))))))))
B(b(b(a(b(c(c(x0))))))) → B(b(b(b(b(c(c(c(a(a(a(a(x0))))))))))))
B(b(b(a(b(a(x0)))))) → B(b(b(b(b(b(b(b(b(b(b(b(b(x0)))))))))))))
B(b(b(a(b(a(x0)))))) → B(b(b(b(b(b(b(b(b(b(b(x0)))))))))))
B(b(b(a(b(b(a(x0))))))) → B(b(b(b(b(b(b(b(b(b(b(b(b(x0)))))))))))))
C(c(x1)) → B(x1)
B(b(b(a(c(c(x0)))))) → B(b(b(b(b(b(c(c(c(a(a(a(a(x0)))))))))))))
B(b(b(a(b(b(a(x0))))))) → B(b(b(b(b(b(b(b(b(b(b(b(b(b(b(x0)))))))))))))))
B(b(b(a(c(c(x0)))))) → B(b(b(b(b(c(c(c(a(a(a(a(x0))))))))))))
B(b(b(a(b(c(c(x0))))))) → B(b(b(c(c(c(a(a(a(a(x0))))))))))
B(b(b(a(b(a(x0)))))) → B(b(b(b(b(b(b(b(b(b(b(b(b(b(x0))))))))))))))
B(b(b(a(c(c(x0)))))) → B(b(b(c(c(c(a(a(a(a(x0))))))))))
B(b(c(c(x1)))) → C(c(a(a(a(a(x1))))))
B(b(b(a(a(x0))))) → B(b(b(b(b(b(b(b(b(b(b(x0)))))))))))
B(b(b(a(c(c(x0)))))) → B(b(c(c(c(a(a(a(a(x0)))))))))
C(c(x1)) → B(b(x1))
B(b(b(a(x1)))) → B(b(x1))
B(b(c(c(x1)))) → C(c(c(a(a(a(a(x1)))))))
B(b(b(a(b(c(c(x0))))))) → B(b(b(b(c(c(c(a(a(a(a(x0)))))))))))
B(b(b(a(a(x0))))) → B(b(b(b(b(b(b(b(b(x0)))))))))
C(c(x1)) → B(b(b(x1)))
The TRS R consists of the following rules:
c(c(x1)) → a(a(a(b(b(b(x1))))))
b(b(b(a(x1)))) → b(b(b(b(b(b(b(b(x1))))))))
b(b(c(c(x1)))) → c(c(c(a(a(a(a(x1)))))))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We found the following quasi-model for the rules of the TRS R.
Interpretation over the domain with elements from 0 to 1.C: 0
c: 0
B: 0
a: 0
b: 1
By semantic labelling [33] we obtain the following labelled TRS:Q DP problem:
The TRS P consists of the following rules:
B.1(b.1(b.0(a.1(b.1(b.0(a.0(x0))))))) → B.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.0(x0))))))))))))
B.1(b.0(c.0(c.1(x1)))) → C.0(c.0(a.0(a.0(a.0(a.1(x1))))))
B.1(b.1(b.0(a.0(c.0(c.1(x0)))))) → B.0(b.1(b.0(c.0(c.0(c.0(a.0(a.0(a.0(a.1(x0))))))))))
B.1(b.1(b.0(a.1(b.0(c.0(c.1(x0))))))) → B.0(b.1(b.1(b.0(c.0(c.0(c.0(a.0(a.0(a.0(a.1(x0)))))))))))
B.1(b.1(b.0(a.0(c.0(c.0(x0)))))) → B.0(b.1(b.1(b.1(b.1(b.0(c.0(c.0(c.0(a.0(a.0(a.0(a.0(x0)))))))))))))
B.1(b.1(b.0(a.1(b.0(c.0(c.0(x0))))))) → B.1(b.1(b.1(b.0(c.0(c.0(c.0(a.0(a.0(a.0(a.0(x0)))))))))))
B.1(b.1(b.0(a.1(b.0(a.1(x0)))))) → B.0(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(x0))))))))))))))
B.1(b.1(b.0(a.0(a.0(x0))))) → B.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.0(x0))))))))))))
B.1(b.1(b.0(a.0(c.0(c.1(x0)))))) → B.1(b.1(b.1(b.1(b.0(c.0(c.0(c.0(a.0(a.0(a.0(a.1(x0))))))))))))
B.1(b.1(b.0(a.1(b.1(b.0(a.0(x0))))))) → B.0(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.0(x0)))))))))))))))
B.1(b.1(b.0(a.1(b.0(c.0(c.1(x0))))))) → B.0(b.1(b.1(b.1(b.0(c.0(c.0(c.0(a.0(a.0(a.0(a.1(x0))))))))))))
B.1(b.1(b.0(a.0(a.1(x0))))) → B.0(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(x0)))))))))))
B.1(b.1(b.0(a.1(b.1(b.0(a.0(x0))))))) → B.0(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.0(x0))))))))))))
B.1(b.1(b.0(a.0(a.0(x0))))) → B.0(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.0(x0))))))))))
B.1(b.1(b.0(a.0(c.0(c.0(x0)))))) → B.0(b.1(b.1(b.1(b.0(c.0(c.0(c.0(a.0(a.0(a.0(a.0(x0))))))))))))
B.1(b.1(b.0(a.1(b.0(a.0(x0)))))) → B.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.0(x0))))))))))))))
B.1(b.1(b.0(a.0(c.0(c.0(x0)))))) → B.0(b.1(b.1(b.0(c.0(c.0(c.0(a.0(a.0(a.0(a.0(x0)))))))))))
C.0(c.0(x1)) → B.0(b.0(x1))
B.1(b.1(b.0(a.0(c.0(c.0(x0)))))) → B.1(b.0(c.0(c.0(c.0(a.0(a.0(a.0(a.0(x0)))))))))
B.1(b.1(b.0(a.0(a.0(x0))))) → B.0(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.0(x0))))))))))))
B.1(b.1(b.0(a.0(a.1(x0))))) → B.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(x0)))))))))))))
B.1(b.1(b.0(a.1(b.0(c.0(c.1(x0))))))) → B.1(b.1(b.1(b.0(c.0(c.0(c.0(a.0(a.0(a.0(a.1(x0)))))))))))
B.1(b.1(b.0(a.0(c.0(c.1(x0)))))) → B.0(b.0(c.0(c.0(c.0(a.0(a.0(a.0(a.1(x0)))))))))
B.1(b.1(b.0(a.1(b.0(c.0(c.0(x0))))))) → B.1(b.1(b.0(c.0(c.0(c.0(a.0(a.0(a.0(a.0(x0))))))))))
B.1(b.1(b.0(a.1(b.0(c.0(c.1(x0))))))) → B.0(b.1(b.0(c.0(c.0(c.0(a.0(a.0(a.0(a.1(x0))))))))))
B.1(b.1(b.0(a.0(a.1(x0))))) → B.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(x0)))))))))
B.1(b.1(b.0(a.1(b.1(b.0(a.1(x0))))))) → B.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(x0)))))))))))))
B.1(b.1(b.0(a.1(x1)))) → B.1(b.1(x1))
B.1(b.1(b.0(a.1(b.0(c.0(c.0(x0))))))) → B.1(b.1(b.1(b.1(b.1(b.1(b.0(c.0(c.0(c.0(a.0(a.0(a.0(a.0(x0))))))))))))))
B.1(b.1(b.0(a.0(a.0(x0))))) → B.0(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.0(x0)))))))))
B.1(b.1(b.0(a.1(b.0(c.0(c.0(x0))))))) → B.1(b.1(b.1(b.1(b.1(b.0(c.0(c.0(c.0(a.0(a.0(a.0(a.0(x0)))))))))))))
B.1(b.1(b.0(a.0(x1)))) → B.0(x1)
B.1(b.1(b.0(a.1(b.0(a.1(x0)))))) → B.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(x0)))))))))))))
B.1(b.1(b.0(a.0(a.0(x0))))) → B.0(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.0(x0)))))))))))
B.1(b.1(b.0(a.1(b.0(a.0(x0)))))) → B.0(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.0(x0))))))))))
B.1(b.1(b.0(a.0(c.0(c.1(x0)))))) → B.1(b.0(c.0(c.0(c.0(a.0(a.0(a.0(a.1(x0)))))))))
B.1(b.1(b.0(a.1(b.1(b.0(a.0(x0))))))) → B.0(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.0(x0)))))))))))))
C.0(c.0(x1)) → B.0(b.1(b.0(x1)))
B.1(b.1(b.0(a.0(x1)))) → B.1(b.1(b.0(x1)))
B.1(b.1(b.0(a.0(a.0(x0))))) → B.0(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.0(x0)))))))))))))
C.0(c.1(x1)) → B.0(b.1(x1))
B.1(b.0(c.0(c.0(x1)))) → C.0(c.0(c.0(a.0(a.0(a.0(a.0(x1)))))))
B.1(b.1(b.0(a.0(a.1(x0))))) → B.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(x0))))))))))))
B.1(b.1(b.0(a.1(b.1(b.0(a.0(x0))))))) → B.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.0(x0)))))))))))))))
B.1(b.1(b.0(a.1(b.0(c.0(c.0(x0))))))) → B.1(b.1(b.1(b.1(b.0(c.0(c.0(c.0(a.0(a.0(a.0(a.0(x0))))))))))))
B.1(b.1(b.0(a.0(c.0(c.0(x0)))))) → B.0(b.1(b.0(c.0(c.0(c.0(a.0(a.0(a.0(a.0(x0))))))))))
B.1(b.1(b.0(a.1(b.0(a.1(x0)))))) → B.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(x0)))))))))))
B.1(b.1(b.0(a.1(x1)))) → B.0(b.1(x1))
B.1(b.1(b.0(a.1(b.1(b.0(a.1(x0))))))) → B.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(x0))))))))))))
B.1(b.1(b.0(a.0(c.0(c.0(x0)))))) → B.1(b.1(b.1(b.1(b.0(c.0(c.0(c.0(a.0(a.0(a.0(a.0(x0))))))))))))
C.0(c.0(x1)) → B.1(b.1(b.0(x1)))
B.1(b.1(b.0(a.1(b.0(a.1(x0)))))) → B.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(x0))))))))))
B.1(b.1(b.0(a.1(b.0(a.0(x0)))))) → B.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.0(x0))))))))))
B.1(b.0(c.0(c.1(x1)))) → C.0(c.0(c.0(a.0(a.0(a.0(a.1(x1)))))))
B.1(b.1(b.0(a.0(c.0(c.1(x0)))))) → B.1(b.1(b.1(b.1(b.1(b.0(c.0(c.0(c.0(a.0(a.0(a.0(a.1(x0)))))))))))))
B.1(b.1(b.0(a.0(a.0(x0))))) → B.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.0(x0))))))))))
B.1(b.1(b.0(a.1(b.1(b.0(a.1(x0))))))) → B.0(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(x0)))))))))))))
B.1(b.1(b.0(a.0(a.1(x0))))) → B.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(x0))))))))))
B.1(b.1(b.0(a.1(x1)))) → B.0(b.1(b.1(x1)))
C.0(c.1(x1)) → B.1(x1)
B.1(b.1(b.0(a.1(b.0(a.0(x0)))))) → B.0(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.0(x0))))))))))))))
B.1(b.1(b.0(a.1(b.0(a.0(x0)))))) → B.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.0(x0)))))))))))))
B.1(b.1(b.0(a.1(b.0(c.0(c.1(x0))))))) → B.0(b.1(b.1(b.1(b.1(b.0(c.0(c.0(c.0(a.0(a.0(a.0(a.1(x0)))))))))))))
B.1(b.1(b.0(a.0(a.0(x0))))) → B.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.0(x0)))))))))))))
C.0(c.0(x1)) → B.1(b.0(x1))
C.0(c.1(x1)) → B.0(x1)
B.1(b.1(b.0(a.0(a.1(x0))))) → B.0(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(x0)))))))))))))
B.1(b.1(b.0(a.0(x1)))) → B.0(b.1(b.0(x1)))
B.1(b.0(c.0(c.0(x1)))) → C.0(c.0(a.0(a.0(a.0(a.0(x1))))))
B.1(b.1(b.0(a.1(b.1(b.0(a.0(x0))))))) → B.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.0(x0))))))))))))))
B.1(b.1(b.0(a.0(c.0(c.0(x0)))))) → B.1(b.1(b.0(c.0(c.0(c.0(a.0(a.0(a.0(a.0(x0))))))))))
B.1(b.1(b.0(a.1(b.0(a.1(x0)))))) → B.0(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(x0)))))))))))
B.1(b.1(b.0(a.1(b.0(a.0(x0)))))) → B.0(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.0(x0))))))))))))
B.1(b.1(b.0(a.1(b.1(b.0(a.1(x0))))))) → B.0(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(x0))))))))))))
B.1(b.1(b.0(a.1(b.0(a.1(x0)))))) → B.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(x0))))))))))))))
B.1(b.1(b.0(a.1(b.1(b.0(a.0(x0))))))) → B.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.0(x0)))))))))))
B.1(b.1(b.0(a.1(x1)))) → B.1(b.1(b.1(x1)))
B.1(b.1(b.0(a.0(c.0(c.0(x0)))))) → B.1(b.1(b.1(b.1(b.1(b.0(c.0(c.0(c.0(a.0(a.0(a.0(a.0(x0)))))))))))))
B.1(b.1(b.0(a.1(b.0(a.0(x0)))))) → B.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.0(x0))))))))))))
B.1(b.1(b.0(a.0(c.0(c.1(x0)))))) → B.1(b.1(b.1(b.0(c.0(c.0(c.0(a.0(a.0(a.0(a.1(x0)))))))))))
B.1(b.1(b.0(a.0(c.0(c.1(x0)))))) → B.1(b.1(b.0(c.0(c.0(c.0(a.0(a.0(a.0(a.1(x0))))))))))
B.1(b.1(b.0(a.1(b.0(a.1(x0)))))) → B.0(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(x0)))))))))))))
B.1(b.1(b.0(a.1(b.0(a.0(x0)))))) → B.0(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.0(x0)))))))))))
C.0(c.1(x1)) → B.1(b.1(x1))
B.1(b.1(b.0(a.0(a.0(x0))))) → B.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.0(x0)))))))))
B.1(b.1(b.0(a.1(b.0(c.0(c.1(x0))))))) → B.1(b.1(b.1(b.1(b.1(b.1(b.0(c.0(c.0(c.0(a.0(a.0(a.0(a.1(x0))))))))))))))
B.1(b.1(b.0(a.0(c.0(c.1(x0)))))) → B.0(b.1(b.1(b.1(b.0(c.0(c.0(c.0(a.0(a.0(a.0(a.1(x0))))))))))))
B.1(b.1(b.0(a.0(c.0(c.1(x0)))))) → B.0(b.1(b.1(b.1(b.1(b.0(c.0(c.0(c.0(a.0(a.0(a.0(a.1(x0)))))))))))))
B.1(b.1(b.0(a.1(b.1(b.0(a.1(x0))))))) → B.0(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(x0)))))))))))
B.1(b.1(b.0(a.1(b.0(a.1(x0)))))) → B.0(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(x0))))))))))))
B.1(b.1(b.0(a.0(a.1(x0))))) → B.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(x0)))))))))))
B.1(b.1(b.0(a.1(b.1(b.0(a.1(x0))))))) → B.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(x0)))))))))))))))
B.1(b.1(b.0(a.1(b.1(b.0(a.1(x0))))))) → B.0(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(x0)))))))))))))))
B.1(b.1(b.0(a.1(b.0(c.0(c.1(x0))))))) → B.1(b.1(b.1(b.1(b.0(c.0(c.0(c.0(a.0(a.0(a.0(a.1(x0))))))))))))
B.1(b.1(b.0(a.1(b.0(c.0(c.0(x0))))))) → B.0(b.1(b.0(c.0(c.0(c.0(a.0(a.0(a.0(a.0(x0))))))))))
B.1(b.1(b.0(a.0(a.1(x0))))) → B.0(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(x0)))))))))
B.1(b.1(b.0(a.1(b.0(a.0(x0)))))) → B.0(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.0(x0)))))))))))))
B.1(b.1(b.0(a.1(b.1(b.0(a.0(x0))))))) → B.0(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.0(x0))))))))))))))
B.1(b.1(b.0(a.1(b.0(c.0(c.1(x0))))))) → B.1(b.1(b.1(b.1(b.1(b.0(c.0(c.0(c.0(a.0(a.0(a.0(a.1(x0)))))))))))))
B.1(b.1(b.0(a.1(b.1(b.0(a.0(x0))))))) → B.0(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.0(x0)))))))))))
B.1(b.1(b.0(a.0(x1)))) → B.0(b.0(x1))
B.1(b.1(b.0(a.0(a.0(x0))))) → B.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.0(x0)))))))))))
B.1(b.1(b.0(a.0(c.0(c.1(x0)))))) → B.0(b.1(b.1(b.0(c.0(c.0(c.0(a.0(a.0(a.0(a.1(x0)))))))))))
B.1(b.1(b.0(a.1(x1)))) → B.1(x1)
C.0(c.1(x1)) → B.0(b.1(b.1(x1)))
B.1(b.1(b.0(a.0(a.1(x0))))) → B.0(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(x0))))))))))
B.1(b.1(b.0(a.1(b.1(b.0(a.1(x0))))))) → B.0(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(x0))))))))))))))
B.1(b.1(b.0(a.0(x1)))) → B.1(b.0(x1))
B.1(b.1(b.0(a.1(b.0(c.0(c.0(x0))))))) → B.0(b.1(b.1(b.1(b.1(b.1(b.0(c.0(c.0(c.0(a.0(a.0(a.0(a.0(x0))))))))))))))
B.1(b.1(b.0(a.1(b.1(b.0(a.0(x0))))))) → B.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.0(x0)))))))))))))
B.1(b.1(b.0(a.1(x1)))) → B.0(x1)
B.1(b.1(b.0(a.1(b.0(c.0(c.0(x0))))))) → B.0(b.1(b.1(b.0(c.0(c.0(c.0(a.0(a.0(a.0(a.0(x0)))))))))))
C.0(c.0(x1)) → B.0(x1)
B.1(b.1(b.0(a.1(b.1(b.0(a.1(x0))))))) → B.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(x0)))))))))))
B.1(b.1(b.0(a.1(b.1(b.0(a.1(x0))))))) → B.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(x0))))))))))))))
B.1(b.1(b.0(a.0(c.0(c.0(x0)))))) → B.0(b.0(c.0(c.0(c.0(a.0(a.0(a.0(a.0(x0)))))))))
B.1(b.1(b.0(a.1(b.0(c.0(c.1(x0))))))) → B.1(b.1(b.0(c.0(c.0(c.0(a.0(a.0(a.0(a.1(x0))))))))))
B.1(b.1(b.0(a.1(b.0(a.1(x0)))))) → B.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(x0))))))))))))
B.1(b.1(b.0(a.1(b.0(a.1(x0)))))) → B.0(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(x0))))))))))
B.1(b.1(b.0(a.1(b.0(c.0(c.0(x0))))))) → B.0(b.1(b.1(b.1(b.0(c.0(c.0(c.0(a.0(a.0(a.0(a.0(x0))))))))))))
B.1(b.1(b.0(a.0(a.1(x0))))) → B.0(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(x0))))))))))))
C.0(c.1(x1)) → B.1(b.1(b.1(x1)))
B.1(b.1(b.0(a.1(b.0(c.0(c.1(x0))))))) → B.0(b.1(b.1(b.1(b.1(b.1(b.0(c.0(c.0(c.0(a.0(a.0(a.0(a.1(x0))))))))))))))
B.1(b.1(b.0(a.0(c.0(c.0(x0)))))) → B.1(b.1(b.1(b.0(c.0(c.0(c.0(a.0(a.0(a.0(a.0(x0)))))))))))
B.1(b.1(b.0(a.1(b.0(c.0(c.0(x0))))))) → B.0(b.1(b.1(b.1(b.1(b.0(c.0(c.0(c.0(a.0(a.0(a.0(a.0(x0)))))))))))))
B.1(b.1(b.0(a.1(b.0(a.0(x0)))))) → B.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.0(x0)))))))))))
The TRS R consists of the following rules:
c.1(x0) → c.0(x0)
c.0(c.0(x1)) → a.0(a.0(a.1(b.1(b.1(b.0(x1))))))
b.1(b.0(c.0(c.1(x1)))) → c.0(c.0(c.0(a.0(a.0(a.0(a.1(x1)))))))
b.1(b.1(b.0(a.0(x1)))) → b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.0(x1))))))))
b.1(x0) → b.0(x0)
c.0(c.1(x1)) → a.0(a.0(a.1(b.1(b.1(b.1(x1))))))
a.1(x0) → a.0(x0)
b.1(b.1(b.0(a.1(x1)))) → b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(x1))))))))
b.1(b.0(c.0(c.0(x1)))) → c.0(c.0(c.0(a.0(a.0(a.0(a.0(x1)))))))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
↳ QTRS
↳ QTRS Reverse
↳ QTRS Reverse
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ SemLabProof
↳ QDP
↳ DependencyGraphProof
↳ SemLabProof2
Q DP problem:
The TRS P consists of the following rules:
B.1(b.1(b.0(a.1(b.1(b.0(a.0(x0))))))) → B.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.0(x0))))))))))))
B.1(b.0(c.0(c.1(x1)))) → C.0(c.0(a.0(a.0(a.0(a.1(x1))))))
B.1(b.1(b.0(a.0(c.0(c.1(x0)))))) → B.0(b.1(b.0(c.0(c.0(c.0(a.0(a.0(a.0(a.1(x0))))))))))
B.1(b.1(b.0(a.1(b.0(c.0(c.1(x0))))))) → B.0(b.1(b.1(b.0(c.0(c.0(c.0(a.0(a.0(a.0(a.1(x0)))))))))))
B.1(b.1(b.0(a.0(c.0(c.0(x0)))))) → B.0(b.1(b.1(b.1(b.1(b.0(c.0(c.0(c.0(a.0(a.0(a.0(a.0(x0)))))))))))))
B.1(b.1(b.0(a.1(b.0(c.0(c.0(x0))))))) → B.1(b.1(b.1(b.0(c.0(c.0(c.0(a.0(a.0(a.0(a.0(x0)))))))))))
B.1(b.1(b.0(a.1(b.0(a.1(x0)))))) → B.0(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(x0))))))))))))))
B.1(b.1(b.0(a.0(a.0(x0))))) → B.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.0(x0))))))))))))
B.1(b.1(b.0(a.0(c.0(c.1(x0)))))) → B.1(b.1(b.1(b.1(b.0(c.0(c.0(c.0(a.0(a.0(a.0(a.1(x0))))))))))))
B.1(b.1(b.0(a.1(b.1(b.0(a.0(x0))))))) → B.0(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.0(x0)))))))))))))))
B.1(b.1(b.0(a.1(b.0(c.0(c.1(x0))))))) → B.0(b.1(b.1(b.1(b.0(c.0(c.0(c.0(a.0(a.0(a.0(a.1(x0))))))))))))
B.1(b.1(b.0(a.0(a.1(x0))))) → B.0(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(x0)))))))))))
B.1(b.1(b.0(a.1(b.1(b.0(a.0(x0))))))) → B.0(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.0(x0))))))))))))
B.1(b.1(b.0(a.0(a.0(x0))))) → B.0(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.0(x0))))))))))
B.1(b.1(b.0(a.0(c.0(c.0(x0)))))) → B.0(b.1(b.1(b.1(b.0(c.0(c.0(c.0(a.0(a.0(a.0(a.0(x0))))))))))))
B.1(b.1(b.0(a.1(b.0(a.0(x0)))))) → B.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.0(x0))))))))))))))
B.1(b.1(b.0(a.0(c.0(c.0(x0)))))) → B.0(b.1(b.1(b.0(c.0(c.0(c.0(a.0(a.0(a.0(a.0(x0)))))))))))
C.0(c.0(x1)) → B.0(b.0(x1))
B.1(b.1(b.0(a.0(c.0(c.0(x0)))))) → B.1(b.0(c.0(c.0(c.0(a.0(a.0(a.0(a.0(x0)))))))))
B.1(b.1(b.0(a.0(a.0(x0))))) → B.0(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.0(x0))))))))))))
B.1(b.1(b.0(a.0(a.1(x0))))) → B.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(x0)))))))))))))
B.1(b.1(b.0(a.1(b.0(c.0(c.1(x0))))))) → B.1(b.1(b.1(b.0(c.0(c.0(c.0(a.0(a.0(a.0(a.1(x0)))))))))))
B.1(b.1(b.0(a.0(c.0(c.1(x0)))))) → B.0(b.0(c.0(c.0(c.0(a.0(a.0(a.0(a.1(x0)))))))))
B.1(b.1(b.0(a.1(b.0(c.0(c.0(x0))))))) → B.1(b.1(b.0(c.0(c.0(c.0(a.0(a.0(a.0(a.0(x0))))))))))
B.1(b.1(b.0(a.1(b.0(c.0(c.1(x0))))))) → B.0(b.1(b.0(c.0(c.0(c.0(a.0(a.0(a.0(a.1(x0))))))))))
B.1(b.1(b.0(a.0(a.1(x0))))) → B.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(x0)))))))))
B.1(b.1(b.0(a.1(b.1(b.0(a.1(x0))))))) → B.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(x0)))))))))))))
B.1(b.1(b.0(a.1(x1)))) → B.1(b.1(x1))
B.1(b.1(b.0(a.1(b.0(c.0(c.0(x0))))))) → B.1(b.1(b.1(b.1(b.1(b.1(b.0(c.0(c.0(c.0(a.0(a.0(a.0(a.0(x0))))))))))))))
B.1(b.1(b.0(a.0(a.0(x0))))) → B.0(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.0(x0)))))))))
B.1(b.1(b.0(a.1(b.0(c.0(c.0(x0))))))) → B.1(b.1(b.1(b.1(b.1(b.0(c.0(c.0(c.0(a.0(a.0(a.0(a.0(x0)))))))))))))
B.1(b.1(b.0(a.0(x1)))) → B.0(x1)
B.1(b.1(b.0(a.1(b.0(a.1(x0)))))) → B.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(x0)))))))))))))
B.1(b.1(b.0(a.0(a.0(x0))))) → B.0(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.0(x0)))))))))))
B.1(b.1(b.0(a.1(b.0(a.0(x0)))))) → B.0(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.0(x0))))))))))
B.1(b.1(b.0(a.0(c.0(c.1(x0)))))) → B.1(b.0(c.0(c.0(c.0(a.0(a.0(a.0(a.1(x0)))))))))
B.1(b.1(b.0(a.1(b.1(b.0(a.0(x0))))))) → B.0(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.0(x0)))))))))))))
C.0(c.0(x1)) → B.0(b.1(b.0(x1)))
B.1(b.1(b.0(a.0(x1)))) → B.1(b.1(b.0(x1)))
B.1(b.1(b.0(a.0(a.0(x0))))) → B.0(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.0(x0)))))))))))))
C.0(c.1(x1)) → B.0(b.1(x1))
B.1(b.0(c.0(c.0(x1)))) → C.0(c.0(c.0(a.0(a.0(a.0(a.0(x1)))))))
B.1(b.1(b.0(a.0(a.1(x0))))) → B.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(x0))))))))))))
B.1(b.1(b.0(a.1(b.1(b.0(a.0(x0))))))) → B.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.0(x0)))))))))))))))
B.1(b.1(b.0(a.1(b.0(c.0(c.0(x0))))))) → B.1(b.1(b.1(b.1(b.0(c.0(c.0(c.0(a.0(a.0(a.0(a.0(x0))))))))))))
B.1(b.1(b.0(a.0(c.0(c.0(x0)))))) → B.0(b.1(b.0(c.0(c.0(c.0(a.0(a.0(a.0(a.0(x0))))))))))
B.1(b.1(b.0(a.1(b.0(a.1(x0)))))) → B.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(x0)))))))))))
B.1(b.1(b.0(a.1(x1)))) → B.0(b.1(x1))
B.1(b.1(b.0(a.1(b.1(b.0(a.1(x0))))))) → B.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(x0))))))))))))
B.1(b.1(b.0(a.0(c.0(c.0(x0)))))) → B.1(b.1(b.1(b.1(b.0(c.0(c.0(c.0(a.0(a.0(a.0(a.0(x0))))))))))))
C.0(c.0(x1)) → B.1(b.1(b.0(x1)))
B.1(b.1(b.0(a.1(b.0(a.1(x0)))))) → B.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(x0))))))))))
B.1(b.1(b.0(a.1(b.0(a.0(x0)))))) → B.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.0(x0))))))))))
B.1(b.0(c.0(c.1(x1)))) → C.0(c.0(c.0(a.0(a.0(a.0(a.1(x1)))))))
B.1(b.1(b.0(a.0(c.0(c.1(x0)))))) → B.1(b.1(b.1(b.1(b.1(b.0(c.0(c.0(c.0(a.0(a.0(a.0(a.1(x0)))))))))))))
B.1(b.1(b.0(a.0(a.0(x0))))) → B.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.0(x0))))))))))
B.1(b.1(b.0(a.1(b.1(b.0(a.1(x0))))))) → B.0(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(x0)))))))))))))
B.1(b.1(b.0(a.0(a.1(x0))))) → B.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(x0))))))))))
B.1(b.1(b.0(a.1(x1)))) → B.0(b.1(b.1(x1)))
C.0(c.1(x1)) → B.1(x1)
B.1(b.1(b.0(a.1(b.0(a.0(x0)))))) → B.0(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.0(x0))))))))))))))
B.1(b.1(b.0(a.1(b.0(a.0(x0)))))) → B.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.0(x0)))))))))))))
B.1(b.1(b.0(a.1(b.0(c.0(c.1(x0))))))) → B.0(b.1(b.1(b.1(b.1(b.0(c.0(c.0(c.0(a.0(a.0(a.0(a.1(x0)))))))))))))
B.1(b.1(b.0(a.0(a.0(x0))))) → B.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.0(x0)))))))))))))
C.0(c.0(x1)) → B.1(b.0(x1))
C.0(c.1(x1)) → B.0(x1)
B.1(b.1(b.0(a.0(a.1(x0))))) → B.0(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(x0)))))))))))))
B.1(b.1(b.0(a.0(x1)))) → B.0(b.1(b.0(x1)))
B.1(b.0(c.0(c.0(x1)))) → C.0(c.0(a.0(a.0(a.0(a.0(x1))))))
B.1(b.1(b.0(a.1(b.1(b.0(a.0(x0))))))) → B.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.0(x0))))))))))))))
B.1(b.1(b.0(a.0(c.0(c.0(x0)))))) → B.1(b.1(b.0(c.0(c.0(c.0(a.0(a.0(a.0(a.0(x0))))))))))
B.1(b.1(b.0(a.1(b.0(a.1(x0)))))) → B.0(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(x0)))))))))))
B.1(b.1(b.0(a.1(b.0(a.0(x0)))))) → B.0(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.0(x0))))))))))))
B.1(b.1(b.0(a.1(b.1(b.0(a.1(x0))))))) → B.0(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(x0))))))))))))
B.1(b.1(b.0(a.1(b.0(a.1(x0)))))) → B.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(x0))))))))))))))
B.1(b.1(b.0(a.1(b.1(b.0(a.0(x0))))))) → B.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.0(x0)))))))))))
B.1(b.1(b.0(a.1(x1)))) → B.1(b.1(b.1(x1)))
B.1(b.1(b.0(a.0(c.0(c.0(x0)))))) → B.1(b.1(b.1(b.1(b.1(b.0(c.0(c.0(c.0(a.0(a.0(a.0(a.0(x0)))))))))))))
B.1(b.1(b.0(a.1(b.0(a.0(x0)))))) → B.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.0(x0))))))))))))
B.1(b.1(b.0(a.0(c.0(c.1(x0)))))) → B.1(b.1(b.1(b.0(c.0(c.0(c.0(a.0(a.0(a.0(a.1(x0)))))))))))
B.1(b.1(b.0(a.0(c.0(c.1(x0)))))) → B.1(b.1(b.0(c.0(c.0(c.0(a.0(a.0(a.0(a.1(x0))))))))))
B.1(b.1(b.0(a.1(b.0(a.1(x0)))))) → B.0(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(x0)))))))))))))
B.1(b.1(b.0(a.1(b.0(a.0(x0)))))) → B.0(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.0(x0)))))))))))
C.0(c.1(x1)) → B.1(b.1(x1))
B.1(b.1(b.0(a.0(a.0(x0))))) → B.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.0(x0)))))))))
B.1(b.1(b.0(a.1(b.0(c.0(c.1(x0))))))) → B.1(b.1(b.1(b.1(b.1(b.1(b.0(c.0(c.0(c.0(a.0(a.0(a.0(a.1(x0))))))))))))))
B.1(b.1(b.0(a.0(c.0(c.1(x0)))))) → B.0(b.1(b.1(b.1(b.0(c.0(c.0(c.0(a.0(a.0(a.0(a.1(x0))))))))))))
B.1(b.1(b.0(a.0(c.0(c.1(x0)))))) → B.0(b.1(b.1(b.1(b.1(b.0(c.0(c.0(c.0(a.0(a.0(a.0(a.1(x0)))))))))))))
B.1(b.1(b.0(a.1(b.1(b.0(a.1(x0))))))) → B.0(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(x0)))))))))))
B.1(b.1(b.0(a.1(b.0(a.1(x0)))))) → B.0(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(x0))))))))))))
B.1(b.1(b.0(a.0(a.1(x0))))) → B.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(x0)))))))))))
B.1(b.1(b.0(a.1(b.1(b.0(a.1(x0))))))) → B.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(x0)))))))))))))))
B.1(b.1(b.0(a.1(b.1(b.0(a.1(x0))))))) → B.0(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(x0)))))))))))))))
B.1(b.1(b.0(a.1(b.0(c.0(c.1(x0))))))) → B.1(b.1(b.1(b.1(b.0(c.0(c.0(c.0(a.0(a.0(a.0(a.1(x0))))))))))))
B.1(b.1(b.0(a.1(b.0(c.0(c.0(x0))))))) → B.0(b.1(b.0(c.0(c.0(c.0(a.0(a.0(a.0(a.0(x0))))))))))
B.1(b.1(b.0(a.0(a.1(x0))))) → B.0(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(x0)))))))))
B.1(b.1(b.0(a.1(b.0(a.0(x0)))))) → B.0(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.0(x0)))))))))))))
B.1(b.1(b.0(a.1(b.1(b.0(a.0(x0))))))) → B.0(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.0(x0))))))))))))))
B.1(b.1(b.0(a.1(b.0(c.0(c.1(x0))))))) → B.1(b.1(b.1(b.1(b.1(b.0(c.0(c.0(c.0(a.0(a.0(a.0(a.1(x0)))))))))))))
B.1(b.1(b.0(a.1(b.1(b.0(a.0(x0))))))) → B.0(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.0(x0)))))))))))
B.1(b.1(b.0(a.0(x1)))) → B.0(b.0(x1))
B.1(b.1(b.0(a.0(a.0(x0))))) → B.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.0(x0)))))))))))
B.1(b.1(b.0(a.0(c.0(c.1(x0)))))) → B.0(b.1(b.1(b.0(c.0(c.0(c.0(a.0(a.0(a.0(a.1(x0)))))))))))
B.1(b.1(b.0(a.1(x1)))) → B.1(x1)
C.0(c.1(x1)) → B.0(b.1(b.1(x1)))
B.1(b.1(b.0(a.0(a.1(x0))))) → B.0(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(x0))))))))))
B.1(b.1(b.0(a.1(b.1(b.0(a.1(x0))))))) → B.0(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(x0))))))))))))))
B.1(b.1(b.0(a.0(x1)))) → B.1(b.0(x1))
B.1(b.1(b.0(a.1(b.0(c.0(c.0(x0))))))) → B.0(b.1(b.1(b.1(b.1(b.1(b.0(c.0(c.0(c.0(a.0(a.0(a.0(a.0(x0))))))))))))))
B.1(b.1(b.0(a.1(b.1(b.0(a.0(x0))))))) → B.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.0(x0)))))))))))))
B.1(b.1(b.0(a.1(x1)))) → B.0(x1)
B.1(b.1(b.0(a.1(b.0(c.0(c.0(x0))))))) → B.0(b.1(b.1(b.0(c.0(c.0(c.0(a.0(a.0(a.0(a.0(x0)))))))))))
C.0(c.0(x1)) → B.0(x1)
B.1(b.1(b.0(a.1(b.1(b.0(a.1(x0))))))) → B.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(x0)))))))))))
B.1(b.1(b.0(a.1(b.1(b.0(a.1(x0))))))) → B.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(x0))))))))))))))
B.1(b.1(b.0(a.0(c.0(c.0(x0)))))) → B.0(b.0(c.0(c.0(c.0(a.0(a.0(a.0(a.0(x0)))))))))
B.1(b.1(b.0(a.1(b.0(c.0(c.1(x0))))))) → B.1(b.1(b.0(c.0(c.0(c.0(a.0(a.0(a.0(a.1(x0))))))))))
B.1(b.1(b.0(a.1(b.0(a.1(x0)))))) → B.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(x0))))))))))))
B.1(b.1(b.0(a.1(b.0(a.1(x0)))))) → B.0(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(x0))))))))))
B.1(b.1(b.0(a.1(b.0(c.0(c.0(x0))))))) → B.0(b.1(b.1(b.1(b.0(c.0(c.0(c.0(a.0(a.0(a.0(a.0(x0))))))))))))
B.1(b.1(b.0(a.0(a.1(x0))))) → B.0(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(x0))))))))))))
C.0(c.1(x1)) → B.1(b.1(b.1(x1)))
B.1(b.1(b.0(a.1(b.0(c.0(c.1(x0))))))) → B.0(b.1(b.1(b.1(b.1(b.1(b.0(c.0(c.0(c.0(a.0(a.0(a.0(a.1(x0))))))))))))))
B.1(b.1(b.0(a.0(c.0(c.0(x0)))))) → B.1(b.1(b.1(b.0(c.0(c.0(c.0(a.0(a.0(a.0(a.0(x0)))))))))))
B.1(b.1(b.0(a.1(b.0(c.0(c.0(x0))))))) → B.0(b.1(b.1(b.1(b.1(b.0(c.0(c.0(c.0(a.0(a.0(a.0(a.0(x0)))))))))))))
B.1(b.1(b.0(a.1(b.0(a.0(x0)))))) → B.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.0(x0)))))))))))
The TRS R consists of the following rules:
c.1(x0) → c.0(x0)
c.0(c.0(x1)) → a.0(a.0(a.1(b.1(b.1(b.0(x1))))))
b.1(b.0(c.0(c.1(x1)))) → c.0(c.0(c.0(a.0(a.0(a.0(a.1(x1)))))))
b.1(b.1(b.0(a.0(x1)))) → b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.0(x1))))))))
b.1(x0) → b.0(x0)
c.0(c.1(x1)) → a.0(a.0(a.1(b.1(b.1(b.1(x1))))))
a.1(x0) → a.0(x0)
b.1(b.1(b.0(a.1(x1)))) → b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(x1))))))))
b.1(b.0(c.0(c.0(x1)))) → c.0(c.0(c.0(a.0(a.0(a.0(a.0(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 65 less nodes.
↳ QTRS
↳ QTRS Reverse
↳ QTRS Reverse
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ SemLabProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ UsableRulesReductionPairsProof
↳ SemLabProof2
Q DP problem:
The TRS P consists of the following rules:
B.1(b.1(b.0(a.1(b.1(b.0(a.0(x0))))))) → B.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.0(x0))))))))))))
C.0(c.0(x1)) → B.1(b.0(x1))
B.1(b.0(c.0(c.0(x1)))) → C.0(c.0(a.0(a.0(a.0(a.0(x1))))))
B.1(b.0(c.0(c.1(x1)))) → C.0(c.0(a.0(a.0(a.0(a.1(x1))))))
B.1(b.1(b.0(a.1(b.1(b.0(a.0(x0))))))) → B.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.0(x0))))))))))))))
B.1(b.1(b.0(a.0(c.0(c.0(x0)))))) → B.1(b.1(b.0(c.0(c.0(c.0(a.0(a.0(a.0(a.0(x0))))))))))
B.1(b.1(b.0(a.1(b.0(c.0(c.0(x0))))))) → B.1(b.1(b.1(b.0(c.0(c.0(c.0(a.0(a.0(a.0(a.0(x0)))))))))))
B.1(b.1(b.0(a.0(a.0(x0))))) → B.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.0(x0))))))))))))
B.1(b.1(b.0(a.0(c.0(c.1(x0)))))) → B.1(b.1(b.1(b.1(b.0(c.0(c.0(c.0(a.0(a.0(a.0(a.1(x0))))))))))))
B.1(b.1(b.0(a.1(b.0(a.1(x0)))))) → B.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(x0))))))))))))))
B.1(b.1(b.0(a.1(b.1(b.0(a.0(x0))))))) → B.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.0(x0)))))))))))
B.1(b.1(b.0(a.0(c.0(c.0(x0)))))) → B.1(b.1(b.1(b.1(b.1(b.0(c.0(c.0(c.0(a.0(a.0(a.0(a.0(x0)))))))))))))
B.1(b.1(b.0(a.1(x1)))) → B.1(b.1(b.1(x1)))
B.1(b.1(b.0(a.1(b.0(a.0(x0)))))) → B.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.0(x0))))))))))))))
B.1(b.1(b.0(a.1(b.0(a.0(x0)))))) → B.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.0(x0))))))))))))
B.1(b.1(b.0(a.0(c.0(c.1(x0)))))) → B.1(b.1(b.1(b.0(c.0(c.0(c.0(a.0(a.0(a.0(a.1(x0)))))))))))
B.1(b.1(b.0(a.0(c.0(c.1(x0)))))) → B.1(b.1(b.0(c.0(c.0(c.0(a.0(a.0(a.0(a.1(x0))))))))))
B.1(b.1(b.0(a.0(a.0(x0))))) → B.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.0(x0)))))))))
B.1(b.1(b.0(a.1(b.0(c.0(c.1(x0))))))) → B.1(b.1(b.1(b.1(b.1(b.1(b.0(c.0(c.0(c.0(a.0(a.0(a.0(a.1(x0))))))))))))))
B.1(b.1(b.0(a.0(c.0(c.0(x0)))))) → B.1(b.0(c.0(c.0(c.0(a.0(a.0(a.0(a.0(x0)))))))))
B.1(b.1(b.0(a.0(a.1(x0))))) → B.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(x0)))))))))))))
B.1(b.1(b.0(a.0(a.1(x0))))) → B.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(x0)))))))))))
B.1(b.1(b.0(a.1(b.1(b.0(a.1(x0))))))) → B.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(x0)))))))))))))))
B.1(b.1(b.0(a.1(b.0(c.0(c.1(x0))))))) → B.1(b.1(b.1(b.0(c.0(c.0(c.0(a.0(a.0(a.0(a.1(x0)))))))))))
B.1(b.1(b.0(a.1(b.0(c.0(c.0(x0))))))) → B.1(b.1(b.0(c.0(c.0(c.0(a.0(a.0(a.0(a.0(x0))))))))))
B.1(b.1(b.0(a.0(a.1(x0))))) → B.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(x0)))))))))
B.1(b.1(b.0(a.1(b.1(b.0(a.1(x0))))))) → B.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(x0)))))))))))))
B.1(b.1(b.0(a.1(b.0(c.0(c.1(x0))))))) → B.1(b.1(b.1(b.1(b.0(c.0(c.0(c.0(a.0(a.0(a.0(a.1(x0))))))))))))
B.1(b.1(b.0(a.1(x1)))) → B.1(b.1(x1))
B.1(b.1(b.0(a.1(b.0(c.0(c.0(x0))))))) → B.1(b.1(b.1(b.1(b.1(b.1(b.0(c.0(c.0(c.0(a.0(a.0(a.0(a.0(x0))))))))))))))
B.1(b.1(b.0(a.1(b.0(c.0(c.0(x0))))))) → B.1(b.1(b.1(b.1(b.1(b.0(c.0(c.0(c.0(a.0(a.0(a.0(a.0(x0)))))))))))))
B.1(b.1(b.0(a.1(b.0(a.1(x0)))))) → B.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(x0)))))))))))))
B.1(b.1(b.0(a.0(c.0(c.1(x0)))))) → B.1(b.0(c.0(c.0(c.0(a.0(a.0(a.0(a.1(x0)))))))))
B.1(b.1(b.0(a.1(b.0(c.0(c.1(x0))))))) → B.1(b.1(b.1(b.1(b.1(b.0(c.0(c.0(c.0(a.0(a.0(a.0(a.1(x0)))))))))))))
B.1(b.1(b.0(a.0(x1)))) → B.1(b.1(b.0(x1)))
B.1(b.1(b.0(a.0(a.0(x0))))) → B.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.0(x0)))))))))))
B.1(b.1(b.0(a.1(x1)))) → B.1(x1)
B.1(b.1(b.0(a.0(x1)))) → B.1(b.0(x1))
B.1(b.1(b.0(a.1(b.1(b.0(a.0(x0))))))) → B.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.0(x0)))))))))))))
B.1(b.0(c.0(c.0(x1)))) → C.0(c.0(c.0(a.0(a.0(a.0(a.0(x1)))))))
B.1(b.1(b.0(a.0(a.1(x0))))) → B.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(x0))))))))))))
B.1(b.1(b.0(a.1(b.1(b.0(a.0(x0))))))) → B.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.0(x0)))))))))))))))
B.1(b.1(b.0(a.1(b.0(c.0(c.0(x0))))))) → B.1(b.1(b.1(b.1(b.0(c.0(c.0(c.0(a.0(a.0(a.0(a.0(x0))))))))))))
B.1(b.1(b.0(a.1(b.0(a.1(x0)))))) → B.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(x0)))))))))))
B.1(b.1(b.0(a.1(b.1(b.0(a.1(x0))))))) → B.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(x0))))))))))))
B.1(b.1(b.0(a.1(b.1(b.0(a.1(x0))))))) → B.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(x0)))))))))))
B.1(b.1(b.0(a.0(c.0(c.0(x0)))))) → B.1(b.1(b.1(b.1(b.0(c.0(c.0(c.0(a.0(a.0(a.0(a.0(x0))))))))))))
B.1(b.1(b.0(a.1(b.1(b.0(a.1(x0))))))) → B.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(x0))))))))))))))
B.1(b.1(b.0(a.1(b.0(a.1(x0)))))) → B.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(x0))))))))))
C.0(c.0(x1)) → B.1(b.1(b.0(x1)))
B.1(b.1(b.0(a.1(b.0(a.0(x0)))))) → B.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.0(x0))))))))))
B.1(b.0(c.0(c.1(x1)))) → C.0(c.0(c.0(a.0(a.0(a.0(a.1(x1)))))))
B.1(b.1(b.0(a.1(b.0(c.0(c.1(x0))))))) → B.1(b.1(b.0(c.0(c.0(c.0(a.0(a.0(a.0(a.1(x0))))))))))
B.1(b.1(b.0(a.0(c.0(c.1(x0)))))) → B.1(b.1(b.1(b.1(b.1(b.0(c.0(c.0(c.0(a.0(a.0(a.0(a.1(x0)))))))))))))
B.1(b.1(b.0(a.1(b.0(a.1(x0)))))) → B.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(x0))))))))))))
B.1(b.1(b.0(a.0(a.0(x0))))) → B.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.0(x0))))))))))
B.1(b.1(b.0(a.0(a.1(x0))))) → B.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(x0))))))))))
B.1(b.1(b.0(a.0(c.0(c.0(x0)))))) → B.1(b.1(b.1(b.0(c.0(c.0(c.0(a.0(a.0(a.0(a.0(x0)))))))))))
B.1(b.1(b.0(a.1(b.0(a.0(x0)))))) → B.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.0(x0)))))))))))
B.1(b.1(b.0(a.0(a.0(x0))))) → B.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.0(x0)))))))))))))
B.1(b.1(b.0(a.1(b.0(a.0(x0)))))) → B.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.0(x0)))))))))))))
The TRS R consists of the following rules:
c.1(x0) → c.0(x0)
c.0(c.0(x1)) → a.0(a.0(a.1(b.1(b.1(b.0(x1))))))
b.1(b.0(c.0(c.1(x1)))) → c.0(c.0(c.0(a.0(a.0(a.0(a.1(x1)))))))
b.1(b.1(b.0(a.0(x1)))) → b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.0(x1))))))))
b.1(x0) → b.0(x0)
c.0(c.1(x1)) → a.0(a.0(a.1(b.1(b.1(b.1(x1))))))
a.1(x0) → a.0(x0)
b.1(b.1(b.0(a.1(x1)))) → b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(x1))))))))
b.1(b.0(c.0(c.0(x1)))) → c.0(c.0(c.0(a.0(a.0(a.0(a.0(x1)))))))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By using the usable rules with reduction pair processor [15] with a polynomial ordering [25], all dependency pairs and the corresponding usable rules [17] can be oriented non-strictly. All non-usable rules are removed, and those dependency pairs and usable rules that have been oriented strictly or contain non-usable symbols in their left-hand side are removed as well.
The following dependency pairs can be deleted:
B.1(b.0(c.0(c.1(x1)))) → C.0(c.0(a.0(a.0(a.0(a.1(x1))))))
B.1(b.1(b.0(a.0(c.0(c.1(x0)))))) → B.1(b.1(b.1(b.1(b.0(c.0(c.0(c.0(a.0(a.0(a.0(a.1(x0))))))))))))
B.1(b.1(b.0(a.0(c.0(c.1(x0)))))) → B.1(b.1(b.1(b.0(c.0(c.0(c.0(a.0(a.0(a.0(a.1(x0)))))))))))
B.1(b.1(b.0(a.0(c.0(c.1(x0)))))) → B.1(b.1(b.0(c.0(c.0(c.0(a.0(a.0(a.0(a.1(x0))))))))))
B.1(b.1(b.0(a.1(b.0(c.0(c.1(x0))))))) → B.1(b.1(b.1(b.1(b.1(b.1(b.0(c.0(c.0(c.0(a.0(a.0(a.0(a.1(x0))))))))))))))
B.1(b.1(b.0(a.1(b.0(c.0(c.1(x0))))))) → B.1(b.1(b.1(b.0(c.0(c.0(c.0(a.0(a.0(a.0(a.1(x0)))))))))))
B.1(b.1(b.0(a.1(b.0(c.0(c.1(x0))))))) → B.1(b.1(b.1(b.1(b.0(c.0(c.0(c.0(a.0(a.0(a.0(a.1(x0))))))))))))
B.1(b.1(b.0(a.0(c.0(c.1(x0)))))) → B.1(b.0(c.0(c.0(c.0(a.0(a.0(a.0(a.1(x0)))))))))
B.1(b.1(b.0(a.1(b.0(c.0(c.1(x0))))))) → B.1(b.1(b.1(b.1(b.1(b.0(c.0(c.0(c.0(a.0(a.0(a.0(a.1(x0)))))))))))))
B.1(b.0(c.0(c.1(x1)))) → C.0(c.0(c.0(a.0(a.0(a.0(a.1(x1)))))))
B.1(b.1(b.0(a.1(b.0(c.0(c.1(x0))))))) → B.1(b.1(b.0(c.0(c.0(c.0(a.0(a.0(a.0(a.1(x0))))))))))
B.1(b.1(b.0(a.0(c.0(c.1(x0)))))) → B.1(b.1(b.1(b.1(b.1(b.0(c.0(c.0(c.0(a.0(a.0(a.0(a.1(x0)))))))))))))
The following rules are removed from R:
b.1(b.0(c.0(c.1(x1)))) → c.0(c.0(c.0(a.0(a.0(a.0(a.1(x1)))))))
c.0(c.1(x1)) → a.0(a.0(a.1(b.1(b.1(b.1(x1))))))
Used ordering: POLO with Polynomial interpretation [25]:
POL(B.1(x1)) = x1
POL(C.0(x1)) = x1
POL(a.0(x1)) = x1
POL(a.1(x1)) = x1
POL(b.0(x1)) = x1
POL(b.1(x1)) = x1
POL(c.0(x1)) = x1
POL(c.1(x1)) = x1
↳ QTRS
↳ QTRS Reverse
↳ QTRS Reverse
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ SemLabProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ UsableRulesReductionPairsProof
↳ QDP
↳ SemLabProof2
Q DP problem:
The TRS P consists of the following rules:
B.1(b.1(b.0(a.1(b.1(b.0(a.0(x0))))))) → B.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.0(x0))))))))))))
C.0(c.0(x1)) → B.1(b.0(x1))
B.1(b.0(c.0(c.0(x1)))) → C.0(c.0(a.0(a.0(a.0(a.0(x1))))))
B.1(b.1(b.0(a.1(b.1(b.0(a.0(x0))))))) → B.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.0(x0))))))))))))))
B.1(b.1(b.0(a.0(c.0(c.0(x0)))))) → B.1(b.1(b.0(c.0(c.0(c.0(a.0(a.0(a.0(a.0(x0))))))))))
B.1(b.1(b.0(a.1(b.0(c.0(c.0(x0))))))) → B.1(b.1(b.1(b.0(c.0(c.0(c.0(a.0(a.0(a.0(a.0(x0)))))))))))
B.1(b.1(b.0(a.0(a.0(x0))))) → B.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.0(x0))))))))))))
B.1(b.1(b.0(a.1(b.0(a.1(x0)))))) → B.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(x0))))))))))))))
B.1(b.1(b.0(a.1(b.1(b.0(a.0(x0))))))) → B.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.0(x0)))))))))))
B.1(b.1(b.0(a.0(c.0(c.0(x0)))))) → B.1(b.1(b.1(b.1(b.1(b.0(c.0(c.0(c.0(a.0(a.0(a.0(a.0(x0)))))))))))))
B.1(b.1(b.0(a.1(x1)))) → B.1(b.1(b.1(x1)))
B.1(b.1(b.0(a.1(b.0(a.0(x0)))))) → B.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.0(x0))))))))))))))
B.1(b.1(b.0(a.1(b.0(a.0(x0)))))) → B.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.0(x0))))))))))))
B.1(b.1(b.0(a.0(a.0(x0))))) → B.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.0(x0)))))))))
B.1(b.1(b.0(a.0(c.0(c.0(x0)))))) → B.1(b.0(c.0(c.0(c.0(a.0(a.0(a.0(a.0(x0)))))))))
B.1(b.1(b.0(a.0(a.1(x0))))) → B.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(x0)))))))))))))
B.1(b.1(b.0(a.0(a.1(x0))))) → B.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(x0)))))))))))
B.1(b.1(b.0(a.1(b.1(b.0(a.1(x0))))))) → B.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(x0)))))))))))))))
B.1(b.1(b.0(a.1(b.0(c.0(c.0(x0))))))) → B.1(b.1(b.0(c.0(c.0(c.0(a.0(a.0(a.0(a.0(x0))))))))))
B.1(b.1(b.0(a.1(b.1(b.0(a.1(x0))))))) → B.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(x0)))))))))))))
B.1(b.1(b.0(a.0(a.1(x0))))) → B.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(x0)))))))))
B.1(b.1(b.0(a.1(x1)))) → B.1(b.1(x1))
B.1(b.1(b.0(a.1(b.0(c.0(c.0(x0))))))) → B.1(b.1(b.1(b.1(b.1(b.1(b.0(c.0(c.0(c.0(a.0(a.0(a.0(a.0(x0))))))))))))))
B.1(b.1(b.0(a.1(b.0(c.0(c.0(x0))))))) → B.1(b.1(b.1(b.1(b.1(b.0(c.0(c.0(c.0(a.0(a.0(a.0(a.0(x0)))))))))))))
B.1(b.1(b.0(a.1(b.0(a.1(x0)))))) → B.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(x0)))))))))))))
B.1(b.1(b.0(a.0(x1)))) → B.1(b.1(b.0(x1)))
B.1(b.1(b.0(a.0(a.0(x0))))) → B.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.0(x0)))))))))))
B.1(b.1(b.0(a.1(x1)))) → B.1(x1)
B.1(b.1(b.0(a.0(x1)))) → B.1(b.0(x1))
B.1(b.1(b.0(a.1(b.1(b.0(a.0(x0))))))) → B.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.0(x0)))))))))))))
B.1(b.0(c.0(c.0(x1)))) → C.0(c.0(c.0(a.0(a.0(a.0(a.0(x1)))))))
B.1(b.1(b.0(a.0(a.1(x0))))) → B.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(x0))))))))))))
B.1(b.1(b.0(a.1(b.1(b.0(a.0(x0))))))) → B.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.0(x0)))))))))))))))
B.1(b.1(b.0(a.1(b.0(c.0(c.0(x0))))))) → B.1(b.1(b.1(b.1(b.0(c.0(c.0(c.0(a.0(a.0(a.0(a.0(x0))))))))))))
B.1(b.1(b.0(a.1(b.0(a.1(x0)))))) → B.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(x0)))))))))))
B.1(b.1(b.0(a.1(b.1(b.0(a.1(x0))))))) → B.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(x0))))))))))))
B.1(b.1(b.0(a.1(b.1(b.0(a.1(x0))))))) → B.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(x0)))))))))))
B.1(b.1(b.0(a.0(c.0(c.0(x0)))))) → B.1(b.1(b.1(b.1(b.0(c.0(c.0(c.0(a.0(a.0(a.0(a.0(x0))))))))))))
B.1(b.1(b.0(a.1(b.1(b.0(a.1(x0))))))) → B.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(x0))))))))))))))
B.1(b.1(b.0(a.1(b.0(a.1(x0)))))) → B.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(x0))))))))))
C.0(c.0(x1)) → B.1(b.1(b.0(x1)))
B.1(b.1(b.0(a.1(b.0(a.0(x0)))))) → B.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.0(x0))))))))))
B.1(b.1(b.0(a.1(b.0(a.1(x0)))))) → B.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(x0))))))))))))
B.1(b.1(b.0(a.0(a.0(x0))))) → B.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.0(x0))))))))))
B.1(b.1(b.0(a.0(a.1(x0))))) → B.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(x0))))))))))
B.1(b.1(b.0(a.0(c.0(c.0(x0)))))) → B.1(b.1(b.1(b.0(c.0(c.0(c.0(a.0(a.0(a.0(a.0(x0)))))))))))
B.1(b.1(b.0(a.1(b.0(a.0(x0)))))) → B.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.0(x0)))))))))))
B.1(b.1(b.0(a.0(a.0(x0))))) → B.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.0(x0)))))))))))))
B.1(b.1(b.0(a.1(b.0(a.0(x0)))))) → B.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.0(x0)))))))))))))
The TRS R consists of the following rules:
b.1(b.1(b.0(a.1(x1)))) → b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.1(x1))))))))
b.1(b.1(b.0(a.0(x1)))) → b.1(b.1(b.1(b.1(b.1(b.1(b.1(b.0(x1))))))))
b.1(x0) → b.0(x0)
b.1(b.0(c.0(c.0(x1)))) → c.0(c.0(c.0(a.0(a.0(a.0(a.0(x1)))))))
c.0(c.0(x1)) → a.0(a.0(a.1(b.1(b.1(b.0(x1))))))
a.1(x0) → a.0(x0)
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
As can be seen after transforming the QDP problem by semantic labelling [33] and then some rule deleting processors, only certain labelled rules and pairs can be used.
Hence, we only have to consider all unlabelled pairs and rules (without the decreasing rules for quasi-models).
↳ QTRS
↳ QTRS Reverse
↳ QTRS Reverse
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ SemLabProof
↳ SemLabProof2
↳ QDP
↳ QDPToSRSProof
Q DP problem:
The TRS P consists of the following rules:
B(b(b(a(b(c(c(x0))))))) → B(b(b(b(b(b(b(c(c(c(a(a(a(a(x0))))))))))))))
B(b(b(a(a(x0))))) → B(b(b(b(b(b(b(b(b(b(x0))))))))))
B(b(b(a(b(a(x0)))))) → B(b(b(b(b(b(b(b(b(b(x0))))))))))
B(b(b(a(a(x0))))) → B(b(b(b(b(b(b(b(b(b(b(b(x0))))))))))))
B(b(b(a(b(a(x0)))))) → B(b(b(b(b(b(b(b(b(b(b(b(x0))))))))))))
B(b(b(a(b(b(a(x0))))))) → B(b(b(b(b(b(b(b(b(b(b(b(x0))))))))))))
B(b(b(a(b(b(a(x0))))))) → B(b(b(b(b(b(b(b(b(b(b(x0)))))))))))
B(b(b(a(a(x0))))) → B(b(b(b(b(b(b(b(b(b(b(b(b(x0)))))))))))))
B(b(b(a(c(c(x0)))))) → B(b(b(b(c(c(c(a(a(a(a(x0)))))))))))
B(b(b(a(x1)))) → B(b(b(x1)))
B(b(b(a(x1)))) → B(x1)
B(b(b(a(b(c(c(x0))))))) → B(b(b(b(b(b(c(c(c(a(a(a(a(x0)))))))))))))
B(b(b(a(b(b(a(x0))))))) → B(b(b(b(b(b(b(b(b(b(b(b(b(b(x0))))))))))))))
B(b(b(a(b(c(c(x0))))))) → B(b(b(b(b(c(c(c(a(a(a(a(x0))))))))))))
B(b(b(a(b(a(x0)))))) → B(b(b(b(b(b(b(b(b(b(b(b(b(x0)))))))))))))
B(b(b(a(b(a(x0)))))) → B(b(b(b(b(b(b(b(b(b(b(x0)))))))))))
B(b(b(a(b(b(a(x0))))))) → B(b(b(b(b(b(b(b(b(b(b(b(b(x0)))))))))))))
B(b(b(a(c(c(x0)))))) → B(b(b(b(b(b(c(c(c(a(a(a(a(x0)))))))))))))
B(b(b(a(b(b(a(x0))))))) → B(b(b(b(b(b(b(b(b(b(b(b(b(b(b(x0)))))))))))))))
B(b(b(a(c(c(x0)))))) → B(b(b(b(b(c(c(c(a(a(a(a(x0))))))))))))
B(b(b(a(b(a(x0)))))) → B(b(b(b(b(b(b(b(b(b(b(b(b(b(x0))))))))))))))
B(b(b(a(b(c(c(x0))))))) → B(b(b(c(c(c(a(a(a(a(x0))))))))))
B(b(b(a(c(c(x0)))))) → B(b(b(c(c(c(a(a(a(a(x0))))))))))
B(b(c(c(x1)))) → C(c(a(a(a(a(x1))))))
B(b(b(a(a(x0))))) → B(b(b(b(b(b(b(b(b(b(b(x0)))))))))))
C(c(x1)) → B(b(x1))
B(b(b(a(c(c(x0)))))) → B(b(c(c(c(a(a(a(a(x0)))))))))
B(b(b(a(b(c(c(x0))))))) → B(b(b(b(c(c(c(a(a(a(a(x0)))))))))))
B(b(b(a(x1)))) → B(b(x1))
B(b(c(c(x1)))) → C(c(c(a(a(a(a(x1)))))))
C(c(x1)) → B(b(b(x1)))
B(b(b(a(a(x0))))) → B(b(b(b(b(b(b(b(b(x0)))))))))
The TRS R consists of the following rules:
c(c(x1)) → a(a(a(b(b(b(x1))))))
b(b(b(a(x1)))) → b(b(b(b(b(b(b(b(x1))))))))
b(b(c(c(x1)))) → c(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
↳ QTRS Reverse
↳ QTRS Reverse
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ SemLabProof
↳ SemLabProof2
↳ QDP
↳ QDPToSRSProof
↳ QTRS
↳ QTRS Reverse
Q restricted rewrite system:
The TRS R consists of the following rules:
c(c(x1)) → a(a(a(b(b(b(x1))))))
b(b(b(a(x1)))) → b(b(b(b(b(b(b(b(x1))))))))
b(b(c(c(x1)))) → c(c(c(a(a(a(a(x1)))))))
B(b(b(a(b(c(c(x0))))))) → B(b(b(b(b(b(b(c(c(c(a(a(a(a(x0))))))))))))))
B(b(b(a(a(x0))))) → B(b(b(b(b(b(b(b(b(b(x0))))))))))
B(b(b(a(b(a(x0)))))) → B(b(b(b(b(b(b(b(b(b(x0))))))))))
B(b(b(a(a(x0))))) → B(b(b(b(b(b(b(b(b(b(b(b(x0))))))))))))
B(b(b(a(b(a(x0)))))) → B(b(b(b(b(b(b(b(b(b(b(b(x0))))))))))))
B(b(b(a(b(b(a(x0))))))) → B(b(b(b(b(b(b(b(b(b(b(b(x0))))))))))))
B(b(b(a(b(b(a(x0))))))) → B(b(b(b(b(b(b(b(b(b(b(x0)))))))))))
B(b(b(a(a(x0))))) → B(b(b(b(b(b(b(b(b(b(b(b(b(x0)))))))))))))
B(b(b(a(c(c(x0)))))) → B(b(b(b(c(c(c(a(a(a(a(x0)))))))))))
B(b(b(a(x1)))) → B(b(b(x1)))
B(b(b(a(x1)))) → B(x1)
B(b(b(a(b(c(c(x0))))))) → B(b(b(b(b(b(c(c(c(a(a(a(a(x0)))))))))))))
B(b(b(a(b(b(a(x0))))))) → B(b(b(b(b(b(b(b(b(b(b(b(b(b(x0))))))))))))))
B(b(b(a(b(c(c(x0))))))) → B(b(b(b(b(c(c(c(a(a(a(a(x0))))))))))))
B(b(b(a(b(a(x0)))))) → B(b(b(b(b(b(b(b(b(b(b(b(b(x0)))))))))))))
B(b(b(a(b(a(x0)))))) → B(b(b(b(b(b(b(b(b(b(b(x0)))))))))))
B(b(b(a(b(b(a(x0))))))) → B(b(b(b(b(b(b(b(b(b(b(b(b(x0)))))))))))))
B(b(b(a(c(c(x0)))))) → B(b(b(b(b(b(c(c(c(a(a(a(a(x0)))))))))))))
B(b(b(a(b(b(a(x0))))))) → B(b(b(b(b(b(b(b(b(b(b(b(b(b(b(x0)))))))))))))))
B(b(b(a(c(c(x0)))))) → B(b(b(b(b(c(c(c(a(a(a(a(x0))))))))))))
B(b(b(a(b(a(x0)))))) → B(b(b(b(b(b(b(b(b(b(b(b(b(b(x0))))))))))))))
B(b(b(a(b(c(c(x0))))))) → B(b(b(c(c(c(a(a(a(a(x0))))))))))
B(b(b(a(c(c(x0)))))) → B(b(b(c(c(c(a(a(a(a(x0))))))))))
B(b(c(c(x1)))) → C(c(a(a(a(a(x1))))))
B(b(b(a(a(x0))))) → B(b(b(b(b(b(b(b(b(b(b(x0)))))))))))
C(c(x1)) → B(b(x1))
B(b(b(a(c(c(x0)))))) → B(b(c(c(c(a(a(a(a(x0)))))))))
B(b(b(a(b(c(c(x0))))))) → B(b(b(b(c(c(c(a(a(a(a(x0)))))))))))
B(b(b(a(x1)))) → B(b(x1))
B(b(c(c(x1)))) → C(c(c(a(a(a(a(x1)))))))
C(c(x1)) → B(b(b(x1)))
B(b(b(a(a(x0))))) → B(b(b(b(b(b(b(b(b(x0)))))))))
Q is empty.
We have reversed the following QTRS:
The set of rules R is
c(c(x1)) → a(a(a(b(b(b(x1))))))
b(b(b(a(x1)))) → b(b(b(b(b(b(b(b(x1))))))))
b(b(c(c(x1)))) → c(c(c(a(a(a(a(x1)))))))
B(b(b(a(b(c(c(x0))))))) → B(b(b(b(b(b(b(c(c(c(a(a(a(a(x0))))))))))))))
B(b(b(a(a(x0))))) → B(b(b(b(b(b(b(b(b(b(x0))))))))))
B(b(b(a(b(a(x0)))))) → B(b(b(b(b(b(b(b(b(b(x0))))))))))
B(b(b(a(a(x0))))) → B(b(b(b(b(b(b(b(b(b(b(b(x0))))))))))))
B(b(b(a(b(a(x0)))))) → B(b(b(b(b(b(b(b(b(b(b(b(x0))))))))))))
B(b(b(a(b(b(a(x0))))))) → B(b(b(b(b(b(b(b(b(b(b(b(x0))))))))))))
B(b(b(a(b(b(a(x0))))))) → B(b(b(b(b(b(b(b(b(b(b(x0)))))))))))
B(b(b(a(a(x0))))) → B(b(b(b(b(b(b(b(b(b(b(b(b(x0)))))))))))))
B(b(b(a(c(c(x0)))))) → B(b(b(b(c(c(c(a(a(a(a(x0)))))))))))
B(b(b(a(x1)))) → B(b(b(x1)))
B(b(b(a(x1)))) → B(x1)
B(b(b(a(b(c(c(x0))))))) → B(b(b(b(b(b(c(c(c(a(a(a(a(x0)))))))))))))
B(b(b(a(b(b(a(x0))))))) → B(b(b(b(b(b(b(b(b(b(b(b(b(b(x0))))))))))))))
B(b(b(a(b(c(c(x0))))))) → B(b(b(b(b(c(c(c(a(a(a(a(x0))))))))))))
B(b(b(a(b(a(x0)))))) → B(b(b(b(b(b(b(b(b(b(b(b(b(x0)))))))))))))
B(b(b(a(b(a(x0)))))) → B(b(b(b(b(b(b(b(b(b(b(x0)))))))))))
B(b(b(a(b(b(a(x0))))))) → B(b(b(b(b(b(b(b(b(b(b(b(b(x0)))))))))))))
B(b(b(a(c(c(x0)))))) → B(b(b(b(b(b(c(c(c(a(a(a(a(x0)))))))))))))
B(b(b(a(b(b(a(x0))))))) → B(b(b(b(b(b(b(b(b(b(b(b(b(b(b(x0)))))))))))))))
B(b(b(a(c(c(x0)))))) → B(b(b(b(b(c(c(c(a(a(a(a(x0))))))))))))
B(b(b(a(b(a(x0)))))) → B(b(b(b(b(b(b(b(b(b(b(b(b(b(x0))))))))))))))
B(b(b(a(b(c(c(x0))))))) → B(b(b(c(c(c(a(a(a(a(x0))))))))))
B(b(b(a(c(c(x0)))))) → B(b(b(c(c(c(a(a(a(a(x0))))))))))
B(b(c(c(x1)))) → C(c(a(a(a(a(x1))))))
B(b(b(a(a(x0))))) → B(b(b(b(b(b(b(b(b(b(b(x0)))))))))))
C(c(x1)) → B(b(x1))
B(b(b(a(c(c(x0)))))) → B(b(c(c(c(a(a(a(a(x0)))))))))
B(b(b(a(b(c(c(x0))))))) → B(b(b(b(c(c(c(a(a(a(a(x0)))))))))))
B(b(b(a(x1)))) → B(b(x1))
B(b(c(c(x1)))) → C(c(c(a(a(a(a(x1)))))))
C(c(x1)) → B(b(b(x1)))
B(b(b(a(a(x0))))) → B(b(b(b(b(b(b(b(b(x0)))))))))
The set Q is empty.
We have obtained the following QTRS:
c(c(x)) → b(b(b(a(a(a(x))))))
a(b(b(b(x)))) → b(b(b(b(b(b(b(b(x))))))))
c(c(b(b(x)))) → a(a(a(a(c(c(c(x)))))))
c(c(b(a(b(b(B(x))))))) → a(a(a(a(c(c(c(b(b(b(b(b(b(B(x))))))))))))))
a(a(b(b(B(x))))) → b(b(b(b(b(b(b(b(b(B(x))))))))))
a(b(a(b(b(B(x)))))) → b(b(b(b(b(b(b(b(b(B(x))))))))))
a(a(b(b(B(x))))) → b(b(b(b(b(b(b(b(b(b(b(B(x))))))))))))
a(b(a(b(b(B(x)))))) → b(b(b(b(b(b(b(b(b(b(b(B(x))))))))))))
a(b(b(a(b(b(B(x))))))) → b(b(b(b(b(b(b(b(b(b(b(B(x))))))))))))
a(b(b(a(b(b(B(x))))))) → b(b(b(b(b(b(b(b(b(b(B(x)))))))))))
a(a(b(b(B(x))))) → b(b(b(b(b(b(b(b(b(b(b(b(B(x)))))))))))))
c(c(a(b(b(B(x)))))) → a(a(a(a(c(c(c(b(b(b(B(x)))))))))))
a(b(b(B(x)))) → b(b(B(x)))
a(b(b(B(x)))) → B(x)
c(c(b(a(b(b(B(x))))))) → a(a(a(a(c(c(c(b(b(b(b(b(B(x)))))))))))))
a(b(b(a(b(b(B(x))))))) → b(b(b(b(b(b(b(b(b(b(b(b(b(B(x))))))))))))))
c(c(b(a(b(b(B(x))))))) → a(a(a(a(c(c(c(b(b(b(b(B(x))))))))))))
a(b(a(b(b(B(x)))))) → b(b(b(b(b(b(b(b(b(b(b(b(B(x)))))))))))))
a(b(a(b(b(B(x)))))) → b(b(b(b(b(b(b(b(b(b(B(x)))))))))))
a(b(b(a(b(b(B(x))))))) → b(b(b(b(b(b(b(b(b(b(b(b(B(x)))))))))))))
c(c(a(b(b(B(x)))))) → a(a(a(a(c(c(c(b(b(b(b(b(B(x)))))))))))))
a(b(b(a(b(b(B(x))))))) → b(b(b(b(b(b(b(b(b(b(b(b(b(b(B(x)))))))))))))))
c(c(a(b(b(B(x)))))) → a(a(a(a(c(c(c(b(b(b(b(B(x))))))))))))
a(b(a(b(b(B(x)))))) → b(b(b(b(b(b(b(b(b(b(b(b(b(B(x))))))))))))))
c(c(b(a(b(b(B(x))))))) → a(a(a(a(c(c(c(b(b(B(x))))))))))
c(c(a(b(b(B(x)))))) → a(a(a(a(c(c(c(b(b(B(x))))))))))
c(c(b(B(x)))) → a(a(a(a(c(C(x))))))
a(a(b(b(B(x))))) → b(b(b(b(b(b(b(b(b(b(B(x)))))))))))
c(C(x)) → b(B(x))
c(c(a(b(b(B(x)))))) → a(a(a(a(c(c(c(b(B(x)))))))))
c(c(b(a(b(b(B(x))))))) → a(a(a(a(c(c(c(b(b(b(B(x)))))))))))
a(b(b(B(x)))) → b(B(x))
c(c(b(B(x)))) → a(a(a(a(c(c(C(x)))))))
c(C(x)) → b(b(B(x)))
a(a(b(b(B(x))))) → b(b(b(b(b(b(b(b(B(x)))))))))
The set Q is empty.
↳ QTRS
↳ QTRS Reverse
↳ QTRS Reverse
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ SemLabProof
↳ SemLabProof2
↳ QDP
↳ QDPToSRSProof
↳ QTRS
↳ QTRS Reverse
↳ QTRS
↳ QTRS Reverse
↳ QTRS Reverse
↳ DependencyPairsProof
Q restricted rewrite system:
The TRS R consists of the following rules:
c(c(x)) → b(b(b(a(a(a(x))))))
a(b(b(b(x)))) → b(b(b(b(b(b(b(b(x))))))))
c(c(b(b(x)))) → a(a(a(a(c(c(c(x)))))))
c(c(b(a(b(b(B(x))))))) → a(a(a(a(c(c(c(b(b(b(b(b(b(B(x))))))))))))))
a(a(b(b(B(x))))) → b(b(b(b(b(b(b(b(b(B(x))))))))))
a(b(a(b(b(B(x)))))) → b(b(b(b(b(b(b(b(b(B(x))))))))))
a(a(b(b(B(x))))) → b(b(b(b(b(b(b(b(b(b(b(B(x))))))))))))
a(b(a(b(b(B(x)))))) → b(b(b(b(b(b(b(b(b(b(b(B(x))))))))))))
a(b(b(a(b(b(B(x))))))) → b(b(b(b(b(b(b(b(b(b(b(B(x))))))))))))
a(b(b(a(b(b(B(x))))))) → b(b(b(b(b(b(b(b(b(b(B(x)))))))))))
a(a(b(b(B(x))))) → b(b(b(b(b(b(b(b(b(b(b(b(B(x)))))))))))))
c(c(a(b(b(B(x)))))) → a(a(a(a(c(c(c(b(b(b(B(x)))))))))))
a(b(b(B(x)))) → b(b(B(x)))
a(b(b(B(x)))) → B(x)
c(c(b(a(b(b(B(x))))))) → a(a(a(a(c(c(c(b(b(b(b(b(B(x)))))))))))))
a(b(b(a(b(b(B(x))))))) → b(b(b(b(b(b(b(b(b(b(b(b(b(B(x))))))))))))))
c(c(b(a(b(b(B(x))))))) → a(a(a(a(c(c(c(b(b(b(b(B(x))))))))))))
a(b(a(b(b(B(x)))))) → b(b(b(b(b(b(b(b(b(b(b(b(B(x)))))))))))))
a(b(a(b(b(B(x)))))) → b(b(b(b(b(b(b(b(b(b(B(x)))))))))))
a(b(b(a(b(b(B(x))))))) → b(b(b(b(b(b(b(b(b(b(b(b(B(x)))))))))))))
c(c(a(b(b(B(x)))))) → a(a(a(a(c(c(c(b(b(b(b(b(B(x)))))))))))))
a(b(b(a(b(b(B(x))))))) → b(b(b(b(b(b(b(b(b(b(b(b(b(b(B(x)))))))))))))))
c(c(a(b(b(B(x)))))) → a(a(a(a(c(c(c(b(b(b(b(B(x))))))))))))
a(b(a(b(b(B(x)))))) → b(b(b(b(b(b(b(b(b(b(b(b(b(B(x))))))))))))))
c(c(b(a(b(b(B(x))))))) → a(a(a(a(c(c(c(b(b(B(x))))))))))
c(c(a(b(b(B(x)))))) → a(a(a(a(c(c(c(b(b(B(x))))))))))
c(c(b(B(x)))) → a(a(a(a(c(C(x))))))
a(a(b(b(B(x))))) → b(b(b(b(b(b(b(b(b(b(B(x)))))))))))
c(C(x)) → b(B(x))
c(c(a(b(b(B(x)))))) → a(a(a(a(c(c(c(b(B(x)))))))))
c(c(b(a(b(b(B(x))))))) → a(a(a(a(c(c(c(b(b(b(B(x)))))))))))
a(b(b(B(x)))) → b(B(x))
c(c(b(B(x)))) → a(a(a(a(c(c(C(x)))))))
c(C(x)) → b(b(B(x)))
a(a(b(b(B(x))))) → b(b(b(b(b(b(b(b(B(x)))))))))
Q is empty.
We have reversed the following QTRS:
The set of rules R is
c(c(x)) → b(b(b(a(a(a(x))))))
a(b(b(b(x)))) → b(b(b(b(b(b(b(b(x))))))))
c(c(b(b(x)))) → a(a(a(a(c(c(c(x)))))))
c(c(b(a(b(b(B(x))))))) → a(a(a(a(c(c(c(b(b(b(b(b(b(B(x))))))))))))))
a(a(b(b(B(x))))) → b(b(b(b(b(b(b(b(b(B(x))))))))))
a(b(a(b(b(B(x)))))) → b(b(b(b(b(b(b(b(b(B(x))))))))))
a(a(b(b(B(x))))) → b(b(b(b(b(b(b(b(b(b(b(B(x))))))))))))
a(b(a(b(b(B(x)))))) → b(b(b(b(b(b(b(b(b(b(b(B(x))))))))))))
a(b(b(a(b(b(B(x))))))) → b(b(b(b(b(b(b(b(b(b(b(B(x))))))))))))
a(b(b(a(b(b(B(x))))))) → b(b(b(b(b(b(b(b(b(b(B(x)))))))))))
a(a(b(b(B(x))))) → b(b(b(b(b(b(b(b(b(b(b(b(B(x)))))))))))))
c(c(a(b(b(B(x)))))) → a(a(a(a(c(c(c(b(b(b(B(x)))))))))))
a(b(b(B(x)))) → b(b(B(x)))
a(b(b(B(x)))) → B(x)
c(c(b(a(b(b(B(x))))))) → a(a(a(a(c(c(c(b(b(b(b(b(B(x)))))))))))))
a(b(b(a(b(b(B(x))))))) → b(b(b(b(b(b(b(b(b(b(b(b(b(B(x))))))))))))))
c(c(b(a(b(b(B(x))))))) → a(a(a(a(c(c(c(b(b(b(b(B(x))))))))))))
a(b(a(b(b(B(x)))))) → b(b(b(b(b(b(b(b(b(b(b(b(B(x)))))))))))))
a(b(a(b(b(B(x)))))) → b(b(b(b(b(b(b(b(b(b(B(x)))))))))))
a(b(b(a(b(b(B(x))))))) → b(b(b(b(b(b(b(b(b(b(b(b(B(x)))))))))))))
c(c(a(b(b(B(x)))))) → a(a(a(a(c(c(c(b(b(b(b(b(B(x)))))))))))))
a(b(b(a(b(b(B(x))))))) → b(b(b(b(b(b(b(b(b(b(b(b(b(b(B(x)))))))))))))))
c(c(a(b(b(B(x)))))) → a(a(a(a(c(c(c(b(b(b(b(B(x))))))))))))
a(b(a(b(b(B(x)))))) → b(b(b(b(b(b(b(b(b(b(b(b(b(B(x))))))))))))))
c(c(b(a(b(b(B(x))))))) → a(a(a(a(c(c(c(b(b(B(x))))))))))
c(c(a(b(b(B(x)))))) → a(a(a(a(c(c(c(b(b(B(x))))))))))
c(c(b(B(x)))) → a(a(a(a(c(C(x))))))
a(a(b(b(B(x))))) → b(b(b(b(b(b(b(b(b(b(B(x)))))))))))
c(C(x)) → b(B(x))
c(c(a(b(b(B(x)))))) → a(a(a(a(c(c(c(b(B(x)))))))))
c(c(b(a(b(b(B(x))))))) → a(a(a(a(c(c(c(b(b(b(B(x)))))))))))
a(b(b(B(x)))) → b(B(x))
c(c(b(B(x)))) → a(a(a(a(c(c(C(x)))))))
c(C(x)) → b(b(B(x)))
a(a(b(b(B(x))))) → b(b(b(b(b(b(b(b(B(x)))))))))
The set Q is empty.
We have obtained the following QTRS:
c(c(x)) → a(a(a(b(b(b(x))))))
b(b(b(a(x)))) → b(b(b(b(b(b(b(b(x))))))))
b(b(c(c(x)))) → c(c(c(a(a(a(a(x)))))))
B(b(b(a(b(c(c(x))))))) → B(b(b(b(b(b(b(c(c(c(a(a(a(a(x))))))))))))))
B(b(b(a(a(x))))) → B(b(b(b(b(b(b(b(b(b(x))))))))))
B(b(b(a(b(a(x)))))) → B(b(b(b(b(b(b(b(b(b(x))))))))))
B(b(b(a(a(x))))) → B(b(b(b(b(b(b(b(b(b(b(b(x))))))))))))
B(b(b(a(b(a(x)))))) → B(b(b(b(b(b(b(b(b(b(b(b(x))))))))))))
B(b(b(a(b(b(a(x))))))) → B(b(b(b(b(b(b(b(b(b(b(b(x))))))))))))
B(b(b(a(b(b(a(x))))))) → B(b(b(b(b(b(b(b(b(b(b(x)))))))))))
B(b(b(a(a(x))))) → B(b(b(b(b(b(b(b(b(b(b(b(b(x)))))))))))))
B(b(b(a(c(c(x)))))) → B(b(b(b(c(c(c(a(a(a(a(x)))))))))))
B(b(b(a(x)))) → B(b(b(x)))
B(b(b(a(x)))) → B(x)
B(b(b(a(b(c(c(x))))))) → B(b(b(b(b(b(c(c(c(a(a(a(a(x)))))))))))))
B(b(b(a(b(b(a(x))))))) → B(b(b(b(b(b(b(b(b(b(b(b(b(b(x))))))))))))))
B(b(b(a(b(c(c(x))))))) → B(b(b(b(b(c(c(c(a(a(a(a(x))))))))))))
B(b(b(a(b(a(x)))))) → B(b(b(b(b(b(b(b(b(b(b(b(b(x)))))))))))))
B(b(b(a(b(a(x)))))) → B(b(b(b(b(b(b(b(b(b(b(x)))))))))))
B(b(b(a(b(b(a(x))))))) → B(b(b(b(b(b(b(b(b(b(b(b(b(x)))))))))))))
B(b(b(a(c(c(x)))))) → B(b(b(b(b(b(c(c(c(a(a(a(a(x)))))))))))))
B(b(b(a(b(b(a(x))))))) → B(b(b(b(b(b(b(b(b(b(b(b(b(b(b(x)))))))))))))))
B(b(b(a(c(c(x)))))) → B(b(b(b(b(c(c(c(a(a(a(a(x))))))))))))
B(b(b(a(b(a(x)))))) → B(b(b(b(b(b(b(b(b(b(b(b(b(b(x))))))))))))))
B(b(b(a(b(c(c(x))))))) → B(b(b(c(c(c(a(a(a(a(x))))))))))
B(b(b(a(c(c(x)))))) → B(b(b(c(c(c(a(a(a(a(x))))))))))
B(b(c(c(x)))) → C(c(a(a(a(a(x))))))
B(b(b(a(a(x))))) → B(b(b(b(b(b(b(b(b(b(b(x)))))))))))
C(c(x)) → B(b(x))
B(b(b(a(c(c(x)))))) → B(b(c(c(c(a(a(a(a(x)))))))))
B(b(b(a(b(c(c(x))))))) → B(b(b(b(c(c(c(a(a(a(a(x)))))))))))
B(b(b(a(x)))) → B(b(x))
B(b(c(c(x)))) → C(c(c(a(a(a(a(x)))))))
C(c(x)) → B(b(b(x)))
B(b(b(a(a(x))))) → B(b(b(b(b(b(b(b(b(x)))))))))
The set Q is empty.
↳ QTRS
↳ QTRS Reverse
↳ QTRS Reverse
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ SemLabProof
↳ SemLabProof2
↳ QDP
↳ QDPToSRSProof
↳ QTRS
↳ QTRS Reverse
↳ QTRS
↳ QTRS Reverse
↳ QTRS
↳ QTRS Reverse
↳ DependencyPairsProof
Q restricted rewrite system:
The TRS R consists of the following rules:
c(c(x)) → a(a(a(b(b(b(x))))))
b(b(b(a(x)))) → b(b(b(b(b(b(b(b(x))))))))
b(b(c(c(x)))) → c(c(c(a(a(a(a(x)))))))
B(b(b(a(b(c(c(x))))))) → B(b(b(b(b(b(b(c(c(c(a(a(a(a(x))))))))))))))
B(b(b(a(a(x))))) → B(b(b(b(b(b(b(b(b(b(x))))))))))
B(b(b(a(b(a(x)))))) → B(b(b(b(b(b(b(b(b(b(x))))))))))
B(b(b(a(a(x))))) → B(b(b(b(b(b(b(b(b(b(b(b(x))))))))))))
B(b(b(a(b(a(x)))))) → B(b(b(b(b(b(b(b(b(b(b(b(x))))))))))))
B(b(b(a(b(b(a(x))))))) → B(b(b(b(b(b(b(b(b(b(b(b(x))))))))))))
B(b(b(a(b(b(a(x))))))) → B(b(b(b(b(b(b(b(b(b(b(x)))))))))))
B(b(b(a(a(x))))) → B(b(b(b(b(b(b(b(b(b(b(b(b(x)))))))))))))
B(b(b(a(c(c(x)))))) → B(b(b(b(c(c(c(a(a(a(a(x)))))))))))
B(b(b(a(x)))) → B(b(b(x)))
B(b(b(a(x)))) → B(x)
B(b(b(a(b(c(c(x))))))) → B(b(b(b(b(b(c(c(c(a(a(a(a(x)))))))))))))
B(b(b(a(b(b(a(x))))))) → B(b(b(b(b(b(b(b(b(b(b(b(b(b(x))))))))))))))
B(b(b(a(b(c(c(x))))))) → B(b(b(b(b(c(c(c(a(a(a(a(x))))))))))))
B(b(b(a(b(a(x)))))) → B(b(b(b(b(b(b(b(b(b(b(b(b(x)))))))))))))
B(b(b(a(b(a(x)))))) → B(b(b(b(b(b(b(b(b(b(b(x)))))))))))
B(b(b(a(b(b(a(x))))))) → B(b(b(b(b(b(b(b(b(b(b(b(b(x)))))))))))))
B(b(b(a(c(c(x)))))) → B(b(b(b(b(b(c(c(c(a(a(a(a(x)))))))))))))
B(b(b(a(b(b(a(x))))))) → B(b(b(b(b(b(b(b(b(b(b(b(b(b(b(x)))))))))))))))
B(b(b(a(c(c(x)))))) → B(b(b(b(b(c(c(c(a(a(a(a(x))))))))))))
B(b(b(a(b(a(x)))))) → B(b(b(b(b(b(b(b(b(b(b(b(b(b(x))))))))))))))
B(b(b(a(b(c(c(x))))))) → B(b(b(c(c(c(a(a(a(a(x))))))))))
B(b(b(a(c(c(x)))))) → B(b(b(c(c(c(a(a(a(a(x))))))))))
B(b(c(c(x)))) → C(c(a(a(a(a(x))))))
B(b(b(a(a(x))))) → B(b(b(b(b(b(b(b(b(b(b(x)))))))))))
C(c(x)) → B(b(x))
B(b(b(a(c(c(x)))))) → B(b(c(c(c(a(a(a(a(x)))))))))
B(b(b(a(b(c(c(x))))))) → B(b(b(b(c(c(c(a(a(a(a(x)))))))))))
B(b(b(a(x)))) → B(b(x))
B(b(c(c(x)))) → C(c(c(a(a(a(a(x)))))))
C(c(x)) → B(b(b(x)))
B(b(b(a(a(x))))) → B(b(b(b(b(b(b(b(b(x)))))))))
Q is empty.
We have reversed the following QTRS:
The set of rules R is
c(c(x)) → b(b(b(a(a(a(x))))))
a(b(b(b(x)))) → b(b(b(b(b(b(b(b(x))))))))
c(c(b(b(x)))) → a(a(a(a(c(c(c(x)))))))
c(c(b(a(b(b(B(x))))))) → a(a(a(a(c(c(c(b(b(b(b(b(b(B(x))))))))))))))
a(a(b(b(B(x))))) → b(b(b(b(b(b(b(b(b(B(x))))))))))
a(b(a(b(b(B(x)))))) → b(b(b(b(b(b(b(b(b(B(x))))))))))
a(a(b(b(B(x))))) → b(b(b(b(b(b(b(b(b(b(b(B(x))))))))))))
a(b(a(b(b(B(x)))))) → b(b(b(b(b(b(b(b(b(b(b(B(x))))))))))))
a(b(b(a(b(b(B(x))))))) → b(b(b(b(b(b(b(b(b(b(b(B(x))))))))))))
a(b(b(a(b(b(B(x))))))) → b(b(b(b(b(b(b(b(b(b(B(x)))))))))))
a(a(b(b(B(x))))) → b(b(b(b(b(b(b(b(b(b(b(b(B(x)))))))))))))
c(c(a(b(b(B(x)))))) → a(a(a(a(c(c(c(b(b(b(B(x)))))))))))
a(b(b(B(x)))) → b(b(B(x)))
a(b(b(B(x)))) → B(x)
c(c(b(a(b(b(B(x))))))) → a(a(a(a(c(c(c(b(b(b(b(b(B(x)))))))))))))
a(b(b(a(b(b(B(x))))))) → b(b(b(b(b(b(b(b(b(b(b(b(b(B(x))))))))))))))
c(c(b(a(b(b(B(x))))))) → a(a(a(a(c(c(c(b(b(b(b(B(x))))))))))))
a(b(a(b(b(B(x)))))) → b(b(b(b(b(b(b(b(b(b(b(b(B(x)))))))))))))
a(b(a(b(b(B(x)))))) → b(b(b(b(b(b(b(b(b(b(B(x)))))))))))
a(b(b(a(b(b(B(x))))))) → b(b(b(b(b(b(b(b(b(b(b(b(B(x)))))))))))))
c(c(a(b(b(B(x)))))) → a(a(a(a(c(c(c(b(b(b(b(b(B(x)))))))))))))
a(b(b(a(b(b(B(x))))))) → b(b(b(b(b(b(b(b(b(b(b(b(b(b(B(x)))))))))))))))
c(c(a(b(b(B(x)))))) → a(a(a(a(c(c(c(b(b(b(b(B(x))))))))))))
a(b(a(b(b(B(x)))))) → b(b(b(b(b(b(b(b(b(b(b(b(b(B(x))))))))))))))
c(c(b(a(b(b(B(x))))))) → a(a(a(a(c(c(c(b(b(B(x))))))))))
c(c(a(b(b(B(x)))))) → a(a(a(a(c(c(c(b(b(B(x))))))))))
c(c(b(B(x)))) → a(a(a(a(c(C(x))))))
a(a(b(b(B(x))))) → b(b(b(b(b(b(b(b(b(b(B(x)))))))))))
c(C(x)) → b(B(x))
c(c(a(b(b(B(x)))))) → a(a(a(a(c(c(c(b(B(x)))))))))
c(c(b(a(b(b(B(x))))))) → a(a(a(a(c(c(c(b(b(b(B(x)))))))))))
a(b(b(B(x)))) → b(B(x))
c(c(b(B(x)))) → a(a(a(a(c(c(C(x)))))))
c(C(x)) → b(b(B(x)))
a(a(b(b(B(x))))) → b(b(b(b(b(b(b(b(B(x)))))))))
The set Q is empty.
We have obtained the following QTRS:
c(c(x)) → a(a(a(b(b(b(x))))))
b(b(b(a(x)))) → b(b(b(b(b(b(b(b(x))))))))
b(b(c(c(x)))) → c(c(c(a(a(a(a(x)))))))
B(b(b(a(b(c(c(x))))))) → B(b(b(b(b(b(b(c(c(c(a(a(a(a(x))))))))))))))
B(b(b(a(a(x))))) → B(b(b(b(b(b(b(b(b(b(x))))))))))
B(b(b(a(b(a(x)))))) → B(b(b(b(b(b(b(b(b(b(x))))))))))
B(b(b(a(a(x))))) → B(b(b(b(b(b(b(b(b(b(b(b(x))))))))))))
B(b(b(a(b(a(x)))))) → B(b(b(b(b(b(b(b(b(b(b(b(x))))))))))))
B(b(b(a(b(b(a(x))))))) → B(b(b(b(b(b(b(b(b(b(b(b(x))))))))))))
B(b(b(a(b(b(a(x))))))) → B(b(b(b(b(b(b(b(b(b(b(x)))))))))))
B(b(b(a(a(x))))) → B(b(b(b(b(b(b(b(b(b(b(b(b(x)))))))))))))
B(b(b(a(c(c(x)))))) → B(b(b(b(c(c(c(a(a(a(a(x)))))))))))
B(b(b(a(x)))) → B(b(b(x)))
B(b(b(a(x)))) → B(x)
B(b(b(a(b(c(c(x))))))) → B(b(b(b(b(b(c(c(c(a(a(a(a(x)))))))))))))
B(b(b(a(b(b(a(x))))))) → B(b(b(b(b(b(b(b(b(b(b(b(b(b(x))))))))))))))
B(b(b(a(b(c(c(x))))))) → B(b(b(b(b(c(c(c(a(a(a(a(x))))))))))))
B(b(b(a(b(a(x)))))) → B(b(b(b(b(b(b(b(b(b(b(b(b(x)))))))))))))
B(b(b(a(b(a(x)))))) → B(b(b(b(b(b(b(b(b(b(b(x)))))))))))
B(b(b(a(b(b(a(x))))))) → B(b(b(b(b(b(b(b(b(b(b(b(b(x)))))))))))))
B(b(b(a(c(c(x)))))) → B(b(b(b(b(b(c(c(c(a(a(a(a(x)))))))))))))
B(b(b(a(b(b(a(x))))))) → B(b(b(b(b(b(b(b(b(b(b(b(b(b(b(x)))))))))))))))
B(b(b(a(c(c(x)))))) → B(b(b(b(b(c(c(c(a(a(a(a(x))))))))))))
B(b(b(a(b(a(x)))))) → B(b(b(b(b(b(b(b(b(b(b(b(b(b(x))))))))))))))
B(b(b(a(b(c(c(x))))))) → B(b(b(c(c(c(a(a(a(a(x))))))))))
B(b(b(a(c(c(x)))))) → B(b(b(c(c(c(a(a(a(a(x))))))))))
B(b(c(c(x)))) → C(c(a(a(a(a(x))))))
B(b(b(a(a(x))))) → B(b(b(b(b(b(b(b(b(b(b(x)))))))))))
C(c(x)) → B(b(x))
B(b(b(a(c(c(x)))))) → B(b(c(c(c(a(a(a(a(x)))))))))
B(b(b(a(b(c(c(x))))))) → B(b(b(b(c(c(c(a(a(a(a(x)))))))))))
B(b(b(a(x)))) → B(b(x))
B(b(c(c(x)))) → C(c(c(a(a(a(a(x)))))))
C(c(x)) → B(b(b(x)))
B(b(b(a(a(x))))) → B(b(b(b(b(b(b(b(b(x)))))))))
The set Q is empty.
↳ QTRS
↳ QTRS Reverse
↳ QTRS Reverse
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ SemLabProof
↳ SemLabProof2
↳ QDP
↳ QDPToSRSProof
↳ QTRS
↳ QTRS Reverse
↳ QTRS
↳ QTRS Reverse
↳ QTRS Reverse
↳ QTRS
↳ DependencyPairsProof
Q restricted rewrite system:
The TRS R consists of the following rules:
c(c(x)) → a(a(a(b(b(b(x))))))
b(b(b(a(x)))) → b(b(b(b(b(b(b(b(x))))))))
b(b(c(c(x)))) → c(c(c(a(a(a(a(x)))))))
B(b(b(a(b(c(c(x))))))) → B(b(b(b(b(b(b(c(c(c(a(a(a(a(x))))))))))))))
B(b(b(a(a(x))))) → B(b(b(b(b(b(b(b(b(b(x))))))))))
B(b(b(a(b(a(x)))))) → B(b(b(b(b(b(b(b(b(b(x))))))))))
B(b(b(a(a(x))))) → B(b(b(b(b(b(b(b(b(b(b(b(x))))))))))))
B(b(b(a(b(a(x)))))) → B(b(b(b(b(b(b(b(b(b(b(b(x))))))))))))
B(b(b(a(b(b(a(x))))))) → B(b(b(b(b(b(b(b(b(b(b(b(x))))))))))))
B(b(b(a(b(b(a(x))))))) → B(b(b(b(b(b(b(b(b(b(b(x)))))))))))
B(b(b(a(a(x))))) → B(b(b(b(b(b(b(b(b(b(b(b(b(x)))))))))))))
B(b(b(a(c(c(x)))))) → B(b(b(b(c(c(c(a(a(a(a(x)))))))))))
B(b(b(a(x)))) → B(b(b(x)))
B(b(b(a(x)))) → B(x)
B(b(b(a(b(c(c(x))))))) → B(b(b(b(b(b(c(c(c(a(a(a(a(x)))))))))))))
B(b(b(a(b(b(a(x))))))) → B(b(b(b(b(b(b(b(b(b(b(b(b(b(x))))))))))))))
B(b(b(a(b(c(c(x))))))) → B(b(b(b(b(c(c(c(a(a(a(a(x))))))))))))
B(b(b(a(b(a(x)))))) → B(b(b(b(b(b(b(b(b(b(b(b(b(x)))))))))))))
B(b(b(a(b(a(x)))))) → B(b(b(b(b(b(b(b(b(b(b(x)))))))))))
B(b(b(a(b(b(a(x))))))) → B(b(b(b(b(b(b(b(b(b(b(b(b(x)))))))))))))
B(b(b(a(c(c(x)))))) → B(b(b(b(b(b(c(c(c(a(a(a(a(x)))))))))))))
B(b(b(a(b(b(a(x))))))) → B(b(b(b(b(b(b(b(b(b(b(b(b(b(b(x)))))))))))))))
B(b(b(a(c(c(x)))))) → B(b(b(b(b(c(c(c(a(a(a(a(x))))))))))))
B(b(b(a(b(a(x)))))) → B(b(b(b(b(b(b(b(b(b(b(b(b(b(x))))))))))))))
B(b(b(a(b(c(c(x))))))) → B(b(b(c(c(c(a(a(a(a(x))))))))))
B(b(b(a(c(c(x)))))) → B(b(b(c(c(c(a(a(a(a(x))))))))))
B(b(c(c(x)))) → C(c(a(a(a(a(x))))))
B(b(b(a(a(x))))) → B(b(b(b(b(b(b(b(b(b(b(x)))))))))))
C(c(x)) → B(b(x))
B(b(b(a(c(c(x)))))) → B(b(c(c(c(a(a(a(a(x)))))))))
B(b(b(a(b(c(c(x))))))) → B(b(b(b(c(c(c(a(a(a(a(x)))))))))))
B(b(b(a(x)))) → B(b(x))
B(b(c(c(x)))) → C(c(c(a(a(a(a(x)))))))
C(c(x)) → B(b(b(x)))
B(b(b(a(a(x))))) → B(b(b(b(b(b(b(b(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:
C1(c(x)) → A(x)
C1(c(b(a(b(b(B(x))))))) → A(a(a(c(c(c(b(b(b(B(x))))))))))
C1(c(a(b(b(B(x)))))) → C1(b(B(x)))
C1(c(a(b(b(B(x)))))) → A(c(c(c(b(b(b(b(B(x)))))))))
C1(c(a(b(b(B(x)))))) → C1(c(b(b(b(B(x))))))
C1(c(b(b(x)))) → C1(x)
C1(c(b(B(x)))) → A(c(c(C(x))))
C1(c(b(a(b(b(B(x))))))) → C1(b(b(b(b(b(B(x)))))))
C1(c(b(a(b(b(B(x))))))) → C1(c(b(b(b(B(x))))))
C1(c(b(a(b(b(B(x))))))) → C1(b(b(b(b(b(b(B(x))))))))
C1(c(b(a(b(b(B(x))))))) → A(a(c(c(c(b(b(b(b(B(x))))))))))
C1(c(a(b(b(B(x)))))) → A(c(c(c(b(B(x))))))
C1(c(a(b(b(B(x)))))) → C1(b(b(b(b(B(x))))))
C1(c(b(a(b(b(B(x))))))) → A(c(c(c(b(b(b(b(b(b(B(x)))))))))))
C1(c(a(b(b(B(x)))))) → A(a(a(c(c(c(b(b(b(b(B(x)))))))))))
C1(c(a(b(b(B(x)))))) → A(a(c(c(c(b(b(B(x))))))))
C1(c(a(b(b(B(x)))))) → A(a(c(c(c(b(b(b(B(x)))))))))
C1(c(a(b(b(B(x)))))) → A(c(c(c(b(b(b(b(b(B(x))))))))))
C1(c(b(a(b(b(B(x))))))) → A(a(a(c(c(c(b(b(b(b(b(B(x))))))))))))
C1(c(b(B(x)))) → A(a(a(c(C(x)))))
C1(c(b(a(b(b(B(x))))))) → A(a(a(c(c(c(b(b(B(x)))))))))
C1(c(b(a(b(b(B(x))))))) → C1(c(b(b(b(b(b(B(x))))))))
C1(c(b(a(b(b(B(x))))))) → A(a(a(c(c(c(b(b(b(b(B(x)))))))))))
C1(c(b(a(b(b(B(x))))))) → A(a(a(a(c(c(c(b(b(b(B(x)))))))))))
C1(c(a(b(b(B(x)))))) → C1(b(b(B(x))))
C1(c(b(a(b(b(B(x))))))) → A(c(c(c(b(b(b(B(x))))))))
C1(c(a(b(b(B(x)))))) → A(a(c(c(c(b(B(x)))))))
C1(c(b(b(x)))) → C1(c(c(x)))
C1(c(b(b(x)))) → A(c(c(c(x))))
C1(c(a(b(b(B(x)))))) → A(a(a(c(c(c(b(b(b(b(b(B(x))))))))))))
C1(c(b(a(b(b(B(x))))))) → A(c(c(c(b(b(b(b(b(B(x))))))))))
C1(c(a(b(b(B(x)))))) → A(a(a(a(c(c(c(b(b(b(b(B(x))))))))))))
C1(c(b(B(x)))) → A(a(a(a(c(c(C(x)))))))
C1(c(a(b(b(B(x)))))) → C1(c(b(b(B(x)))))
C1(c(a(b(b(B(x)))))) → A(a(a(a(c(c(c(b(B(x)))))))))
C1(c(b(B(x)))) → A(a(a(a(c(C(x))))))
C1(c(b(B(x)))) → A(a(a(c(c(C(x))))))
C1(c(a(b(b(B(x)))))) → A(a(a(a(c(c(c(b(b(b(b(b(B(x)))))))))))))
C1(c(a(b(b(B(x)))))) → A(a(a(a(c(c(c(b(b(b(B(x)))))))))))
C1(c(b(a(b(b(B(x))))))) → C1(c(b(b(B(x)))))
C1(c(b(a(b(b(B(x))))))) → A(a(c(c(c(b(b(b(b(b(B(x)))))))))))
C1(c(b(a(b(b(B(x))))))) → C1(b(b(b(b(B(x))))))
C1(c(b(B(x)))) → C1(C(x))
C1(c(b(a(b(b(B(x))))))) → A(a(c(c(c(b(b(B(x))))))))
C1(c(b(a(b(b(B(x))))))) → C1(c(b(b(b(b(b(b(B(x)))))))))
C1(c(b(a(b(b(B(x))))))) → A(a(a(a(c(c(c(b(b(b(b(B(x))))))))))))
C1(c(a(b(b(B(x)))))) → C1(b(b(b(B(x)))))
C1(c(b(B(x)))) → A(a(c(C(x))))
C1(c(b(a(b(b(B(x))))))) → A(c(c(c(b(b(B(x)))))))
C1(c(b(a(b(b(B(x))))))) → C1(c(c(b(b(B(x))))))
C1(c(b(b(x)))) → A(a(c(c(c(x)))))
C1(c(b(B(x)))) → C1(c(C(x)))
C1(c(b(a(b(b(B(x))))))) → C1(c(c(b(b(b(B(x)))))))
C1(c(b(a(b(b(B(x))))))) → A(a(a(a(c(c(c(b(b(b(b(b(b(B(x))))))))))))))
C1(c(b(B(x)))) → A(c(C(x)))
C1(c(x)) → A(a(x))
C1(c(a(b(b(B(x)))))) → C1(c(c(b(b(b(b(B(x))))))))
C1(c(b(a(b(b(B(x))))))) → A(a(c(c(c(b(b(b(B(x)))))))))
C1(c(b(a(b(b(B(x))))))) → C1(c(c(b(b(b(b(B(x))))))))
C1(c(b(b(x)))) → A(a(a(c(c(c(x))))))
C1(c(b(a(b(b(B(x))))))) → C1(c(b(b(b(b(B(x)))))))
C1(c(b(b(x)))) → C1(c(x))
C1(c(b(a(b(b(B(x))))))) → C1(b(b(B(x))))
C1(c(a(b(b(B(x)))))) → A(a(c(c(c(b(b(b(b(b(B(x)))))))))))
C1(c(a(b(b(B(x)))))) → C1(c(c(b(B(x)))))
C1(c(a(b(b(B(x)))))) → C1(c(b(B(x))))
C1(c(b(a(b(b(B(x))))))) → A(a(a(a(c(c(c(b(b(b(b(b(B(x)))))))))))))
C1(c(a(b(b(B(x)))))) → C1(c(c(b(b(b(B(x)))))))
C1(c(b(a(b(b(B(x))))))) → C1(c(c(b(b(b(b(b(b(B(x))))))))))
C1(c(a(b(b(B(x)))))) → A(a(c(c(c(b(b(b(b(B(x))))))))))
C1(c(a(b(b(B(x)))))) → A(a(a(c(c(c(b(b(b(B(x))))))))))
C1(c(b(b(x)))) → A(a(a(a(c(c(c(x)))))))
C1(c(b(a(b(b(B(x))))))) → C1(b(b(b(B(x)))))
C1(c(a(b(b(B(x)))))) → A(a(a(c(c(c(b(B(x))))))))
C1(c(b(a(b(b(B(x))))))) → A(a(a(c(c(c(b(b(b(b(b(b(B(x)))))))))))))
C1(c(a(b(b(B(x)))))) → C1(c(c(b(b(B(x))))))
C1(c(a(b(b(B(x)))))) → C1(b(b(b(b(b(B(x)))))))
C1(c(b(B(x)))) → A(a(c(c(C(x)))))
C1(c(b(a(b(b(B(x))))))) → A(a(a(a(c(c(c(b(b(B(x))))))))))
C1(c(x)) → A(a(a(x)))
C1(c(a(b(b(B(x)))))) → A(a(a(c(c(c(b(b(B(x)))))))))
C1(c(a(b(b(B(x)))))) → A(a(a(a(c(c(c(b(b(B(x))))))))))
C1(c(b(a(b(b(B(x))))))) → A(a(c(c(c(b(b(b(b(b(b(B(x))))))))))))
C1(c(a(b(b(B(x)))))) → A(c(c(c(b(b(B(x)))))))
C1(c(b(a(b(b(B(x))))))) → A(c(c(c(b(b(b(b(B(x)))))))))
C1(c(a(b(b(B(x)))))) → C1(c(b(b(b(b(B(x)))))))
C1(c(a(b(b(B(x)))))) → C1(c(c(b(b(b(b(b(B(x)))))))))
C1(c(a(b(b(B(x)))))) → A(c(c(c(b(b(b(B(x))))))))
C1(c(a(b(b(B(x)))))) → C1(c(b(b(b(b(b(B(x))))))))
C1(c(b(a(b(b(B(x))))))) → C1(c(c(b(b(b(b(b(B(x)))))))))
The TRS R consists of the following rules:
c(c(x)) → b(b(b(a(a(a(x))))))
a(b(b(b(x)))) → b(b(b(b(b(b(b(b(x))))))))
c(c(b(b(x)))) → a(a(a(a(c(c(c(x)))))))
c(c(b(a(b(b(B(x))))))) → a(a(a(a(c(c(c(b(b(b(b(b(b(B(x))))))))))))))
a(a(b(b(B(x))))) → b(b(b(b(b(b(b(b(b(B(x))))))))))
a(b(a(b(b(B(x)))))) → b(b(b(b(b(b(b(b(b(B(x))))))))))
a(a(b(b(B(x))))) → b(b(b(b(b(b(b(b(b(b(b(B(x))))))))))))
a(b(a(b(b(B(x)))))) → b(b(b(b(b(b(b(b(b(b(b(B(x))))))))))))
a(b(b(a(b(b(B(x))))))) → b(b(b(b(b(b(b(b(b(b(b(B(x))))))))))))
a(b(b(a(b(b(B(x))))))) → b(b(b(b(b(b(b(b(b(b(B(x)))))))))))
a(a(b(b(B(x))))) → b(b(b(b(b(b(b(b(b(b(b(b(B(x)))))))))))))
c(c(a(b(b(B(x)))))) → a(a(a(a(c(c(c(b(b(b(B(x)))))))))))
a(b(b(B(x)))) → b(b(B(x)))
a(b(b(B(x)))) → B(x)
c(c(b(a(b(b(B(x))))))) → a(a(a(a(c(c(c(b(b(b(b(b(B(x)))))))))))))
a(b(b(a(b(b(B(x))))))) → b(b(b(b(b(b(b(b(b(b(b(b(b(B(x))))))))))))))
c(c(b(a(b(b(B(x))))))) → a(a(a(a(c(c(c(b(b(b(b(B(x))))))))))))
a(b(a(b(b(B(x)))))) → b(b(b(b(b(b(b(b(b(b(b(b(B(x)))))))))))))
a(b(a(b(b(B(x)))))) → b(b(b(b(b(b(b(b(b(b(B(x)))))))))))
a(b(b(a(b(b(B(x))))))) → b(b(b(b(b(b(b(b(b(b(b(b(B(x)))))))))))))
c(c(a(b(b(B(x)))))) → a(a(a(a(c(c(c(b(b(b(b(b(B(x)))))))))))))
a(b(b(a(b(b(B(x))))))) → b(b(b(b(b(b(b(b(b(b(b(b(b(b(B(x)))))))))))))))
c(c(a(b(b(B(x)))))) → a(a(a(a(c(c(c(b(b(b(b(B(x))))))))))))
a(b(a(b(b(B(x)))))) → b(b(b(b(b(b(b(b(b(b(b(b(b(B(x))))))))))))))
c(c(b(a(b(b(B(x))))))) → a(a(a(a(c(c(c(b(b(B(x))))))))))
c(c(a(b(b(B(x)))))) → a(a(a(a(c(c(c(b(b(B(x))))))))))
c(c(b(B(x)))) → a(a(a(a(c(C(x))))))
a(a(b(b(B(x))))) → b(b(b(b(b(b(b(b(b(b(B(x)))))))))))
c(C(x)) → b(B(x))
c(c(a(b(b(B(x)))))) → a(a(a(a(c(c(c(b(B(x)))))))))
c(c(b(a(b(b(B(x))))))) → a(a(a(a(c(c(c(b(b(b(B(x)))))))))))
a(b(b(B(x)))) → b(B(x))
c(c(b(B(x)))) → a(a(a(a(c(c(C(x)))))))
c(C(x)) → b(b(B(x)))
a(a(b(b(B(x))))) → b(b(b(b(b(b(b(b(B(x)))))))))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
↳ QTRS
↳ QTRS Reverse
↳ QTRS Reverse
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ SemLabProof
↳ SemLabProof2
↳ QDP
↳ QDPToSRSProof
↳ QTRS
↳ QTRS Reverse
↳ QTRS
↳ QTRS Reverse
↳ QTRS Reverse
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
Q DP problem:
The TRS P consists of the following rules:
C1(c(x)) → A(x)
C1(c(b(a(b(b(B(x))))))) → A(a(a(c(c(c(b(b(b(B(x))))))))))
C1(c(a(b(b(B(x)))))) → C1(b(B(x)))
C1(c(a(b(b(B(x)))))) → A(c(c(c(b(b(b(b(B(x)))))))))
C1(c(a(b(b(B(x)))))) → C1(c(b(b(b(B(x))))))
C1(c(b(b(x)))) → C1(x)
C1(c(b(B(x)))) → A(c(c(C(x))))
C1(c(b(a(b(b(B(x))))))) → C1(b(b(b(b(b(B(x)))))))
C1(c(b(a(b(b(B(x))))))) → C1(c(b(b(b(B(x))))))
C1(c(b(a(b(b(B(x))))))) → C1(b(b(b(b(b(b(B(x))))))))
C1(c(b(a(b(b(B(x))))))) → A(a(c(c(c(b(b(b(b(B(x))))))))))
C1(c(a(b(b(B(x)))))) → A(c(c(c(b(B(x))))))
C1(c(a(b(b(B(x)))))) → C1(b(b(b(b(B(x))))))
C1(c(b(a(b(b(B(x))))))) → A(c(c(c(b(b(b(b(b(b(B(x)))))))))))
C1(c(a(b(b(B(x)))))) → A(a(a(c(c(c(b(b(b(b(B(x)))))))))))
C1(c(a(b(b(B(x)))))) → A(a(c(c(c(b(b(B(x))))))))
C1(c(a(b(b(B(x)))))) → A(a(c(c(c(b(b(b(B(x)))))))))
C1(c(a(b(b(B(x)))))) → A(c(c(c(b(b(b(b(b(B(x))))))))))
C1(c(b(a(b(b(B(x))))))) → A(a(a(c(c(c(b(b(b(b(b(B(x))))))))))))
C1(c(b(B(x)))) → A(a(a(c(C(x)))))
C1(c(b(a(b(b(B(x))))))) → A(a(a(c(c(c(b(b(B(x)))))))))
C1(c(b(a(b(b(B(x))))))) → C1(c(b(b(b(b(b(B(x))))))))
C1(c(b(a(b(b(B(x))))))) → A(a(a(c(c(c(b(b(b(b(B(x)))))))))))
C1(c(b(a(b(b(B(x))))))) → A(a(a(a(c(c(c(b(b(b(B(x)))))))))))
C1(c(a(b(b(B(x)))))) → C1(b(b(B(x))))
C1(c(b(a(b(b(B(x))))))) → A(c(c(c(b(b(b(B(x))))))))
C1(c(a(b(b(B(x)))))) → A(a(c(c(c(b(B(x)))))))
C1(c(b(b(x)))) → C1(c(c(x)))
C1(c(b(b(x)))) → A(c(c(c(x))))
C1(c(a(b(b(B(x)))))) → A(a(a(c(c(c(b(b(b(b(b(B(x))))))))))))
C1(c(b(a(b(b(B(x))))))) → A(c(c(c(b(b(b(b(b(B(x))))))))))
C1(c(a(b(b(B(x)))))) → A(a(a(a(c(c(c(b(b(b(b(B(x))))))))))))
C1(c(b(B(x)))) → A(a(a(a(c(c(C(x)))))))
C1(c(a(b(b(B(x)))))) → C1(c(b(b(B(x)))))
C1(c(a(b(b(B(x)))))) → A(a(a(a(c(c(c(b(B(x)))))))))
C1(c(b(B(x)))) → A(a(a(a(c(C(x))))))
C1(c(b(B(x)))) → A(a(a(c(c(C(x))))))
C1(c(a(b(b(B(x)))))) → A(a(a(a(c(c(c(b(b(b(b(b(B(x)))))))))))))
C1(c(a(b(b(B(x)))))) → A(a(a(a(c(c(c(b(b(b(B(x)))))))))))
C1(c(b(a(b(b(B(x))))))) → C1(c(b(b(B(x)))))
C1(c(b(a(b(b(B(x))))))) → A(a(c(c(c(b(b(b(b(b(B(x)))))))))))
C1(c(b(a(b(b(B(x))))))) → C1(b(b(b(b(B(x))))))
C1(c(b(B(x)))) → C1(C(x))
C1(c(b(a(b(b(B(x))))))) → A(a(c(c(c(b(b(B(x))))))))
C1(c(b(a(b(b(B(x))))))) → C1(c(b(b(b(b(b(b(B(x)))))))))
C1(c(b(a(b(b(B(x))))))) → A(a(a(a(c(c(c(b(b(b(b(B(x))))))))))))
C1(c(a(b(b(B(x)))))) → C1(b(b(b(B(x)))))
C1(c(b(B(x)))) → A(a(c(C(x))))
C1(c(b(a(b(b(B(x))))))) → A(c(c(c(b(b(B(x)))))))
C1(c(b(a(b(b(B(x))))))) → C1(c(c(b(b(B(x))))))
C1(c(b(b(x)))) → A(a(c(c(c(x)))))
C1(c(b(B(x)))) → C1(c(C(x)))
C1(c(b(a(b(b(B(x))))))) → C1(c(c(b(b(b(B(x)))))))
C1(c(b(a(b(b(B(x))))))) → A(a(a(a(c(c(c(b(b(b(b(b(b(B(x))))))))))))))
C1(c(b(B(x)))) → A(c(C(x)))
C1(c(x)) → A(a(x))
C1(c(a(b(b(B(x)))))) → C1(c(c(b(b(b(b(B(x))))))))
C1(c(b(a(b(b(B(x))))))) → A(a(c(c(c(b(b(b(B(x)))))))))
C1(c(b(a(b(b(B(x))))))) → C1(c(c(b(b(b(b(B(x))))))))
C1(c(b(b(x)))) → A(a(a(c(c(c(x))))))
C1(c(b(a(b(b(B(x))))))) → C1(c(b(b(b(b(B(x)))))))
C1(c(b(b(x)))) → C1(c(x))
C1(c(b(a(b(b(B(x))))))) → C1(b(b(B(x))))
C1(c(a(b(b(B(x)))))) → A(a(c(c(c(b(b(b(b(b(B(x)))))))))))
C1(c(a(b(b(B(x)))))) → C1(c(c(b(B(x)))))
C1(c(a(b(b(B(x)))))) → C1(c(b(B(x))))
C1(c(b(a(b(b(B(x))))))) → A(a(a(a(c(c(c(b(b(b(b(b(B(x)))))))))))))
C1(c(a(b(b(B(x)))))) → C1(c(c(b(b(b(B(x)))))))
C1(c(b(a(b(b(B(x))))))) → C1(c(c(b(b(b(b(b(b(B(x))))))))))
C1(c(a(b(b(B(x)))))) → A(a(c(c(c(b(b(b(b(B(x))))))))))
C1(c(a(b(b(B(x)))))) → A(a(a(c(c(c(b(b(b(B(x))))))))))
C1(c(b(b(x)))) → A(a(a(a(c(c(c(x)))))))
C1(c(b(a(b(b(B(x))))))) → C1(b(b(b(B(x)))))
C1(c(a(b(b(B(x)))))) → A(a(a(c(c(c(b(B(x))))))))
C1(c(b(a(b(b(B(x))))))) → A(a(a(c(c(c(b(b(b(b(b(b(B(x)))))))))))))
C1(c(a(b(b(B(x)))))) → C1(c(c(b(b(B(x))))))
C1(c(a(b(b(B(x)))))) → C1(b(b(b(b(b(B(x)))))))
C1(c(b(B(x)))) → A(a(c(c(C(x)))))
C1(c(b(a(b(b(B(x))))))) → A(a(a(a(c(c(c(b(b(B(x))))))))))
C1(c(x)) → A(a(a(x)))
C1(c(a(b(b(B(x)))))) → A(a(a(c(c(c(b(b(B(x)))))))))
C1(c(a(b(b(B(x)))))) → A(a(a(a(c(c(c(b(b(B(x))))))))))
C1(c(b(a(b(b(B(x))))))) → A(a(c(c(c(b(b(b(b(b(b(B(x))))))))))))
C1(c(a(b(b(B(x)))))) → A(c(c(c(b(b(B(x)))))))
C1(c(b(a(b(b(B(x))))))) → A(c(c(c(b(b(b(b(B(x)))))))))
C1(c(a(b(b(B(x)))))) → C1(c(b(b(b(b(B(x)))))))
C1(c(a(b(b(B(x)))))) → C1(c(c(b(b(b(b(b(B(x)))))))))
C1(c(a(b(b(B(x)))))) → A(c(c(c(b(b(b(B(x))))))))
C1(c(a(b(b(B(x)))))) → C1(c(b(b(b(b(b(B(x))))))))
C1(c(b(a(b(b(B(x))))))) → C1(c(c(b(b(b(b(b(B(x)))))))))
The TRS R consists of the following rules:
c(c(x)) → b(b(b(a(a(a(x))))))
a(b(b(b(x)))) → b(b(b(b(b(b(b(b(x))))))))
c(c(b(b(x)))) → a(a(a(a(c(c(c(x)))))))
c(c(b(a(b(b(B(x))))))) → a(a(a(a(c(c(c(b(b(b(b(b(b(B(x))))))))))))))
a(a(b(b(B(x))))) → b(b(b(b(b(b(b(b(b(B(x))))))))))
a(b(a(b(b(B(x)))))) → b(b(b(b(b(b(b(b(b(B(x))))))))))
a(a(b(b(B(x))))) → b(b(b(b(b(b(b(b(b(b(b(B(x))))))))))))
a(b(a(b(b(B(x)))))) → b(b(b(b(b(b(b(b(b(b(b(B(x))))))))))))
a(b(b(a(b(b(B(x))))))) → b(b(b(b(b(b(b(b(b(b(b(B(x))))))))))))
a(b(b(a(b(b(B(x))))))) → b(b(b(b(b(b(b(b(b(b(B(x)))))))))))
a(a(b(b(B(x))))) → b(b(b(b(b(b(b(b(b(b(b(b(B(x)))))))))))))
c(c(a(b(b(B(x)))))) → a(a(a(a(c(c(c(b(b(b(B(x)))))))))))
a(b(b(B(x)))) → b(b(B(x)))
a(b(b(B(x)))) → B(x)
c(c(b(a(b(b(B(x))))))) → a(a(a(a(c(c(c(b(b(b(b(b(B(x)))))))))))))
a(b(b(a(b(b(B(x))))))) → b(b(b(b(b(b(b(b(b(b(b(b(b(B(x))))))))))))))
c(c(b(a(b(b(B(x))))))) → a(a(a(a(c(c(c(b(b(b(b(B(x))))))))))))
a(b(a(b(b(B(x)))))) → b(b(b(b(b(b(b(b(b(b(b(b(B(x)))))))))))))
a(b(a(b(b(B(x)))))) → b(b(b(b(b(b(b(b(b(b(B(x)))))))))))
a(b(b(a(b(b(B(x))))))) → b(b(b(b(b(b(b(b(b(b(b(b(B(x)))))))))))))
c(c(a(b(b(B(x)))))) → a(a(a(a(c(c(c(b(b(b(b(b(B(x)))))))))))))
a(b(b(a(b(b(B(x))))))) → b(b(b(b(b(b(b(b(b(b(b(b(b(b(B(x)))))))))))))))
c(c(a(b(b(B(x)))))) → a(a(a(a(c(c(c(b(b(b(b(B(x))))))))))))
a(b(a(b(b(B(x)))))) → b(b(b(b(b(b(b(b(b(b(b(b(b(B(x))))))))))))))
c(c(b(a(b(b(B(x))))))) → a(a(a(a(c(c(c(b(b(B(x))))))))))
c(c(a(b(b(B(x)))))) → a(a(a(a(c(c(c(b(b(B(x))))))))))
c(c(b(B(x)))) → a(a(a(a(c(C(x))))))
a(a(b(b(B(x))))) → b(b(b(b(b(b(b(b(b(b(B(x)))))))))))
c(C(x)) → b(B(x))
c(c(a(b(b(B(x)))))) → a(a(a(a(c(c(c(b(B(x)))))))))
c(c(b(a(b(b(B(x))))))) → a(a(a(a(c(c(c(b(b(b(B(x)))))))))))
a(b(b(B(x)))) → b(B(x))
c(c(b(B(x)))) → a(a(a(a(c(c(C(x)))))))
c(C(x)) → b(b(B(x)))
a(a(b(b(B(x))))) → b(b(b(b(b(b(b(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 66 less nodes.
↳ QTRS
↳ QTRS Reverse
↳ QTRS Reverse
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ SemLabProof
↳ SemLabProof2
↳ QDP
↳ QDPToSRSProof
↳ QTRS
↳ QTRS Reverse
↳ QTRS
↳ QTRS Reverse
↳ QTRS Reverse
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
Q DP problem:
The TRS P consists of the following rules:
C1(c(b(b(x)))) → C1(c(c(x)))
C1(c(b(a(b(b(B(x))))))) → C1(c(c(b(b(B(x))))))
C1(c(b(a(b(b(B(x))))))) → C1(c(c(b(b(b(B(x)))))))
C1(c(b(B(x)))) → C1(c(C(x)))
C1(c(a(b(b(B(x)))))) → C1(c(b(b(b(B(x))))))
C1(c(a(b(b(B(x)))))) → C1(c(c(b(b(b(b(B(x))))))))
C1(c(b(b(x)))) → C1(x)
C1(c(a(b(b(B(x)))))) → C1(c(b(b(B(x)))))
C1(c(b(a(b(b(B(x))))))) → C1(c(c(b(b(b(b(B(x))))))))
C1(c(b(a(b(b(B(x))))))) → C1(c(b(b(b(B(x))))))
C1(c(b(a(b(b(B(x))))))) → C1(c(b(b(b(b(B(x)))))))
C1(c(b(b(x)))) → C1(c(x))
C1(c(a(b(b(B(x)))))) → C1(c(b(b(b(b(B(x)))))))
C1(c(a(b(b(B(x)))))) → C1(c(c(b(b(b(b(b(B(x)))))))))
C1(c(a(b(b(B(x)))))) → C1(c(c(b(B(x)))))
C1(c(b(a(b(b(B(x))))))) → C1(c(b(b(B(x)))))
C1(c(a(b(b(B(x)))))) → C1(c(b(B(x))))
C1(c(b(a(b(b(B(x))))))) → C1(c(c(b(b(b(b(b(b(B(x))))))))))
C1(c(a(b(b(B(x)))))) → C1(c(c(b(b(b(B(x)))))))
C1(c(b(a(b(b(B(x))))))) → C1(c(b(b(b(b(b(b(B(x)))))))))
C1(c(b(a(b(b(B(x))))))) → C1(c(b(b(b(b(b(B(x))))))))
C1(c(a(b(b(B(x)))))) → C1(c(b(b(b(b(b(B(x))))))))
C1(c(b(a(b(b(B(x))))))) → C1(c(c(b(b(b(b(b(B(x)))))))))
C1(c(a(b(b(B(x)))))) → C1(c(c(b(b(B(x))))))
The TRS R consists of the following rules:
c(c(x)) → b(b(b(a(a(a(x))))))
a(b(b(b(x)))) → b(b(b(b(b(b(b(b(x))))))))
c(c(b(b(x)))) → a(a(a(a(c(c(c(x)))))))
c(c(b(a(b(b(B(x))))))) → a(a(a(a(c(c(c(b(b(b(b(b(b(B(x))))))))))))))
a(a(b(b(B(x))))) → b(b(b(b(b(b(b(b(b(B(x))))))))))
a(b(a(b(b(B(x)))))) → b(b(b(b(b(b(b(b(b(B(x))))))))))
a(a(b(b(B(x))))) → b(b(b(b(b(b(b(b(b(b(b(B(x))))))))))))
a(b(a(b(b(B(x)))))) → b(b(b(b(b(b(b(b(b(b(b(B(x))))))))))))
a(b(b(a(b(b(B(x))))))) → b(b(b(b(b(b(b(b(b(b(b(B(x))))))))))))
a(b(b(a(b(b(B(x))))))) → b(b(b(b(b(b(b(b(b(b(B(x)))))))))))
a(a(b(b(B(x))))) → b(b(b(b(b(b(b(b(b(b(b(b(B(x)))))))))))))
c(c(a(b(b(B(x)))))) → a(a(a(a(c(c(c(b(b(b(B(x)))))))))))
a(b(b(B(x)))) → b(b(B(x)))
a(b(b(B(x)))) → B(x)
c(c(b(a(b(b(B(x))))))) → a(a(a(a(c(c(c(b(b(b(b(b(B(x)))))))))))))
a(b(b(a(b(b(B(x))))))) → b(b(b(b(b(b(b(b(b(b(b(b(b(B(x))))))))))))))
c(c(b(a(b(b(B(x))))))) → a(a(a(a(c(c(c(b(b(b(b(B(x))))))))))))
a(b(a(b(b(B(x)))))) → b(b(b(b(b(b(b(b(b(b(b(b(B(x)))))))))))))
a(b(a(b(b(B(x)))))) → b(b(b(b(b(b(b(b(b(b(B(x)))))))))))
a(b(b(a(b(b(B(x))))))) → b(b(b(b(b(b(b(b(b(b(b(b(B(x)))))))))))))
c(c(a(b(b(B(x)))))) → a(a(a(a(c(c(c(b(b(b(b(b(B(x)))))))))))))
a(b(b(a(b(b(B(x))))))) → b(b(b(b(b(b(b(b(b(b(b(b(b(b(B(x)))))))))))))))
c(c(a(b(b(B(x)))))) → a(a(a(a(c(c(c(b(b(b(b(B(x))))))))))))
a(b(a(b(b(B(x)))))) → b(b(b(b(b(b(b(b(b(b(b(b(b(B(x))))))))))))))
c(c(b(a(b(b(B(x))))))) → a(a(a(a(c(c(c(b(b(B(x))))))))))
c(c(a(b(b(B(x)))))) → a(a(a(a(c(c(c(b(b(B(x))))))))))
c(c(b(B(x)))) → a(a(a(a(c(C(x))))))
a(a(b(b(B(x))))) → b(b(b(b(b(b(b(b(b(b(B(x)))))))))))
c(C(x)) → b(B(x))
c(c(a(b(b(B(x)))))) → a(a(a(a(c(c(c(b(B(x)))))))))
c(c(b(a(b(b(B(x))))))) → a(a(a(a(c(c(c(b(b(b(B(x)))))))))))
a(b(b(B(x)))) → b(B(x))
c(c(b(B(x)))) → a(a(a(a(c(c(C(x)))))))
c(C(x)) → b(b(B(x)))
a(a(b(b(B(x))))) → b(b(b(b(b(b(b(b(B(x)))))))))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule C1(c(b(a(b(b(B(x))))))) → C1(c(c(b(b(b(b(b(b(B(x)))))))))) at position [0] we obtained the following new rules:
C1(c(b(a(b(b(B(y0))))))) → C1(a(a(a(a(c(c(c(b(b(b(b(B(y0)))))))))))))
C1(c(b(a(b(b(B(y0))))))) → C1(b(b(b(a(a(a(b(b(b(b(b(b(B(y0))))))))))))))
↳ QTRS
↳ QTRS Reverse
↳ QTRS Reverse
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ SemLabProof
↳ SemLabProof2
↳ QDP
↳ QDPToSRSProof
↳ QTRS
↳ QTRS Reverse
↳ QTRS
↳ QTRS Reverse
↳ QTRS Reverse
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
Q DP problem:
The TRS P consists of the following rules:
C1(c(b(a(b(b(B(x))))))) → C1(c(c(b(b(B(x))))))
C1(c(b(B(x)))) → C1(c(C(x)))
C1(c(b(a(b(b(B(x))))))) → C1(c(c(b(b(b(B(x)))))))
C1(c(a(b(b(B(x)))))) → C1(c(b(b(b(B(x))))))
C1(c(b(b(x)))) → C1(x)
C1(c(a(b(b(B(x)))))) → C1(c(c(b(b(b(b(B(x))))))))
C1(c(b(a(b(b(B(x))))))) → C1(c(c(b(b(b(b(B(x))))))))
C1(c(b(a(b(b(B(x))))))) → C1(c(b(b(b(B(x))))))
C1(c(b(a(b(b(B(x))))))) → C1(c(b(b(b(b(B(x)))))))
C1(c(b(b(x)))) → C1(c(x))
C1(c(a(b(b(B(x)))))) → C1(c(c(b(B(x)))))
C1(c(a(b(b(B(x)))))) → C1(c(c(b(b(b(B(x)))))))
C1(c(a(b(b(B(x)))))) → C1(c(b(B(x))))
C1(c(b(a(b(b(B(x))))))) → C1(c(b(b(b(b(b(B(x))))))))
C1(c(b(a(b(b(B(y0))))))) → C1(b(b(b(a(a(a(b(b(b(b(b(b(B(y0))))))))))))))
C1(c(a(b(b(B(x)))))) → C1(c(c(b(b(B(x))))))
C1(c(b(b(x)))) → C1(c(c(x)))
C1(c(a(b(b(B(x)))))) → C1(c(b(b(B(x)))))
C1(c(a(b(b(B(x)))))) → C1(c(c(b(b(b(b(b(B(x)))))))))
C1(c(a(b(b(B(x)))))) → C1(c(b(b(b(b(B(x)))))))
C1(c(b(a(b(b(B(y0))))))) → C1(a(a(a(a(c(c(c(b(b(b(b(B(y0)))))))))))))
C1(c(b(a(b(b(B(x))))))) → C1(c(b(b(B(x)))))
C1(c(b(a(b(b(B(x))))))) → C1(c(b(b(b(b(b(b(B(x)))))))))
C1(c(a(b(b(B(x)))))) → C1(c(b(b(b(b(b(B(x))))))))
C1(c(b(a(b(b(B(x))))))) → C1(c(c(b(b(b(b(b(B(x)))))))))
The TRS R consists of the following rules:
c(c(x)) → b(b(b(a(a(a(x))))))
a(b(b(b(x)))) → b(b(b(b(b(b(b(b(x))))))))
c(c(b(b(x)))) → a(a(a(a(c(c(c(x)))))))
c(c(b(a(b(b(B(x))))))) → a(a(a(a(c(c(c(b(b(b(b(b(b(B(x))))))))))))))
a(a(b(b(B(x))))) → b(b(b(b(b(b(b(b(b(B(x))))))))))
a(b(a(b(b(B(x)))))) → b(b(b(b(b(b(b(b(b(B(x))))))))))
a(a(b(b(B(x))))) → b(b(b(b(b(b(b(b(b(b(b(B(x))))))))))))
a(b(a(b(b(B(x)))))) → b(b(b(b(b(b(b(b(b(b(b(B(x))))))))))))
a(b(b(a(b(b(B(x))))))) → b(b(b(b(b(b(b(b(b(b(b(B(x))))))))))))
a(b(b(a(b(b(B(x))))))) → b(b(b(b(b(b(b(b(b(b(B(x)))))))))))
a(a(b(b(B(x))))) → b(b(b(b(b(b(b(b(b(b(b(b(B(x)))))))))))))
c(c(a(b(b(B(x)))))) → a(a(a(a(c(c(c(b(b(b(B(x)))))))))))
a(b(b(B(x)))) → b(b(B(x)))
a(b(b(B(x)))) → B(x)
c(c(b(a(b(b(B(x))))))) → a(a(a(a(c(c(c(b(b(b(b(b(B(x)))))))))))))
a(b(b(a(b(b(B(x))))))) → b(b(b(b(b(b(b(b(b(b(b(b(b(B(x))))))))))))))
c(c(b(a(b(b(B(x))))))) → a(a(a(a(c(c(c(b(b(b(b(B(x))))))))))))
a(b(a(b(b(B(x)))))) → b(b(b(b(b(b(b(b(b(b(b(b(B(x)))))))))))))
a(b(a(b(b(B(x)))))) → b(b(b(b(b(b(b(b(b(b(B(x)))))))))))
a(b(b(a(b(b(B(x))))))) → b(b(b(b(b(b(b(b(b(b(b(b(B(x)))))))))))))
c(c(a(b(b(B(x)))))) → a(a(a(a(c(c(c(b(b(b(b(b(B(x)))))))))))))
a(b(b(a(b(b(B(x))))))) → b(b(b(b(b(b(b(b(b(b(b(b(b(b(B(x)))))))))))))))
c(c(a(b(b(B(x)))))) → a(a(a(a(c(c(c(b(b(b(b(B(x))))))))))))
a(b(a(b(b(B(x)))))) → b(b(b(b(b(b(b(b(b(b(b(b(b(B(x))))))))))))))
c(c(b(a(b(b(B(x))))))) → a(a(a(a(c(c(c(b(b(B(x))))))))))
c(c(a(b(b(B(x)))))) → a(a(a(a(c(c(c(b(b(B(x))))))))))
c(c(b(B(x)))) → a(a(a(a(c(C(x))))))
a(a(b(b(B(x))))) → b(b(b(b(b(b(b(b(b(b(B(x)))))))))))
c(C(x)) → b(B(x))
c(c(a(b(b(B(x)))))) → a(a(a(a(c(c(c(b(B(x)))))))))
c(c(b(a(b(b(B(x))))))) → a(a(a(a(c(c(c(b(b(b(B(x)))))))))))
a(b(b(B(x)))) → b(B(x))
c(c(b(B(x)))) → a(a(a(a(c(c(C(x)))))))
c(C(x)) → b(b(B(x)))
a(a(b(b(B(x))))) → b(b(b(b(b(b(b(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 2 less nodes.
↳ QTRS
↳ QTRS Reverse
↳ QTRS Reverse
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ SemLabProof
↳ SemLabProof2
↳ QDP
↳ QDPToSRSProof
↳ QTRS
↳ QTRS Reverse
↳ QTRS
↳ QTRS Reverse
↳ QTRS Reverse
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
Q DP problem:
The TRS P consists of the following rules:
C1(c(b(b(x)))) → C1(c(c(x)))
C1(c(b(a(b(b(B(x))))))) → C1(c(c(b(b(B(x))))))
C1(c(b(a(b(b(B(x))))))) → C1(c(c(b(b(b(B(x)))))))
C1(c(b(B(x)))) → C1(c(C(x)))
C1(c(a(b(b(B(x)))))) → C1(c(b(b(b(B(x))))))
C1(c(b(b(x)))) → C1(x)
C1(c(a(b(b(B(x)))))) → C1(c(c(b(b(b(b(B(x))))))))
C1(c(a(b(b(B(x)))))) → C1(c(b(b(B(x)))))
C1(c(b(a(b(b(B(x))))))) → C1(c(c(b(b(b(b(B(x))))))))
C1(c(b(a(b(b(B(x))))))) → C1(c(b(b(b(B(x))))))
C1(c(b(a(b(b(B(x))))))) → C1(c(b(b(b(b(B(x)))))))
C1(c(b(b(x)))) → C1(c(x))
C1(c(a(b(b(B(x)))))) → C1(c(b(b(b(b(B(x)))))))
C1(c(a(b(b(B(x)))))) → C1(c(c(b(b(b(b(b(B(x)))))))))
C1(c(a(b(b(B(x)))))) → C1(c(c(b(B(x)))))
C1(c(b(a(b(b(B(x))))))) → C1(c(b(b(B(x)))))
C1(c(a(b(b(B(x)))))) → C1(c(b(B(x))))
C1(c(a(b(b(B(x)))))) → C1(c(c(b(b(b(B(x)))))))
C1(c(b(a(b(b(B(x))))))) → C1(c(b(b(b(b(b(b(B(x)))))))))
C1(c(b(a(b(b(B(x))))))) → C1(c(b(b(b(b(b(B(x))))))))
C1(c(a(b(b(B(x)))))) → C1(c(b(b(b(b(b(B(x))))))))
C1(c(b(a(b(b(B(x))))))) → C1(c(c(b(b(b(b(b(B(x)))))))))
C1(c(a(b(b(B(x)))))) → C1(c(c(b(b(B(x))))))
The TRS R consists of the following rules:
c(c(x)) → b(b(b(a(a(a(x))))))
a(b(b(b(x)))) → b(b(b(b(b(b(b(b(x))))))))
c(c(b(b(x)))) → a(a(a(a(c(c(c(x)))))))
c(c(b(a(b(b(B(x))))))) → a(a(a(a(c(c(c(b(b(b(b(b(b(B(x))))))))))))))
a(a(b(b(B(x))))) → b(b(b(b(b(b(b(b(b(B(x))))))))))
a(b(a(b(b(B(x)))))) → b(b(b(b(b(b(b(b(b(B(x))))))))))
a(a(b(b(B(x))))) → b(b(b(b(b(b(b(b(b(b(b(B(x))))))))))))
a(b(a(b(b(B(x)))))) → b(b(b(b(b(b(b(b(b(b(b(B(x))))))))))))
a(b(b(a(b(b(B(x))))))) → b(b(b(b(b(b(b(b(b(b(b(B(x))))))))))))
a(b(b(a(b(b(B(x))))))) → b(b(b(b(b(b(b(b(b(b(B(x)))))))))))
a(a(b(b(B(x))))) → b(b(b(b(b(b(b(b(b(b(b(b(B(x)))))))))))))
c(c(a(b(b(B(x)))))) → a(a(a(a(c(c(c(b(b(b(B(x)))))))))))
a(b(b(B(x)))) → b(b(B(x)))
a(b(b(B(x)))) → B(x)
c(c(b(a(b(b(B(x))))))) → a(a(a(a(c(c(c(b(b(b(b(b(B(x)))))))))))))
a(b(b(a(b(b(B(x))))))) → b(b(b(b(b(b(b(b(b(b(b(b(b(B(x))))))))))))))
c(c(b(a(b(b(B(x))))))) → a(a(a(a(c(c(c(b(b(b(b(B(x))))))))))))
a(b(a(b(b(B(x)))))) → b(b(b(b(b(b(b(b(b(b(b(b(B(x)))))))))))))
a(b(a(b(b(B(x)))))) → b(b(b(b(b(b(b(b(b(b(B(x)))))))))))
a(b(b(a(b(b(B(x))))))) → b(b(b(b(b(b(b(b(b(b(b(b(B(x)))))))))))))
c(c(a(b(b(B(x)))))) → a(a(a(a(c(c(c(b(b(b(b(b(B(x)))))))))))))
a(b(b(a(b(b(B(x))))))) → b(b(b(b(b(b(b(b(b(b(b(b(b(b(B(x)))))))))))))))
c(c(a(b(b(B(x)))))) → a(a(a(a(c(c(c(b(b(b(b(B(x))))))))))))
a(b(a(b(b(B(x)))))) → b(b(b(b(b(b(b(b(b(b(b(b(b(B(x))))))))))))))
c(c(b(a(b(b(B(x))))))) → a(a(a(a(c(c(c(b(b(B(x))))))))))
c(c(a(b(b(B(x)))))) → a(a(a(a(c(c(c(b(b(B(x))))))))))
c(c(b(B(x)))) → a(a(a(a(c(C(x))))))
a(a(b(b(B(x))))) → b(b(b(b(b(b(b(b(b(b(B(x)))))))))))
c(C(x)) → b(B(x))
c(c(a(b(b(B(x)))))) → a(a(a(a(c(c(c(b(B(x)))))))))
c(c(b(a(b(b(B(x))))))) → a(a(a(a(c(c(c(b(b(b(B(x)))))))))))
a(b(b(B(x)))) → b(B(x))
c(c(b(B(x)))) → a(a(a(a(c(c(C(x)))))))
c(C(x)) → b(b(B(x)))
a(a(b(b(B(x))))) → b(b(b(b(b(b(b(b(B(x)))))))))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule C1(c(a(b(b(B(x)))))) → C1(c(c(b(b(b(B(x))))))) at position [0] we obtained the following new rules:
C1(c(a(b(b(B(y0)))))) → C1(a(a(a(a(c(c(c(b(B(y0))))))))))
C1(c(a(b(b(B(y0)))))) → C1(b(b(b(a(a(a(b(b(b(B(y0)))))))))))
↳ QTRS
↳ QTRS Reverse
↳ QTRS Reverse
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ SemLabProof
↳ SemLabProof2
↳ QDP
↳ QDPToSRSProof
↳ QTRS
↳ QTRS Reverse
↳ QTRS
↳ QTRS Reverse
↳ QTRS Reverse
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
Q DP problem:
The TRS P consists of the following rules:
C1(c(b(a(b(b(B(x))))))) → C1(c(c(b(b(B(x))))))
C1(c(b(a(b(b(B(x))))))) → C1(c(c(b(b(b(B(x)))))))
C1(c(b(B(x)))) → C1(c(C(x)))
C1(c(a(b(b(B(x)))))) → C1(c(b(b(b(B(x))))))
C1(c(b(b(x)))) → C1(x)
C1(c(a(b(b(B(x)))))) → C1(c(c(b(b(b(b(B(x))))))))
C1(c(b(a(b(b(B(x))))))) → C1(c(c(b(b(b(b(B(x))))))))
C1(c(a(b(b(B(y0)))))) → C1(a(a(a(a(c(c(c(b(B(y0))))))))))
C1(c(b(a(b(b(B(x))))))) → C1(c(b(b(b(B(x))))))
C1(c(b(a(b(b(B(x))))))) → C1(c(b(b(b(b(B(x)))))))
C1(c(b(b(x)))) → C1(c(x))
C1(c(a(b(b(B(x)))))) → C1(c(c(b(B(x)))))
C1(c(a(b(b(B(x)))))) → C1(c(b(B(x))))
C1(c(b(a(b(b(B(x))))))) → C1(c(b(b(b(b(b(B(x))))))))
C1(c(a(b(b(B(x)))))) → C1(c(c(b(b(B(x))))))
C1(c(a(b(b(B(y0)))))) → C1(b(b(b(a(a(a(b(b(b(B(y0)))))))))))
C1(c(b(b(x)))) → C1(c(c(x)))
C1(c(a(b(b(B(x)))))) → C1(c(b(b(B(x)))))
C1(c(a(b(b(B(x)))))) → C1(c(b(b(b(b(B(x)))))))
C1(c(a(b(b(B(x)))))) → C1(c(c(b(b(b(b(b(B(x)))))))))
C1(c(b(a(b(b(B(x))))))) → C1(c(b(b(B(x)))))
C1(c(b(a(b(b(B(x))))))) → C1(c(b(b(b(b(b(b(B(x)))))))))
C1(c(a(b(b(B(x)))))) → C1(c(b(b(b(b(b(B(x))))))))
C1(c(b(a(b(b(B(x))))))) → C1(c(c(b(b(b(b(b(B(x)))))))))
The TRS R consists of the following rules:
c(c(x)) → b(b(b(a(a(a(x))))))
a(b(b(b(x)))) → b(b(b(b(b(b(b(b(x))))))))
c(c(b(b(x)))) → a(a(a(a(c(c(c(x)))))))
c(c(b(a(b(b(B(x))))))) → a(a(a(a(c(c(c(b(b(b(b(b(b(B(x))))))))))))))
a(a(b(b(B(x))))) → b(b(b(b(b(b(b(b(b(B(x))))))))))
a(b(a(b(b(B(x)))))) → b(b(b(b(b(b(b(b(b(B(x))))))))))
a(a(b(b(B(x))))) → b(b(b(b(b(b(b(b(b(b(b(B(x))))))))))))
a(b(a(b(b(B(x)))))) → b(b(b(b(b(b(b(b(b(b(b(B(x))))))))))))
a(b(b(a(b(b(B(x))))))) → b(b(b(b(b(b(b(b(b(b(b(B(x))))))))))))
a(b(b(a(b(b(B(x))))))) → b(b(b(b(b(b(b(b(b(b(B(x)))))))))))
a(a(b(b(B(x))))) → b(b(b(b(b(b(b(b(b(b(b(b(B(x)))))))))))))
c(c(a(b(b(B(x)))))) → a(a(a(a(c(c(c(b(b(b(B(x)))))))))))
a(b(b(B(x)))) → b(b(B(x)))
a(b(b(B(x)))) → B(x)
c(c(b(a(b(b(B(x))))))) → a(a(a(a(c(c(c(b(b(b(b(b(B(x)))))))))))))
a(b(b(a(b(b(B(x))))))) → b(b(b(b(b(b(b(b(b(b(b(b(b(B(x))))))))))))))
c(c(b(a(b(b(B(x))))))) → a(a(a(a(c(c(c(b(b(b(b(B(x))))))))))))
a(b(a(b(b(B(x)))))) → b(b(b(b(b(b(b(b(b(b(b(b(B(x)))))))))))))
a(b(a(b(b(B(x)))))) → b(b(b(b(b(b(b(b(b(b(B(x)))))))))))
a(b(b(a(b(b(B(x))))))) → b(b(b(b(b(b(b(b(b(b(b(b(B(x)))))))))))))
c(c(a(b(b(B(x)))))) → a(a(a(a(c(c(c(b(b(b(b(b(B(x)))))))))))))
a(b(b(a(b(b(B(x))))))) → b(b(b(b(b(b(b(b(b(b(b(b(b(b(B(x)))))))))))))))
c(c(a(b(b(B(x)))))) → a(a(a(a(c(c(c(b(b(b(b(B(x))))))))))))
a(b(a(b(b(B(x)))))) → b(b(b(b(b(b(b(b(b(b(b(b(b(B(x))))))))))))))
c(c(b(a(b(b(B(x))))))) → a(a(a(a(c(c(c(b(b(B(x))))))))))
c(c(a(b(b(B(x)))))) → a(a(a(a(c(c(c(b(b(B(x))))))))))
c(c(b(B(x)))) → a(a(a(a(c(C(x))))))
a(a(b(b(B(x))))) → b(b(b(b(b(b(b(b(b(b(B(x)))))))))))
c(C(x)) → b(B(x))
c(c(a(b(b(B(x)))))) → a(a(a(a(c(c(c(b(B(x)))))))))
c(c(b(a(b(b(B(x))))))) → a(a(a(a(c(c(c(b(b(b(B(x)))))))))))
a(b(b(B(x)))) → b(B(x))
c(c(b(B(x)))) → a(a(a(a(c(c(C(x)))))))
c(C(x)) → b(b(B(x)))
a(a(b(b(B(x))))) → b(b(b(b(b(b(b(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 2 less nodes.
↳ QTRS
↳ QTRS Reverse
↳ QTRS Reverse
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ SemLabProof
↳ SemLabProof2
↳ QDP
↳ QDPToSRSProof
↳ QTRS
↳ QTRS Reverse
↳ QTRS
↳ QTRS Reverse
↳ QTRS Reverse
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
Q DP problem:
The TRS P consists of the following rules:
C1(c(b(b(x)))) → C1(c(c(x)))
C1(c(b(a(b(b(B(x))))))) → C1(c(c(b(b(B(x))))))
C1(c(b(a(b(b(B(x))))))) → C1(c(c(b(b(b(B(x)))))))
C1(c(b(B(x)))) → C1(c(C(x)))
C1(c(a(b(b(B(x)))))) → C1(c(b(b(b(B(x))))))
C1(c(b(b(x)))) → C1(x)
C1(c(a(b(b(B(x)))))) → C1(c(c(b(b(b(b(B(x))))))))
C1(c(a(b(b(B(x)))))) → C1(c(b(b(B(x)))))
C1(c(b(a(b(b(B(x))))))) → C1(c(c(b(b(b(b(B(x))))))))
C1(c(b(a(b(b(B(x))))))) → C1(c(b(b(b(B(x))))))
C1(c(b(a(b(b(B(x))))))) → C1(c(b(b(b(b(B(x)))))))
C1(c(b(b(x)))) → C1(c(x))
C1(c(a(b(b(B(x)))))) → C1(c(c(b(b(b(b(b(B(x)))))))))
C1(c(a(b(b(B(x)))))) → C1(c(b(b(b(b(B(x)))))))
C1(c(a(b(b(B(x)))))) → C1(c(c(b(B(x)))))
C1(c(b(a(b(b(B(x))))))) → C1(c(b(b(B(x)))))
C1(c(a(b(b(B(x)))))) → C1(c(b(B(x))))
C1(c(b(a(b(b(B(x))))))) → C1(c(b(b(b(b(b(b(B(x)))))))))
C1(c(b(a(b(b(B(x))))))) → C1(c(b(b(b(b(b(B(x))))))))
C1(c(a(b(b(B(x)))))) → C1(c(b(b(b(b(b(B(x))))))))
C1(c(b(a(b(b(B(x))))))) → C1(c(c(b(b(b(b(b(B(x)))))))))
C1(c(a(b(b(B(x)))))) → C1(c(c(b(b(B(x))))))
The TRS R consists of the following rules:
c(c(x)) → b(b(b(a(a(a(x))))))
a(b(b(b(x)))) → b(b(b(b(b(b(b(b(x))))))))
c(c(b(b(x)))) → a(a(a(a(c(c(c(x)))))))
c(c(b(a(b(b(B(x))))))) → a(a(a(a(c(c(c(b(b(b(b(b(b(B(x))))))))))))))
a(a(b(b(B(x))))) → b(b(b(b(b(b(b(b(b(B(x))))))))))
a(b(a(b(b(B(x)))))) → b(b(b(b(b(b(b(b(b(B(x))))))))))
a(a(b(b(B(x))))) → b(b(b(b(b(b(b(b(b(b(b(B(x))))))))))))
a(b(a(b(b(B(x)))))) → b(b(b(b(b(b(b(b(b(b(b(B(x))))))))))))
a(b(b(a(b(b(B(x))))))) → b(b(b(b(b(b(b(b(b(b(b(B(x))))))))))))
a(b(b(a(b(b(B(x))))))) → b(b(b(b(b(b(b(b(b(b(B(x)))))))))))
a(a(b(b(B(x))))) → b(b(b(b(b(b(b(b(b(b(b(b(B(x)))))))))))))
c(c(a(b(b(B(x)))))) → a(a(a(a(c(c(c(b(b(b(B(x)))))))))))
a(b(b(B(x)))) → b(b(B(x)))
a(b(b(B(x)))) → B(x)
c(c(b(a(b(b(B(x))))))) → a(a(a(a(c(c(c(b(b(b(b(b(B(x)))))))))))))
a(b(b(a(b(b(B(x))))))) → b(b(b(b(b(b(b(b(b(b(b(b(b(B(x))))))))))))))
c(c(b(a(b(b(B(x))))))) → a(a(a(a(c(c(c(b(b(b(b(B(x))))))))))))
a(b(a(b(b(B(x)))))) → b(b(b(b(b(b(b(b(b(b(b(b(B(x)))))))))))))
a(b(a(b(b(B(x)))))) → b(b(b(b(b(b(b(b(b(b(B(x)))))))))))
a(b(b(a(b(b(B(x))))))) → b(b(b(b(b(b(b(b(b(b(b(b(B(x)))))))))))))
c(c(a(b(b(B(x)))))) → a(a(a(a(c(c(c(b(b(b(b(b(B(x)))))))))))))
a(b(b(a(b(b(B(x))))))) → b(b(b(b(b(b(b(b(b(b(b(b(b(b(B(x)))))))))))))))
c(c(a(b(b(B(x)))))) → a(a(a(a(c(c(c(b(b(b(b(B(x))))))))))))
a(b(a(b(b(B(x)))))) → b(b(b(b(b(b(b(b(b(b(b(b(b(B(x))))))))))))))
c(c(b(a(b(b(B(x))))))) → a(a(a(a(c(c(c(b(b(B(x))))))))))
c(c(a(b(b(B(x)))))) → a(a(a(a(c(c(c(b(b(B(x))))))))))
c(c(b(B(x)))) → a(a(a(a(c(C(x))))))
a(a(b(b(B(x))))) → b(b(b(b(b(b(b(b(b(b(B(x)))))))))))
c(C(x)) → b(B(x))
c(c(a(b(b(B(x)))))) → a(a(a(a(c(c(c(b(B(x)))))))))
c(c(b(a(b(b(B(x))))))) → a(a(a(a(c(c(c(b(b(b(B(x)))))))))))
a(b(b(B(x)))) → b(B(x))
c(c(b(B(x)))) → a(a(a(a(c(c(C(x)))))))
c(C(x)) → b(b(B(x)))
a(a(b(b(B(x))))) → b(b(b(b(b(b(b(b(B(x)))))))))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule C1(c(b(a(b(b(B(x))))))) → C1(c(c(b(b(b(b(b(B(x))))))))) at position [0] we obtained the following new rules:
C1(c(b(a(b(b(B(y0))))))) → C1(b(b(b(a(a(a(b(b(b(b(b(B(y0)))))))))))))
C1(c(b(a(b(b(B(y0))))))) → C1(a(a(a(a(c(c(c(b(b(b(B(y0))))))))))))
↳ QTRS
↳ QTRS Reverse
↳ QTRS Reverse
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ SemLabProof
↳ SemLabProof2
↳ QDP
↳ QDPToSRSProof
↳ QTRS
↳ QTRS Reverse
↳ QTRS
↳ QTRS Reverse
↳ QTRS Reverse
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
Q DP problem:
The TRS P consists of the following rules:
C1(c(b(b(x)))) → C1(c(c(x)))
C1(c(b(a(b(b(B(x))))))) → C1(c(c(b(b(B(x))))))
C1(c(b(a(b(b(B(y0))))))) → C1(b(b(b(a(a(a(b(b(b(b(b(B(y0)))))))))))))
C1(c(b(B(x)))) → C1(c(C(x)))
C1(c(b(a(b(b(B(x))))))) → C1(c(c(b(b(b(B(x)))))))
C1(c(a(b(b(B(x)))))) → C1(c(b(b(b(B(x))))))
C1(c(a(b(b(B(x)))))) → C1(c(c(b(b(b(b(B(x))))))))
C1(c(b(b(x)))) → C1(x)
C1(c(a(b(b(B(x)))))) → C1(c(b(b(B(x)))))
C1(c(b(a(b(b(B(x))))))) → C1(c(c(b(b(b(b(B(x))))))))
C1(c(b(a(b(b(B(x))))))) → C1(c(b(b(b(B(x))))))
C1(c(b(a(b(b(B(x))))))) → C1(c(b(b(b(b(B(x)))))))
C1(c(b(b(x)))) → C1(c(x))
C1(c(a(b(b(B(x)))))) → C1(c(b(b(b(b(B(x)))))))
C1(c(a(b(b(B(x)))))) → C1(c(c(b(b(b(b(b(B(x)))))))))
C1(c(a(b(b(B(x)))))) → C1(c(c(b(B(x)))))
C1(c(b(a(b(b(B(x))))))) → C1(c(b(b(B(x)))))
C1(c(a(b(b(B(x)))))) → C1(c(b(B(x))))
C1(c(b(a(b(b(B(y0))))))) → C1(a(a(a(a(c(c(c(b(b(b(B(y0))))))))))))
C1(c(b(a(b(b(B(x))))))) → C1(c(b(b(b(b(b(B(x))))))))
C1(c(b(a(b(b(B(x))))))) → C1(c(b(b(b(b(b(b(B(x)))))))))
C1(c(a(b(b(B(x)))))) → C1(c(b(b(b(b(b(B(x))))))))
C1(c(a(b(b(B(x)))))) → C1(c(c(b(b(B(x))))))
The TRS R consists of the following rules:
c(c(x)) → b(b(b(a(a(a(x))))))
a(b(b(b(x)))) → b(b(b(b(b(b(b(b(x))))))))
c(c(b(b(x)))) → a(a(a(a(c(c(c(x)))))))
c(c(b(a(b(b(B(x))))))) → a(a(a(a(c(c(c(b(b(b(b(b(b(B(x))))))))))))))
a(a(b(b(B(x))))) → b(b(b(b(b(b(b(b(b(B(x))))))))))
a(b(a(b(b(B(x)))))) → b(b(b(b(b(b(b(b(b(B(x))))))))))
a(a(b(b(B(x))))) → b(b(b(b(b(b(b(b(b(b(b(B(x))))))))))))
a(b(a(b(b(B(x)))))) → b(b(b(b(b(b(b(b(b(b(b(B(x))))))))))))
a(b(b(a(b(b(B(x))))))) → b(b(b(b(b(b(b(b(b(b(b(B(x))))))))))))
a(b(b(a(b(b(B(x))))))) → b(b(b(b(b(b(b(b(b(b(B(x)))))))))))
a(a(b(b(B(x))))) → b(b(b(b(b(b(b(b(b(b(b(b(B(x)))))))))))))
c(c(a(b(b(B(x)))))) → a(a(a(a(c(c(c(b(b(b(B(x)))))))))))
a(b(b(B(x)))) → b(b(B(x)))
a(b(b(B(x)))) → B(x)
c(c(b(a(b(b(B(x))))))) → a(a(a(a(c(c(c(b(b(b(b(b(B(x)))))))))))))
a(b(b(a(b(b(B(x))))))) → b(b(b(b(b(b(b(b(b(b(b(b(b(B(x))))))))))))))
c(c(b(a(b(b(B(x))))))) → a(a(a(a(c(c(c(b(b(b(b(B(x))))))))))))
a(b(a(b(b(B(x)))))) → b(b(b(b(b(b(b(b(b(b(b(b(B(x)))))))))))))
a(b(a(b(b(B(x)))))) → b(b(b(b(b(b(b(b(b(b(B(x)))))))))))
a(b(b(a(b(b(B(x))))))) → b(b(b(b(b(b(b(b(b(b(b(b(B(x)))))))))))))
c(c(a(b(b(B(x)))))) → a(a(a(a(c(c(c(b(b(b(b(b(B(x)))))))))))))
a(b(b(a(b(b(B(x))))))) → b(b(b(b(b(b(b(b(b(b(b(b(b(b(B(x)))))))))))))))
c(c(a(b(b(B(x)))))) → a(a(a(a(c(c(c(b(b(b(b(B(x))))))))))))
a(b(a(b(b(B(x)))))) → b(b(b(b(b(b(b(b(b(b(b(b(b(B(x))))))))))))))
c(c(b(a(b(b(B(x))))))) → a(a(a(a(c(c(c(b(b(B(x))))))))))
c(c(a(b(b(B(x)))))) → a(a(a(a(c(c(c(b(b(B(x))))))))))
c(c(b(B(x)))) → a(a(a(a(c(C(x))))))
a(a(b(b(B(x))))) → b(b(b(b(b(b(b(b(b(b(B(x)))))))))))
c(C(x)) → b(B(x))
c(c(a(b(b(B(x)))))) → a(a(a(a(c(c(c(b(B(x)))))))))
c(c(b(a(b(b(B(x))))))) → a(a(a(a(c(c(c(b(b(b(B(x)))))))))))
a(b(b(B(x)))) → b(B(x))
c(c(b(B(x)))) → a(a(a(a(c(c(C(x)))))))
c(C(x)) → b(b(B(x)))
a(a(b(b(B(x))))) → b(b(b(b(b(b(b(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 2 less nodes.
↳ QTRS
↳ QTRS Reverse
↳ QTRS Reverse
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ SemLabProof
↳ SemLabProof2
↳ QDP
↳ QDPToSRSProof
↳ QTRS
↳ QTRS Reverse
↳ QTRS
↳ QTRS Reverse
↳ QTRS Reverse
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
Q DP problem:
The TRS P consists of the following rules:
C1(c(b(b(x)))) → C1(c(c(x)))
C1(c(b(a(b(b(B(x))))))) → C1(c(c(b(b(B(x))))))
C1(c(b(a(b(b(B(x))))))) → C1(c(c(b(b(b(B(x)))))))
C1(c(b(B(x)))) → C1(c(C(x)))
C1(c(a(b(b(B(x)))))) → C1(c(b(b(b(B(x))))))
C1(c(b(b(x)))) → C1(x)
C1(c(a(b(b(B(x)))))) → C1(c(c(b(b(b(b(B(x))))))))
C1(c(a(b(b(B(x)))))) → C1(c(b(b(B(x)))))
C1(c(b(a(b(b(B(x))))))) → C1(c(c(b(b(b(b(B(x))))))))
C1(c(b(a(b(b(B(x))))))) → C1(c(b(b(b(B(x))))))
C1(c(b(a(b(b(B(x))))))) → C1(c(b(b(b(b(B(x)))))))
C1(c(b(b(x)))) → C1(c(x))
C1(c(a(b(b(B(x)))))) → C1(c(c(b(b(b(b(b(B(x)))))))))
C1(c(a(b(b(B(x)))))) → C1(c(b(b(b(b(B(x)))))))
C1(c(a(b(b(B(x)))))) → C1(c(c(b(B(x)))))
C1(c(b(a(b(b(B(x))))))) → C1(c(b(b(B(x)))))
C1(c(a(b(b(B(x)))))) → C1(c(b(B(x))))
C1(c(b(a(b(b(B(x))))))) → C1(c(b(b(b(b(b(b(B(x)))))))))
C1(c(b(a(b(b(B(x))))))) → C1(c(b(b(b(b(b(B(x))))))))
C1(c(a(b(b(B(x)))))) → C1(c(b(b(b(b(b(B(x))))))))
C1(c(a(b(b(B(x)))))) → C1(c(c(b(b(B(x))))))
The TRS R consists of the following rules:
c(c(x)) → b(b(b(a(a(a(x))))))
a(b(b(b(x)))) → b(b(b(b(b(b(b(b(x))))))))
c(c(b(b(x)))) → a(a(a(a(c(c(c(x)))))))
c(c(b(a(b(b(B(x))))))) → a(a(a(a(c(c(c(b(b(b(b(b(b(B(x))))))))))))))
a(a(b(b(B(x))))) → b(b(b(b(b(b(b(b(b(B(x))))))))))
a(b(a(b(b(B(x)))))) → b(b(b(b(b(b(b(b(b(B(x))))))))))
a(a(b(b(B(x))))) → b(b(b(b(b(b(b(b(b(b(b(B(x))))))))))))
a(b(a(b(b(B(x)))))) → b(b(b(b(b(b(b(b(b(b(b(B(x))))))))))))
a(b(b(a(b(b(B(x))))))) → b(b(b(b(b(b(b(b(b(b(b(B(x))))))))))))
a(b(b(a(b(b(B(x))))))) → b(b(b(b(b(b(b(b(b(b(B(x)))))))))))
a(a(b(b(B(x))))) → b(b(b(b(b(b(b(b(b(b(b(b(B(x)))))))))))))
c(c(a(b(b(B(x)))))) → a(a(a(a(c(c(c(b(b(b(B(x)))))))))))
a(b(b(B(x)))) → b(b(B(x)))
a(b(b(B(x)))) → B(x)
c(c(b(a(b(b(B(x))))))) → a(a(a(a(c(c(c(b(b(b(b(b(B(x)))))))))))))
a(b(b(a(b(b(B(x))))))) → b(b(b(b(b(b(b(b(b(b(b(b(b(B(x))))))))))))))
c(c(b(a(b(b(B(x))))))) → a(a(a(a(c(c(c(b(b(b(b(B(x))))))))))))
a(b(a(b(b(B(x)))))) → b(b(b(b(b(b(b(b(b(b(b(b(B(x)))))))))))))
a(b(a(b(b(B(x)))))) → b(b(b(b(b(b(b(b(b(b(B(x)))))))))))
a(b(b(a(b(b(B(x))))))) → b(b(b(b(b(b(b(b(b(b(b(b(B(x)))))))))))))
c(c(a(b(b(B(x)))))) → a(a(a(a(c(c(c(b(b(b(b(b(B(x)))))))))))))
a(b(b(a(b(b(B(x))))))) → b(b(b(b(b(b(b(b(b(b(b(b(b(b(B(x)))))))))))))))
c(c(a(b(b(B(x)))))) → a(a(a(a(c(c(c(b(b(b(b(B(x))))))))))))
a(b(a(b(b(B(x)))))) → b(b(b(b(b(b(b(b(b(b(b(b(b(B(x))))))))))))))
c(c(b(a(b(b(B(x))))))) → a(a(a(a(c(c(c(b(b(B(x))))))))))
c(c(a(b(b(B(x)))))) → a(a(a(a(c(c(c(b(b(B(x))))))))))
c(c(b(B(x)))) → a(a(a(a(c(C(x))))))
a(a(b(b(B(x))))) → b(b(b(b(b(b(b(b(b(b(B(x)))))))))))
c(C(x)) → b(B(x))
c(c(a(b(b(B(x)))))) → a(a(a(a(c(c(c(b(B(x)))))))))
c(c(b(a(b(b(B(x))))))) → a(a(a(a(c(c(c(b(b(b(B(x)))))))))))
a(b(b(B(x)))) → b(B(x))
c(c(b(B(x)))) → a(a(a(a(c(c(C(x)))))))
c(C(x)) → b(b(B(x)))
a(a(b(b(B(x))))) → b(b(b(b(b(b(b(b(B(x)))))))))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule C1(c(b(a(b(b(B(x))))))) → C1(c(c(b(b(b(b(B(x)))))))) at position [0] we obtained the following new rules:
C1(c(b(a(b(b(B(y0))))))) → C1(b(b(b(a(a(a(b(b(b(b(B(y0))))))))))))
C1(c(b(a(b(b(B(y0))))))) → C1(a(a(a(a(c(c(c(b(b(B(y0)))))))))))
↳ QTRS
↳ QTRS Reverse
↳ QTRS Reverse
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ SemLabProof
↳ SemLabProof2
↳ QDP
↳ QDPToSRSProof
↳ QTRS
↳ QTRS Reverse
↳ QTRS
↳ QTRS Reverse
↳ QTRS Reverse
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
Q DP problem:
The TRS P consists of the following rules:
C1(c(b(b(x)))) → C1(c(c(x)))
C1(c(b(a(b(b(B(x))))))) → C1(c(c(b(b(B(x))))))
C1(c(b(B(x)))) → C1(c(C(x)))
C1(c(b(a(b(b(B(x))))))) → C1(c(c(b(b(b(B(x)))))))
C1(c(a(b(b(B(x)))))) → C1(c(b(b(b(B(x))))))
C1(c(a(b(b(B(x)))))) → C1(c(c(b(b(b(b(B(x))))))))
C1(c(b(b(x)))) → C1(x)
C1(c(a(b(b(B(x)))))) → C1(c(b(b(B(x)))))
C1(c(b(a(b(b(B(y0))))))) → C1(b(b(b(a(a(a(b(b(b(b(B(y0))))))))))))
C1(c(b(a(b(b(B(x))))))) → C1(c(b(b(b(B(x))))))
C1(c(b(a(b(b(B(x))))))) → C1(c(b(b(b(b(B(x)))))))
C1(c(b(b(x)))) → C1(c(x))
C1(c(a(b(b(B(x)))))) → C1(c(b(b(b(b(B(x)))))))
C1(c(a(b(b(B(x)))))) → C1(c(c(b(b(b(b(b(B(x)))))))))
C1(c(b(a(b(b(B(y0))))))) → C1(a(a(a(a(c(c(c(b(b(B(y0)))))))))))
C1(c(a(b(b(B(x)))))) → C1(c(c(b(B(x)))))
C1(c(b(a(b(b(B(x))))))) → C1(c(b(b(B(x)))))
C1(c(a(b(b(B(x)))))) → C1(c(b(B(x))))
C1(c(b(a(b(b(B(x))))))) → C1(c(b(b(b(b(b(B(x))))))))
C1(c(b(a(b(b(B(x))))))) → C1(c(b(b(b(b(b(b(B(x)))))))))
C1(c(a(b(b(B(x)))))) → C1(c(b(b(b(b(b(B(x))))))))
C1(c(a(b(b(B(x)))))) → C1(c(c(b(b(B(x))))))
The TRS R consists of the following rules:
c(c(x)) → b(b(b(a(a(a(x))))))
a(b(b(b(x)))) → b(b(b(b(b(b(b(b(x))))))))
c(c(b(b(x)))) → a(a(a(a(c(c(c(x)))))))
c(c(b(a(b(b(B(x))))))) → a(a(a(a(c(c(c(b(b(b(b(b(b(B(x))))))))))))))
a(a(b(b(B(x))))) → b(b(b(b(b(b(b(b(b(B(x))))))))))
a(b(a(b(b(B(x)))))) → b(b(b(b(b(b(b(b(b(B(x))))))))))
a(a(b(b(B(x))))) → b(b(b(b(b(b(b(b(b(b(b(B(x))))))))))))
a(b(a(b(b(B(x)))))) → b(b(b(b(b(b(b(b(b(b(b(B(x))))))))))))
a(b(b(a(b(b(B(x))))))) → b(b(b(b(b(b(b(b(b(b(b(B(x))))))))))))
a(b(b(a(b(b(B(x))))))) → b(b(b(b(b(b(b(b(b(b(B(x)))))))))))
a(a(b(b(B(x))))) → b(b(b(b(b(b(b(b(b(b(b(b(B(x)))))))))))))
c(c(a(b(b(B(x)))))) → a(a(a(a(c(c(c(b(b(b(B(x)))))))))))
a(b(b(B(x)))) → b(b(B(x)))
a(b(b(B(x)))) → B(x)
c(c(b(a(b(b(B(x))))))) → a(a(a(a(c(c(c(b(b(b(b(b(B(x)))))))))))))
a(b(b(a(b(b(B(x))))))) → b(b(b(b(b(b(b(b(b(b(b(b(b(B(x))))))))))))))
c(c(b(a(b(b(B(x))))))) → a(a(a(a(c(c(c(b(b(b(b(B(x))))))))))))
a(b(a(b(b(B(x)))))) → b(b(b(b(b(b(b(b(b(b(b(b(B(x)))))))))))))
a(b(a(b(b(B(x)))))) → b(b(b(b(b(b(b(b(b(b(B(x)))))))))))
a(b(b(a(b(b(B(x))))))) → b(b(b(b(b(b(b(b(b(b(b(b(B(x)))))))))))))
c(c(a(b(b(B(x)))))) → a(a(a(a(c(c(c(b(b(b(b(b(B(x)))))))))))))
a(b(b(a(b(b(B(x))))))) → b(b(b(b(b(b(b(b(b(b(b(b(b(b(B(x)))))))))))))))
c(c(a(b(b(B(x)))))) → a(a(a(a(c(c(c(b(b(b(b(B(x))))))))))))
a(b(a(b(b(B(x)))))) → b(b(b(b(b(b(b(b(b(b(b(b(b(B(x))))))))))))))
c(c(b(a(b(b(B(x))))))) → a(a(a(a(c(c(c(b(b(B(x))))))))))
c(c(a(b(b(B(x)))))) → a(a(a(a(c(c(c(b(b(B(x))))))))))
c(c(b(B(x)))) → a(a(a(a(c(C(x))))))
a(a(b(b(B(x))))) → b(b(b(b(b(b(b(b(b(b(B(x)))))))))))
c(C(x)) → b(B(x))
c(c(a(b(b(B(x)))))) → a(a(a(a(c(c(c(b(B(x)))))))))
c(c(b(a(b(b(B(x))))))) → a(a(a(a(c(c(c(b(b(b(B(x)))))))))))
a(b(b(B(x)))) → b(B(x))
c(c(b(B(x)))) → a(a(a(a(c(c(C(x)))))))
c(C(x)) → b(b(B(x)))
a(a(b(b(B(x))))) → b(b(b(b(b(b(b(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 2 less nodes.
↳ QTRS
↳ QTRS Reverse
↳ QTRS Reverse
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ SemLabProof
↳ SemLabProof2
↳ QDP
↳ QDPToSRSProof
↳ QTRS
↳ QTRS Reverse
↳ QTRS
↳ QTRS Reverse
↳ QTRS Reverse
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
Q DP problem:
The TRS P consists of the following rules:
C1(c(b(b(x)))) → C1(c(c(x)))
C1(c(b(a(b(b(B(x))))))) → C1(c(c(b(b(B(x))))))
C1(c(b(a(b(b(B(x))))))) → C1(c(c(b(b(b(B(x)))))))
C1(c(b(B(x)))) → C1(c(C(x)))
C1(c(a(b(b(B(x)))))) → C1(c(b(b(b(B(x))))))
C1(c(b(b(x)))) → C1(x)
C1(c(a(b(b(B(x)))))) → C1(c(c(b(b(b(b(B(x))))))))
C1(c(a(b(b(B(x)))))) → C1(c(b(b(B(x)))))
C1(c(b(a(b(b(B(x))))))) → C1(c(b(b(b(B(x))))))
C1(c(b(a(b(b(B(x))))))) → C1(c(b(b(b(b(B(x)))))))
C1(c(b(b(x)))) → C1(c(x))
C1(c(a(b(b(B(x)))))) → C1(c(c(b(b(b(b(b(B(x)))))))))
C1(c(a(b(b(B(x)))))) → C1(c(b(b(b(b(B(x)))))))
C1(c(a(b(b(B(x)))))) → C1(c(c(b(B(x)))))
C1(c(b(a(b(b(B(x))))))) → C1(c(b(b(B(x)))))
C1(c(a(b(b(B(x)))))) → C1(c(b(B(x))))
C1(c(b(a(b(b(B(x))))))) → C1(c(b(b(b(b(b(b(B(x)))))))))
C1(c(b(a(b(b(B(x))))))) → C1(c(b(b(b(b(b(B(x))))))))
C1(c(a(b(b(B(x)))))) → C1(c(b(b(b(b(b(B(x))))))))
C1(c(a(b(b(B(x)))))) → C1(c(c(b(b(B(x))))))
The TRS R consists of the following rules:
c(c(x)) → b(b(b(a(a(a(x))))))
a(b(b(b(x)))) → b(b(b(b(b(b(b(b(x))))))))
c(c(b(b(x)))) → a(a(a(a(c(c(c(x)))))))
c(c(b(a(b(b(B(x))))))) → a(a(a(a(c(c(c(b(b(b(b(b(b(B(x))))))))))))))
a(a(b(b(B(x))))) → b(b(b(b(b(b(b(b(b(B(x))))))))))
a(b(a(b(b(B(x)))))) → b(b(b(b(b(b(b(b(b(B(x))))))))))
a(a(b(b(B(x))))) → b(b(b(b(b(b(b(b(b(b(b(B(x))))))))))))
a(b(a(b(b(B(x)))))) → b(b(b(b(b(b(b(b(b(b(b(B(x))))))))))))
a(b(b(a(b(b(B(x))))))) → b(b(b(b(b(b(b(b(b(b(b(B(x))))))))))))
a(b(b(a(b(b(B(x))))))) → b(b(b(b(b(b(b(b(b(b(B(x)))))))))))
a(a(b(b(B(x))))) → b(b(b(b(b(b(b(b(b(b(b(b(B(x)))))))))))))
c(c(a(b(b(B(x)))))) → a(a(a(a(c(c(c(b(b(b(B(x)))))))))))
a(b(b(B(x)))) → b(b(B(x)))
a(b(b(B(x)))) → B(x)
c(c(b(a(b(b(B(x))))))) → a(a(a(a(c(c(c(b(b(b(b(b(B(x)))))))))))))
a(b(b(a(b(b(B(x))))))) → b(b(b(b(b(b(b(b(b(b(b(b(b(B(x))))))))))))))
c(c(b(a(b(b(B(x))))))) → a(a(a(a(c(c(c(b(b(b(b(B(x))))))))))))
a(b(a(b(b(B(x)))))) → b(b(b(b(b(b(b(b(b(b(b(b(B(x)))))))))))))
a(b(a(b(b(B(x)))))) → b(b(b(b(b(b(b(b(b(b(B(x)))))))))))
a(b(b(a(b(b(B(x))))))) → b(b(b(b(b(b(b(b(b(b(b(b(B(x)))))))))))))
c(c(a(b(b(B(x)))))) → a(a(a(a(c(c(c(b(b(b(b(b(B(x)))))))))))))
a(b(b(a(b(b(B(x))))))) → b(b(b(b(b(b(b(b(b(b(b(b(b(b(B(x)))))))))))))))
c(c(a(b(b(B(x)))))) → a(a(a(a(c(c(c(b(b(b(b(B(x))))))))))))
a(b(a(b(b(B(x)))))) → b(b(b(b(b(b(b(b(b(b(b(b(b(B(x))))))))))))))
c(c(b(a(b(b(B(x))))))) → a(a(a(a(c(c(c(b(b(B(x))))))))))
c(c(a(b(b(B(x)))))) → a(a(a(a(c(c(c(b(b(B(x))))))))))
c(c(b(B(x)))) → a(a(a(a(c(C(x))))))
a(a(b(b(B(x))))) → b(b(b(b(b(b(b(b(b(b(B(x)))))))))))
c(C(x)) → b(B(x))
c(c(a(b(b(B(x)))))) → a(a(a(a(c(c(c(b(B(x)))))))))
c(c(b(a(b(b(B(x))))))) → a(a(a(a(c(c(c(b(b(b(B(x)))))))))))
a(b(b(B(x)))) → b(B(x))
c(c(b(B(x)))) → a(a(a(a(c(c(C(x)))))))
c(C(x)) → b(b(B(x)))
a(a(b(b(B(x))))) → b(b(b(b(b(b(b(b(B(x)))))))))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule C1(c(a(b(b(B(x)))))) → C1(c(c(b(b(b(b(b(B(x))))))))) at position [0] we obtained the following new rules:
C1(c(a(b(b(B(y0)))))) → C1(a(a(a(a(c(c(c(b(b(b(B(y0))))))))))))
C1(c(a(b(b(B(y0)))))) → C1(b(b(b(a(a(a(b(b(b(b(b(B(y0)))))))))))))
↳ QTRS
↳ QTRS Reverse
↳ QTRS Reverse
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ SemLabProof
↳ SemLabProof2
↳ QDP
↳ QDPToSRSProof
↳ QTRS
↳ QTRS Reverse
↳ QTRS
↳ QTRS Reverse
↳ QTRS Reverse
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
Q DP problem:
The TRS P consists of the following rules:
C1(c(b(b(x)))) → C1(c(c(x)))
C1(c(b(a(b(b(B(x))))))) → C1(c(c(b(b(B(x))))))
C1(c(b(B(x)))) → C1(c(C(x)))
C1(c(b(a(b(b(B(x))))))) → C1(c(c(b(b(b(B(x)))))))
C1(c(a(b(b(B(x)))))) → C1(c(b(b(b(B(x))))))
C1(c(a(b(b(B(y0)))))) → C1(b(b(b(a(a(a(b(b(b(b(b(B(y0)))))))))))))
C1(c(a(b(b(B(x)))))) → C1(c(c(b(b(b(b(B(x))))))))
C1(c(b(b(x)))) → C1(x)
C1(c(a(b(b(B(x)))))) → C1(c(b(b(B(x)))))
C1(c(b(a(b(b(B(x))))))) → C1(c(b(b(b(B(x))))))
C1(c(b(a(b(b(B(x))))))) → C1(c(b(b(b(b(B(x)))))))
C1(c(b(b(x)))) → C1(c(x))
C1(c(a(b(b(B(x)))))) → C1(c(b(b(b(b(B(x)))))))
C1(c(a(b(b(B(x)))))) → C1(c(c(b(B(x)))))
C1(c(b(a(b(b(B(x))))))) → C1(c(b(b(B(x)))))
C1(c(a(b(b(B(x)))))) → C1(c(b(B(x))))
C1(c(b(a(b(b(B(x))))))) → C1(c(b(b(b(b(b(B(x))))))))
C1(c(b(a(b(b(B(x))))))) → C1(c(b(b(b(b(b(b(B(x)))))))))
C1(c(a(b(b(B(x)))))) → C1(c(b(b(b(b(b(B(x))))))))
C1(c(a(b(b(B(x)))))) → C1(c(c(b(b(B(x))))))
C1(c(a(b(b(B(y0)))))) → C1(a(a(a(a(c(c(c(b(b(b(B(y0))))))))))))
The TRS R consists of the following rules:
c(c(x)) → b(b(b(a(a(a(x))))))
a(b(b(b(x)))) → b(b(b(b(b(b(b(b(x))))))))
c(c(b(b(x)))) → a(a(a(a(c(c(c(x)))))))
c(c(b(a(b(b(B(x))))))) → a(a(a(a(c(c(c(b(b(b(b(b(b(B(x))))))))))))))
a(a(b(b(B(x))))) → b(b(b(b(b(b(b(b(b(B(x))))))))))
a(b(a(b(b(B(x)))))) → b(b(b(b(b(b(b(b(b(B(x))))))))))
a(a(b(b(B(x))))) → b(b(b(b(b(b(b(b(b(b(b(B(x))))))))))))
a(b(a(b(b(B(x)))))) → b(b(b(b(b(b(b(b(b(b(b(B(x))))))))))))
a(b(b(a(b(b(B(x))))))) → b(b(b(b(b(b(b(b(b(b(b(B(x))))))))))))
a(b(b(a(b(b(B(x))))))) → b(b(b(b(b(b(b(b(b(b(B(x)))))))))))
a(a(b(b(B(x))))) → b(b(b(b(b(b(b(b(b(b(b(b(B(x)))))))))))))
c(c(a(b(b(B(x)))))) → a(a(a(a(c(c(c(b(b(b(B(x)))))))))))
a(b(b(B(x)))) → b(b(B(x)))
a(b(b(B(x)))) → B(x)
c(c(b(a(b(b(B(x))))))) → a(a(a(a(c(c(c(b(b(b(b(b(B(x)))))))))))))
a(b(b(a(b(b(B(x))))))) → b(b(b(b(b(b(b(b(b(b(b(b(b(B(x))))))))))))))
c(c(b(a(b(b(B(x))))))) → a(a(a(a(c(c(c(b(b(b(b(B(x))))))))))))
a(b(a(b(b(B(x)))))) → b(b(b(b(b(b(b(b(b(b(b(b(B(x)))))))))))))
a(b(a(b(b(B(x)))))) → b(b(b(b(b(b(b(b(b(b(B(x)))))))))))
a(b(b(a(b(b(B(x))))))) → b(b(b(b(b(b(b(b(b(b(b(b(B(x)))))))))))))
c(c(a(b(b(B(x)))))) → a(a(a(a(c(c(c(b(b(b(b(b(B(x)))))))))))))
a(b(b(a(b(b(B(x))))))) → b(b(b(b(b(b(b(b(b(b(b(b(b(b(B(x)))))))))))))))
c(c(a(b(b(B(x)))))) → a(a(a(a(c(c(c(b(b(b(b(B(x))))))))))))
a(b(a(b(b(B(x)))))) → b(b(b(b(b(b(b(b(b(b(b(b(b(B(x))))))))))))))
c(c(b(a(b(b(B(x))))))) → a(a(a(a(c(c(c(b(b(B(x))))))))))
c(c(a(b(b(B(x)))))) → a(a(a(a(c(c(c(b(b(B(x))))))))))
c(c(b(B(x)))) → a(a(a(a(c(C(x))))))
a(a(b(b(B(x))))) → b(b(b(b(b(b(b(b(b(b(B(x)))))))))))
c(C(x)) → b(B(x))
c(c(a(b(b(B(x)))))) → a(a(a(a(c(c(c(b(B(x)))))))))
c(c(b(a(b(b(B(x))))))) → a(a(a(a(c(c(c(b(b(b(B(x)))))))))))
a(b(b(B(x)))) → b(B(x))
c(c(b(B(x)))) → a(a(a(a(c(c(C(x)))))))
c(C(x)) → b(b(B(x)))
a(a(b(b(B(x))))) → b(b(b(b(b(b(b(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 2 less nodes.
↳ QTRS
↳ QTRS Reverse
↳ QTRS Reverse
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ SemLabProof
↳ SemLabProof2
↳ QDP
↳ QDPToSRSProof
↳ QTRS
↳ QTRS Reverse
↳ QTRS
↳ QTRS Reverse
↳ QTRS Reverse
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
Q DP problem:
The TRS P consists of the following rules:
C1(c(b(b(x)))) → C1(c(c(x)))
C1(c(b(a(b(b(B(x))))))) → C1(c(c(b(b(B(x))))))
C1(c(b(a(b(b(B(x))))))) → C1(c(c(b(b(b(B(x)))))))
C1(c(b(B(x)))) → C1(c(C(x)))
C1(c(a(b(b(B(x)))))) → C1(c(b(b(b(B(x))))))
C1(c(b(b(x)))) → C1(x)
C1(c(a(b(b(B(x)))))) → C1(c(c(b(b(b(b(B(x))))))))
C1(c(a(b(b(B(x)))))) → C1(c(b(b(B(x)))))
C1(c(b(a(b(b(B(x))))))) → C1(c(b(b(b(B(x))))))
C1(c(b(a(b(b(B(x))))))) → C1(c(b(b(b(b(B(x)))))))
C1(c(b(b(x)))) → C1(c(x))
C1(c(a(b(b(B(x)))))) → C1(c(b(b(b(b(B(x)))))))
C1(c(a(b(b(B(x)))))) → C1(c(c(b(B(x)))))
C1(c(b(a(b(b(B(x))))))) → C1(c(b(b(B(x)))))
C1(c(a(b(b(B(x)))))) → C1(c(b(B(x))))
C1(c(b(a(b(b(B(x))))))) → C1(c(b(b(b(b(b(b(B(x)))))))))
C1(c(b(a(b(b(B(x))))))) → C1(c(b(b(b(b(b(B(x))))))))
C1(c(a(b(b(B(x)))))) → C1(c(b(b(b(b(b(B(x))))))))
C1(c(a(b(b(B(x)))))) → C1(c(c(b(b(B(x))))))
The TRS R consists of the following rules:
c(c(x)) → b(b(b(a(a(a(x))))))
a(b(b(b(x)))) → b(b(b(b(b(b(b(b(x))))))))
c(c(b(b(x)))) → a(a(a(a(c(c(c(x)))))))
c(c(b(a(b(b(B(x))))))) → a(a(a(a(c(c(c(b(b(b(b(b(b(B(x))))))))))))))
a(a(b(b(B(x))))) → b(b(b(b(b(b(b(b(b(B(x))))))))))
a(b(a(b(b(B(x)))))) → b(b(b(b(b(b(b(b(b(B(x))))))))))
a(a(b(b(B(x))))) → b(b(b(b(b(b(b(b(b(b(b(B(x))))))))))))
a(b(a(b(b(B(x)))))) → b(b(b(b(b(b(b(b(b(b(b(B(x))))))))))))
a(b(b(a(b(b(B(x))))))) → b(b(b(b(b(b(b(b(b(b(b(B(x))))))))))))
a(b(b(a(b(b(B(x))))))) → b(b(b(b(b(b(b(b(b(b(B(x)))))))))))
a(a(b(b(B(x))))) → b(b(b(b(b(b(b(b(b(b(b(b(B(x)))))))))))))
c(c(a(b(b(B(x)))))) → a(a(a(a(c(c(c(b(b(b(B(x)))))))))))
a(b(b(B(x)))) → b(b(B(x)))
a(b(b(B(x)))) → B(x)
c(c(b(a(b(b(B(x))))))) → a(a(a(a(c(c(c(b(b(b(b(b(B(x)))))))))))))
a(b(b(a(b(b(B(x))))))) → b(b(b(b(b(b(b(b(b(b(b(b(b(B(x))))))))))))))
c(c(b(a(b(b(B(x))))))) → a(a(a(a(c(c(c(b(b(b(b(B(x))))))))))))
a(b(a(b(b(B(x)))))) → b(b(b(b(b(b(b(b(b(b(b(b(B(x)))))))))))))
a(b(a(b(b(B(x)))))) → b(b(b(b(b(b(b(b(b(b(B(x)))))))))))
a(b(b(a(b(b(B(x))))))) → b(b(b(b(b(b(b(b(b(b(b(b(B(x)))))))))))))
c(c(a(b(b(B(x)))))) → a(a(a(a(c(c(c(b(b(b(b(b(B(x)))))))))))))
a(b(b(a(b(b(B(x))))))) → b(b(b(b(b(b(b(b(b(b(b(b(b(b(B(x)))))))))))))))
c(c(a(b(b(B(x)))))) → a(a(a(a(c(c(c(b(b(b(b(B(x))))))))))))
a(b(a(b(b(B(x)))))) → b(b(b(b(b(b(b(b(b(b(b(b(b(B(x))))))))))))))
c(c(b(a(b(b(B(x))))))) → a(a(a(a(c(c(c(b(b(B(x))))))))))
c(c(a(b(b(B(x)))))) → a(a(a(a(c(c(c(b(b(B(x))))))))))
c(c(b(B(x)))) → a(a(a(a(c(C(x))))))
a(a(b(b(B(x))))) → b(b(b(b(b(b(b(b(b(b(B(x)))))))))))
c(C(x)) → b(B(x))
c(c(a(b(b(B(x)))))) → a(a(a(a(c(c(c(b(B(x)))))))))
c(c(b(a(b(b(B(x))))))) → a(a(a(a(c(c(c(b(b(b(B(x)))))))))))
a(b(b(B(x)))) → b(B(x))
c(c(b(B(x)))) → a(a(a(a(c(c(C(x)))))))
c(C(x)) → b(b(B(x)))
a(a(b(b(B(x))))) → b(b(b(b(b(b(b(b(B(x)))))))))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule C1(c(a(b(b(B(x)))))) → C1(c(c(b(b(b(b(B(x)))))))) at position [0] we obtained the following new rules:
C1(c(a(b(b(B(y0)))))) → C1(a(a(a(a(c(c(c(b(b(B(y0)))))))))))
C1(c(a(b(b(B(y0)))))) → C1(b(b(b(a(a(a(b(b(b(b(B(y0))))))))))))
↳ QTRS
↳ QTRS Reverse
↳ QTRS Reverse
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ SemLabProof
↳ SemLabProof2
↳ QDP
↳ QDPToSRSProof
↳ QTRS
↳ QTRS Reverse
↳ QTRS
↳ QTRS Reverse
↳ QTRS Reverse
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
Q DP problem:
The TRS P consists of the following rules:
C1(c(b(b(x)))) → C1(c(c(x)))
C1(c(b(a(b(b(B(x))))))) → C1(c(c(b(b(B(x))))))
C1(c(b(B(x)))) → C1(c(C(x)))
C1(c(b(a(b(b(B(x))))))) → C1(c(c(b(b(b(B(x)))))))
C1(c(a(b(b(B(x)))))) → C1(c(b(b(b(B(x))))))
C1(c(b(b(x)))) → C1(x)
C1(c(a(b(b(B(x)))))) → C1(c(b(b(B(x)))))
C1(c(b(a(b(b(B(x))))))) → C1(c(b(b(b(B(x))))))
C1(c(b(a(b(b(B(x))))))) → C1(c(b(b(b(b(B(x)))))))
C1(c(b(b(x)))) → C1(c(x))
C1(c(a(b(b(B(x)))))) → C1(c(b(b(b(b(B(x)))))))
C1(c(a(b(b(B(x)))))) → C1(c(c(b(B(x)))))
C1(c(a(b(b(B(y0)))))) → C1(a(a(a(a(c(c(c(b(b(B(y0)))))))))))
C1(c(b(a(b(b(B(x))))))) → C1(c(b(b(B(x)))))
C1(c(a(b(b(B(x)))))) → C1(c(b(B(x))))
C1(c(b(a(b(b(B(x))))))) → C1(c(b(b(b(b(b(B(x))))))))
C1(c(b(a(b(b(B(x))))))) → C1(c(b(b(b(b(b(b(B(x)))))))))
C1(c(a(b(b(B(x)))))) → C1(c(b(b(b(b(b(B(x))))))))
C1(c(a(b(b(B(y0)))))) → C1(b(b(b(a(a(a(b(b(b(b(B(y0))))))))))))
C1(c(a(b(b(B(x)))))) → C1(c(c(b(b(B(x))))))
The TRS R consists of the following rules:
c(c(x)) → b(b(b(a(a(a(x))))))
a(b(b(b(x)))) → b(b(b(b(b(b(b(b(x))))))))
c(c(b(b(x)))) → a(a(a(a(c(c(c(x)))))))
c(c(b(a(b(b(B(x))))))) → a(a(a(a(c(c(c(b(b(b(b(b(b(B(x))))))))))))))
a(a(b(b(B(x))))) → b(b(b(b(b(b(b(b(b(B(x))))))))))
a(b(a(b(b(B(x)))))) → b(b(b(b(b(b(b(b(b(B(x))))))))))
a(a(b(b(B(x))))) → b(b(b(b(b(b(b(b(b(b(b(B(x))))))))))))
a(b(a(b(b(B(x)))))) → b(b(b(b(b(b(b(b(b(b(b(B(x))))))))))))
a(b(b(a(b(b(B(x))))))) → b(b(b(b(b(b(b(b(b(b(b(B(x))))))))))))
a(b(b(a(b(b(B(x))))))) → b(b(b(b(b(b(b(b(b(b(B(x)))))))))))
a(a(b(b(B(x))))) → b(b(b(b(b(b(b(b(b(b(b(b(B(x)))))))))))))
c(c(a(b(b(B(x)))))) → a(a(a(a(c(c(c(b(b(b(B(x)))))))))))
a(b(b(B(x)))) → b(b(B(x)))
a(b(b(B(x)))) → B(x)
c(c(b(a(b(b(B(x))))))) → a(a(a(a(c(c(c(b(b(b(b(b(B(x)))))))))))))
a(b(b(a(b(b(B(x))))))) → b(b(b(b(b(b(b(b(b(b(b(b(b(B(x))))))))))))))
c(c(b(a(b(b(B(x))))))) → a(a(a(a(c(c(c(b(b(b(b(B(x))))))))))))
a(b(a(b(b(B(x)))))) → b(b(b(b(b(b(b(b(b(b(b(b(B(x)))))))))))))
a(b(a(b(b(B(x)))))) → b(b(b(b(b(b(b(b(b(b(B(x)))))))))))
a(b(b(a(b(b(B(x))))))) → b(b(b(b(b(b(b(b(b(b(b(b(B(x)))))))))))))
c(c(a(b(b(B(x)))))) → a(a(a(a(c(c(c(b(b(b(b(b(B(x)))))))))))))
a(b(b(a(b(b(B(x))))))) → b(b(b(b(b(b(b(b(b(b(b(b(b(b(B(x)))))))))))))))
c(c(a(b(b(B(x)))))) → a(a(a(a(c(c(c(b(b(b(b(B(x))))))))))))
a(b(a(b(b(B(x)))))) → b(b(b(b(b(b(b(b(b(b(b(b(b(B(x))))))))))))))
c(c(b(a(b(b(B(x))))))) → a(a(a(a(c(c(c(b(b(B(x))))))))))
c(c(a(b(b(B(x)))))) → a(a(a(a(c(c(c(b(b(B(x))))))))))
c(c(b(B(x)))) → a(a(a(a(c(C(x))))))
a(a(b(b(B(x))))) → b(b(b(b(b(b(b(b(b(b(B(x)))))))))))
c(C(x)) → b(B(x))
c(c(a(b(b(B(x)))))) → a(a(a(a(c(c(c(b(B(x)))))))))
c(c(b(a(b(b(B(x))))))) → a(a(a(a(c(c(c(b(b(b(B(x)))))))))))
a(b(b(B(x)))) → b(B(x))
c(c(b(B(x)))) → a(a(a(a(c(c(C(x)))))))
c(C(x)) → b(b(B(x)))
a(a(b(b(B(x))))) → b(b(b(b(b(b(b(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 2 less nodes.
↳ QTRS
↳ QTRS Reverse
↳ QTRS Reverse
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ SemLabProof
↳ SemLabProof2
↳ QDP
↳ QDPToSRSProof
↳ QTRS
↳ QTRS Reverse
↳ QTRS
↳ QTRS Reverse
↳ QTRS Reverse
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
Q DP problem:
The TRS P consists of the following rules:
C1(c(b(b(x)))) → C1(c(c(x)))
C1(c(b(a(b(b(B(x))))))) → C1(c(c(b(b(B(x))))))
C1(c(b(a(b(b(B(x))))))) → C1(c(c(b(b(b(B(x)))))))
C1(c(b(B(x)))) → C1(c(C(x)))
C1(c(a(b(b(B(x)))))) → C1(c(b(b(b(B(x))))))
C1(c(b(b(x)))) → C1(x)
C1(c(a(b(b(B(x)))))) → C1(c(b(b(B(x)))))
C1(c(b(a(b(b(B(x))))))) → C1(c(b(b(b(B(x))))))
C1(c(b(a(b(b(B(x))))))) → C1(c(b(b(b(b(B(x)))))))
C1(c(b(b(x)))) → C1(c(x))
C1(c(a(b(b(B(x)))))) → C1(c(b(b(b(b(B(x)))))))
C1(c(a(b(b(B(x)))))) → C1(c(c(b(B(x)))))
C1(c(b(a(b(b(B(x))))))) → C1(c(b(b(B(x)))))
C1(c(a(b(b(B(x)))))) → C1(c(b(B(x))))
C1(c(b(a(b(b(B(x))))))) → C1(c(b(b(b(b(b(b(B(x)))))))))
C1(c(b(a(b(b(B(x))))))) → C1(c(b(b(b(b(b(B(x))))))))
C1(c(a(b(b(B(x)))))) → C1(c(b(b(b(b(b(B(x))))))))
C1(c(a(b(b(B(x)))))) → C1(c(c(b(b(B(x))))))
The TRS R consists of the following rules:
c(c(x)) → b(b(b(a(a(a(x))))))
a(b(b(b(x)))) → b(b(b(b(b(b(b(b(x))))))))
c(c(b(b(x)))) → a(a(a(a(c(c(c(x)))))))
c(c(b(a(b(b(B(x))))))) → a(a(a(a(c(c(c(b(b(b(b(b(b(B(x))))))))))))))
a(a(b(b(B(x))))) → b(b(b(b(b(b(b(b(b(B(x))))))))))
a(b(a(b(b(B(x)))))) → b(b(b(b(b(b(b(b(b(B(x))))))))))
a(a(b(b(B(x))))) → b(b(b(b(b(b(b(b(b(b(b(B(x))))))))))))
a(b(a(b(b(B(x)))))) → b(b(b(b(b(b(b(b(b(b(b(B(x))))))))))))
a(b(b(a(b(b(B(x))))))) → b(b(b(b(b(b(b(b(b(b(b(B(x))))))))))))
a(b(b(a(b(b(B(x))))))) → b(b(b(b(b(b(b(b(b(b(B(x)))))))))))
a(a(b(b(B(x))))) → b(b(b(b(b(b(b(b(b(b(b(b(B(x)))))))))))))
c(c(a(b(b(B(x)))))) → a(a(a(a(c(c(c(b(b(b(B(x)))))))))))
a(b(b(B(x)))) → b(b(B(x)))
a(b(b(B(x)))) → B(x)
c(c(b(a(b(b(B(x))))))) → a(a(a(a(c(c(c(b(b(b(b(b(B(x)))))))))))))
a(b(b(a(b(b(B(x))))))) → b(b(b(b(b(b(b(b(b(b(b(b(b(B(x))))))))))))))
c(c(b(a(b(b(B(x))))))) → a(a(a(a(c(c(c(b(b(b(b(B(x))))))))))))
a(b(a(b(b(B(x)))))) → b(b(b(b(b(b(b(b(b(b(b(b(B(x)))))))))))))
a(b(a(b(b(B(x)))))) → b(b(b(b(b(b(b(b(b(b(B(x)))))))))))
a(b(b(a(b(b(B(x))))))) → b(b(b(b(b(b(b(b(b(b(b(b(B(x)))))))))))))
c(c(a(b(b(B(x)))))) → a(a(a(a(c(c(c(b(b(b(b(b(B(x)))))))))))))
a(b(b(a(b(b(B(x))))))) → b(b(b(b(b(b(b(b(b(b(b(b(b(b(B(x)))))))))))))))
c(c(a(b(b(B(x)))))) → a(a(a(a(c(c(c(b(b(b(b(B(x))))))))))))
a(b(a(b(b(B(x)))))) → b(b(b(b(b(b(b(b(b(b(b(b(b(B(x))))))))))))))
c(c(b(a(b(b(B(x))))))) → a(a(a(a(c(c(c(b(b(B(x))))))))))
c(c(a(b(b(B(x)))))) → a(a(a(a(c(c(c(b(b(B(x))))))))))
c(c(b(B(x)))) → a(a(a(a(c(C(x))))))
a(a(b(b(B(x))))) → b(b(b(b(b(b(b(b(b(b(B(x)))))))))))
c(C(x)) → b(B(x))
c(c(a(b(b(B(x)))))) → a(a(a(a(c(c(c(b(B(x)))))))))
c(c(b(a(b(b(B(x))))))) → a(a(a(a(c(c(c(b(b(b(B(x)))))))))))
a(b(b(B(x)))) → b(B(x))
c(c(b(B(x)))) → a(a(a(a(c(c(C(x)))))))
c(C(x)) → b(b(B(x)))
a(a(b(b(B(x))))) → b(b(b(b(b(b(b(b(B(x)))))))))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule C1(c(b(a(b(b(B(x))))))) → C1(c(c(b(b(B(x)))))) at position [0] we obtained the following new rules:
C1(c(b(a(b(b(B(y0))))))) → C1(b(b(b(a(a(a(b(b(B(y0))))))))))
C1(c(b(a(b(b(B(y0))))))) → C1(a(a(a(a(c(c(c(B(y0)))))))))
↳ QTRS
↳ QTRS Reverse
↳ QTRS Reverse
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ SemLabProof
↳ SemLabProof2
↳ QDP
↳ QDPToSRSProof
↳ QTRS
↳ QTRS Reverse
↳ QTRS
↳ QTRS Reverse
↳ QTRS Reverse
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
Q DP problem:
The TRS P consists of the following rules:
C1(c(b(b(x)))) → C1(c(c(x)))
C1(c(b(B(x)))) → C1(c(C(x)))
C1(c(b(a(b(b(B(x))))))) → C1(c(c(b(b(b(B(x)))))))
C1(c(a(b(b(B(x)))))) → C1(c(b(b(b(B(x))))))
C1(c(b(b(x)))) → C1(x)
C1(c(a(b(b(B(x)))))) → C1(c(b(b(B(x)))))
C1(c(b(a(b(b(B(y0))))))) → C1(b(b(b(a(a(a(b(b(B(y0))))))))))
C1(c(b(a(b(b(B(x))))))) → C1(c(b(b(b(B(x))))))
C1(c(b(a(b(b(B(x))))))) → C1(c(b(b(b(b(B(x)))))))
C1(c(b(b(x)))) → C1(c(x))
C1(c(a(b(b(B(x)))))) → C1(c(b(b(b(b(B(x)))))))
C1(c(a(b(b(B(x)))))) → C1(c(c(b(B(x)))))
C1(c(b(a(b(b(B(x))))))) → C1(c(b(b(B(x)))))
C1(c(a(b(b(B(x)))))) → C1(c(b(B(x))))
C1(c(b(a(b(b(B(y0))))))) → C1(a(a(a(a(c(c(c(B(y0)))))))))
C1(c(b(a(b(b(B(x))))))) → C1(c(b(b(b(b(b(B(x))))))))
C1(c(b(a(b(b(B(x))))))) → C1(c(b(b(b(b(b(b(B(x)))))))))
C1(c(a(b(b(B(x)))))) → C1(c(b(b(b(b(b(B(x))))))))
C1(c(a(b(b(B(x)))))) → C1(c(c(b(b(B(x))))))
The TRS R consists of the following rules:
c(c(x)) → b(b(b(a(a(a(x))))))
a(b(b(b(x)))) → b(b(b(b(b(b(b(b(x))))))))
c(c(b(b(x)))) → a(a(a(a(c(c(c(x)))))))
c(c(b(a(b(b(B(x))))))) → a(a(a(a(c(c(c(b(b(b(b(b(b(B(x))))))))))))))
a(a(b(b(B(x))))) → b(b(b(b(b(b(b(b(b(B(x))))))))))
a(b(a(b(b(B(x)))))) → b(b(b(b(b(b(b(b(b(B(x))))))))))
a(a(b(b(B(x))))) → b(b(b(b(b(b(b(b(b(b(b(B(x))))))))))))
a(b(a(b(b(B(x)))))) → b(b(b(b(b(b(b(b(b(b(b(B(x))))))))))))
a(b(b(a(b(b(B(x))))))) → b(b(b(b(b(b(b(b(b(b(b(B(x))))))))))))
a(b(b(a(b(b(B(x))))))) → b(b(b(b(b(b(b(b(b(b(B(x)))))))))))
a(a(b(b(B(x))))) → b(b(b(b(b(b(b(b(b(b(b(b(B(x)))))))))))))
c(c(a(b(b(B(x)))))) → a(a(a(a(c(c(c(b(b(b(B(x)))))))))))
a(b(b(B(x)))) → b(b(B(x)))
a(b(b(B(x)))) → B(x)
c(c(b(a(b(b(B(x))))))) → a(a(a(a(c(c(c(b(b(b(b(b(B(x)))))))))))))
a(b(b(a(b(b(B(x))))))) → b(b(b(b(b(b(b(b(b(b(b(b(b(B(x))))))))))))))
c(c(b(a(b(b(B(x))))))) → a(a(a(a(c(c(c(b(b(b(b(B(x))))))))))))
a(b(a(b(b(B(x)))))) → b(b(b(b(b(b(b(b(b(b(b(b(B(x)))))))))))))
a(b(a(b(b(B(x)))))) → b(b(b(b(b(b(b(b(b(b(B(x)))))))))))
a(b(b(a(b(b(B(x))))))) → b(b(b(b(b(b(b(b(b(b(b(b(B(x)))))))))))))
c(c(a(b(b(B(x)))))) → a(a(a(a(c(c(c(b(b(b(b(b(B(x)))))))))))))
a(b(b(a(b(b(B(x))))))) → b(b(b(b(b(b(b(b(b(b(b(b(b(b(B(x)))))))))))))))
c(c(a(b(b(B(x)))))) → a(a(a(a(c(c(c(b(b(b(b(B(x))))))))))))
a(b(a(b(b(B(x)))))) → b(b(b(b(b(b(b(b(b(b(b(b(b(B(x))))))))))))))
c(c(b(a(b(b(B(x))))))) → a(a(a(a(c(c(c(b(b(B(x))))))))))
c(c(a(b(b(B(x)))))) → a(a(a(a(c(c(c(b(b(B(x))))))))))
c(c(b(B(x)))) → a(a(a(a(c(C(x))))))
a(a(b(b(B(x))))) → b(b(b(b(b(b(b(b(b(b(B(x)))))))))))
c(C(x)) → b(B(x))
c(c(a(b(b(B(x)))))) → a(a(a(a(c(c(c(b(B(x)))))))))
c(c(b(a(b(b(B(x))))))) → a(a(a(a(c(c(c(b(b(b(B(x)))))))))))
a(b(b(B(x)))) → b(B(x))
c(c(b(B(x)))) → a(a(a(a(c(c(C(x)))))))
c(C(x)) → b(b(B(x)))
a(a(b(b(B(x))))) → b(b(b(b(b(b(b(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 2 less nodes.
↳ QTRS
↳ QTRS Reverse
↳ QTRS Reverse
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ SemLabProof
↳ SemLabProof2
↳ QDP
↳ QDPToSRSProof
↳ QTRS
↳ QTRS Reverse
↳ QTRS
↳ QTRS Reverse
↳ QTRS Reverse
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
Q DP problem:
The TRS P consists of the following rules:
C1(c(b(b(x)))) → C1(c(c(x)))
C1(c(b(a(b(b(B(x))))))) → C1(c(c(b(b(b(B(x)))))))
C1(c(b(B(x)))) → C1(c(C(x)))
C1(c(a(b(b(B(x)))))) → C1(c(b(b(b(B(x))))))
C1(c(b(b(x)))) → C1(x)
C1(c(a(b(b(B(x)))))) → C1(c(b(b(B(x)))))
C1(c(b(a(b(b(B(x))))))) → C1(c(b(b(b(B(x))))))
C1(c(b(a(b(b(B(x))))))) → C1(c(b(b(b(b(B(x)))))))
C1(c(b(b(x)))) → C1(c(x))
C1(c(a(b(b(B(x)))))) → C1(c(b(b(b(b(B(x)))))))
C1(c(a(b(b(B(x)))))) → C1(c(c(b(B(x)))))
C1(c(b(a(b(b(B(x))))))) → C1(c(b(b(B(x)))))
C1(c(a(b(b(B(x)))))) → C1(c(b(B(x))))
C1(c(b(a(b(b(B(x))))))) → C1(c(b(b(b(b(b(b(B(x)))))))))
C1(c(b(a(b(b(B(x))))))) → C1(c(b(b(b(b(b(B(x))))))))
C1(c(a(b(b(B(x)))))) → C1(c(b(b(b(b(b(B(x))))))))
C1(c(a(b(b(B(x)))))) → C1(c(c(b(b(B(x))))))
The TRS R consists of the following rules:
c(c(x)) → b(b(b(a(a(a(x))))))
a(b(b(b(x)))) → b(b(b(b(b(b(b(b(x))))))))
c(c(b(b(x)))) → a(a(a(a(c(c(c(x)))))))
c(c(b(a(b(b(B(x))))))) → a(a(a(a(c(c(c(b(b(b(b(b(b(B(x))))))))))))))
a(a(b(b(B(x))))) → b(b(b(b(b(b(b(b(b(B(x))))))))))
a(b(a(b(b(B(x)))))) → b(b(b(b(b(b(b(b(b(B(x))))))))))
a(a(b(b(B(x))))) → b(b(b(b(b(b(b(b(b(b(b(B(x))))))))))))
a(b(a(b(b(B(x)))))) → b(b(b(b(b(b(b(b(b(b(b(B(x))))))))))))
a(b(b(a(b(b(B(x))))))) → b(b(b(b(b(b(b(b(b(b(b(B(x))))))))))))
a(b(b(a(b(b(B(x))))))) → b(b(b(b(b(b(b(b(b(b(B(x)))))))))))
a(a(b(b(B(x))))) → b(b(b(b(b(b(b(b(b(b(b(b(B(x)))))))))))))
c(c(a(b(b(B(x)))))) → a(a(a(a(c(c(c(b(b(b(B(x)))))))))))
a(b(b(B(x)))) → b(b(B(x)))
a(b(b(B(x)))) → B(x)
c(c(b(a(b(b(B(x))))))) → a(a(a(a(c(c(c(b(b(b(b(b(B(x)))))))))))))
a(b(b(a(b(b(B(x))))))) → b(b(b(b(b(b(b(b(b(b(b(b(b(B(x))))))))))))))
c(c(b(a(b(b(B(x))))))) → a(a(a(a(c(c(c(b(b(b(b(B(x))))))))))))
a(b(a(b(b(B(x)))))) → b(b(b(b(b(b(b(b(b(b(b(b(B(x)))))))))))))
a(b(a(b(b(B(x)))))) → b(b(b(b(b(b(b(b(b(b(B(x)))))))))))
a(b(b(a(b(b(B(x))))))) → b(b(b(b(b(b(b(b(b(b(b(b(B(x)))))))))))))
c(c(a(b(b(B(x)))))) → a(a(a(a(c(c(c(b(b(b(b(b(B(x)))))))))))))
a(b(b(a(b(b(B(x))))))) → b(b(b(b(b(b(b(b(b(b(b(b(b(b(B(x)))))))))))))))
c(c(a(b(b(B(x)))))) → a(a(a(a(c(c(c(b(b(b(b(B(x))))))))))))
a(b(a(b(b(B(x)))))) → b(b(b(b(b(b(b(b(b(b(b(b(b(B(x))))))))))))))
c(c(b(a(b(b(B(x))))))) → a(a(a(a(c(c(c(b(b(B(x))))))))))
c(c(a(b(b(B(x)))))) → a(a(a(a(c(c(c(b(b(B(x))))))))))
c(c(b(B(x)))) → a(a(a(a(c(C(x))))))
a(a(b(b(B(x))))) → b(b(b(b(b(b(b(b(b(b(B(x)))))))))))
c(C(x)) → b(B(x))
c(c(a(b(b(B(x)))))) → a(a(a(a(c(c(c(b(B(x)))))))))
c(c(b(a(b(b(B(x))))))) → a(a(a(a(c(c(c(b(b(b(B(x)))))))))))
a(b(b(B(x)))) → b(B(x))
c(c(b(B(x)))) → a(a(a(a(c(c(C(x)))))))
c(C(x)) → b(b(B(x)))
a(a(b(b(B(x))))) → b(b(b(b(b(b(b(b(B(x)))))))))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule C1(c(a(b(b(B(x)))))) → C1(c(c(b(b(B(x)))))) at position [0] we obtained the following new rules:
C1(c(a(b(b(B(y0)))))) → C1(a(a(a(a(c(c(c(B(y0)))))))))
C1(c(a(b(b(B(y0)))))) → C1(b(b(b(a(a(a(b(b(B(y0))))))))))
↳ QTRS
↳ QTRS Reverse
↳ QTRS Reverse
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ SemLabProof
↳ SemLabProof2
↳ QDP
↳ QDPToSRSProof
↳ QTRS
↳ QTRS Reverse
↳ QTRS
↳ QTRS Reverse
↳ QTRS Reverse
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
Q DP problem:
The TRS P consists of the following rules:
C1(c(b(b(x)))) → C1(c(c(x)))
C1(c(a(b(b(B(y0)))))) → C1(a(a(a(a(c(c(c(B(y0)))))))))
C1(c(b(B(x)))) → C1(c(C(x)))
C1(c(b(a(b(b(B(x))))))) → C1(c(c(b(b(b(B(x)))))))
C1(c(a(b(b(B(y0)))))) → C1(b(b(b(a(a(a(b(b(B(y0))))))))))
C1(c(a(b(b(B(x)))))) → C1(c(b(b(b(B(x))))))
C1(c(b(b(x)))) → C1(x)
C1(c(a(b(b(B(x)))))) → C1(c(b(b(B(x)))))
C1(c(b(a(b(b(B(x))))))) → C1(c(b(b(b(B(x))))))
C1(c(b(a(b(b(B(x))))))) → C1(c(b(b(b(b(B(x)))))))
C1(c(b(b(x)))) → C1(c(x))
C1(c(a(b(b(B(x)))))) → C1(c(b(b(b(b(B(x)))))))
C1(c(a(b(b(B(x)))))) → C1(c(c(b(B(x)))))
C1(c(b(a(b(b(B(x))))))) → C1(c(b(b(B(x)))))
C1(c(a(b(b(B(x)))))) → C1(c(b(B(x))))
C1(c(b(a(b(b(B(x))))))) → C1(c(b(b(b(b(b(B(x))))))))
C1(c(b(a(b(b(B(x))))))) → C1(c(b(b(b(b(b(b(B(x)))))))))
C1(c(a(b(b(B(x)))))) → C1(c(b(b(b(b(b(B(x))))))))
The TRS R consists of the following rules:
c(c(x)) → b(b(b(a(a(a(x))))))
a(b(b(b(x)))) → b(b(b(b(b(b(b(b(x))))))))
c(c(b(b(x)))) → a(a(a(a(c(c(c(x)))))))
c(c(b(a(b(b(B(x))))))) → a(a(a(a(c(c(c(b(b(b(b(b(b(B(x))))))))))))))
a(a(b(b(B(x))))) → b(b(b(b(b(b(b(b(b(B(x))))))))))
a(b(a(b(b(B(x)))))) → b(b(b(b(b(b(b(b(b(B(x))))))))))
a(a(b(b(B(x))))) → b(b(b(b(b(b(b(b(b(b(b(B(x))))))))))))
a(b(a(b(b(B(x)))))) → b(b(b(b(b(b(b(b(b(b(b(B(x))))))))))))
a(b(b(a(b(b(B(x))))))) → b(b(b(b(b(b(b(b(b(b(b(B(x))))))))))))
a(b(b(a(b(b(B(x))))))) → b(b(b(b(b(b(b(b(b(b(B(x)))))))))))
a(a(b(b(B(x))))) → b(b(b(b(b(b(b(b(b(b(b(b(B(x)))))))))))))
c(c(a(b(b(B(x)))))) → a(a(a(a(c(c(c(b(b(b(B(x)))))))))))
a(b(b(B(x)))) → b(b(B(x)))
a(b(b(B(x)))) → B(x)
c(c(b(a(b(b(B(x))))))) → a(a(a(a(c(c(c(b(b(b(b(b(B(x)))))))))))))
a(b(b(a(b(b(B(x))))))) → b(b(b(b(b(b(b(b(b(b(b(b(b(B(x))))))))))))))
c(c(b(a(b(b(B(x))))))) → a(a(a(a(c(c(c(b(b(b(b(B(x))))))))))))
a(b(a(b(b(B(x)))))) → b(b(b(b(b(b(b(b(b(b(b(b(B(x)))))))))))))
a(b(a(b(b(B(x)))))) → b(b(b(b(b(b(b(b(b(b(B(x)))))))))))
a(b(b(a(b(b(B(x))))))) → b(b(b(b(b(b(b(b(b(b(b(b(B(x)))))))))))))
c(c(a(b(b(B(x)))))) → a(a(a(a(c(c(c(b(b(b(b(b(B(x)))))))))))))
a(b(b(a(b(b(B(x))))))) → b(b(b(b(b(b(b(b(b(b(b(b(b(b(B(x)))))))))))))))
c(c(a(b(b(B(x)))))) → a(a(a(a(c(c(c(b(b(b(b(B(x))))))))))))
a(b(a(b(b(B(x)))))) → b(b(b(b(b(b(b(b(b(b(b(b(b(B(x))))))))))))))
c(c(b(a(b(b(B(x))))))) → a(a(a(a(c(c(c(b(b(B(x))))))))))
c(c(a(b(b(B(x)))))) → a(a(a(a(c(c(c(b(b(B(x))))))))))
c(c(b(B(x)))) → a(a(a(a(c(C(x))))))
a(a(b(b(B(x))))) → b(b(b(b(b(b(b(b(b(b(B(x)))))))))))
c(C(x)) → b(B(x))
c(c(a(b(b(B(x)))))) → a(a(a(a(c(c(c(b(B(x)))))))))
c(c(b(a(b(b(B(x))))))) → a(a(a(a(c(c(c(b(b(b(B(x)))))))))))
a(b(b(B(x)))) → b(B(x))
c(c(b(B(x)))) → a(a(a(a(c(c(C(x)))))))
c(C(x)) → b(b(B(x)))
a(a(b(b(B(x))))) → b(b(b(b(b(b(b(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 2 less nodes.
↳ QTRS
↳ QTRS Reverse
↳ QTRS Reverse
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ SemLabProof
↳ SemLabProof2
↳ QDP
↳ QDPToSRSProof
↳ QTRS
↳ QTRS Reverse
↳ QTRS
↳ QTRS Reverse
↳ QTRS Reverse
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
Q DP problem:
The TRS P consists of the following rules:
C1(c(b(b(x)))) → C1(c(c(x)))
C1(c(b(a(b(b(B(x))))))) → C1(c(c(b(b(b(B(x)))))))
C1(c(b(B(x)))) → C1(c(C(x)))
C1(c(a(b(b(B(x)))))) → C1(c(b(b(b(B(x))))))
C1(c(b(b(x)))) → C1(x)
C1(c(a(b(b(B(x)))))) → C1(c(b(b(B(x)))))
C1(c(b(a(b(b(B(x))))))) → C1(c(b(b(b(B(x))))))
C1(c(b(a(b(b(B(x))))))) → C1(c(b(b(b(b(B(x)))))))
C1(c(b(b(x)))) → C1(c(x))
C1(c(a(b(b(B(x)))))) → C1(c(b(b(b(b(B(x)))))))
C1(c(a(b(b(B(x)))))) → C1(c(c(b(B(x)))))
C1(c(b(a(b(b(B(x))))))) → C1(c(b(b(B(x)))))
C1(c(a(b(b(B(x)))))) → C1(c(b(B(x))))
C1(c(b(a(b(b(B(x))))))) → C1(c(b(b(b(b(b(b(B(x)))))))))
C1(c(b(a(b(b(B(x))))))) → C1(c(b(b(b(b(b(B(x))))))))
C1(c(a(b(b(B(x)))))) → C1(c(b(b(b(b(b(B(x))))))))
The TRS R consists of the following rules:
c(c(x)) → b(b(b(a(a(a(x))))))
a(b(b(b(x)))) → b(b(b(b(b(b(b(b(x))))))))
c(c(b(b(x)))) → a(a(a(a(c(c(c(x)))))))
c(c(b(a(b(b(B(x))))))) → a(a(a(a(c(c(c(b(b(b(b(b(b(B(x))))))))))))))
a(a(b(b(B(x))))) → b(b(b(b(b(b(b(b(b(B(x))))))))))
a(b(a(b(b(B(x)))))) → b(b(b(b(b(b(b(b(b(B(x))))))))))
a(a(b(b(B(x))))) → b(b(b(b(b(b(b(b(b(b(b(B(x))))))))))))
a(b(a(b(b(B(x)))))) → b(b(b(b(b(b(b(b(b(b(b(B(x))))))))))))
a(b(b(a(b(b(B(x))))))) → b(b(b(b(b(b(b(b(b(b(b(B(x))))))))))))
a(b(b(a(b(b(B(x))))))) → b(b(b(b(b(b(b(b(b(b(B(x)))))))))))
a(a(b(b(B(x))))) → b(b(b(b(b(b(b(b(b(b(b(b(B(x)))))))))))))
c(c(a(b(b(B(x)))))) → a(a(a(a(c(c(c(b(b(b(B(x)))))))))))
a(b(b(B(x)))) → b(b(B(x)))
a(b(b(B(x)))) → B(x)
c(c(b(a(b(b(B(x))))))) → a(a(a(a(c(c(c(b(b(b(b(b(B(x)))))))))))))
a(b(b(a(b(b(B(x))))))) → b(b(b(b(b(b(b(b(b(b(b(b(b(B(x))))))))))))))
c(c(b(a(b(b(B(x))))))) → a(a(a(a(c(c(c(b(b(b(b(B(x))))))))))))
a(b(a(b(b(B(x)))))) → b(b(b(b(b(b(b(b(b(b(b(b(B(x)))))))))))))
a(b(a(b(b(B(x)))))) → b(b(b(b(b(b(b(b(b(b(B(x)))))))))))
a(b(b(a(b(b(B(x))))))) → b(b(b(b(b(b(b(b(b(b(b(b(B(x)))))))))))))
c(c(a(b(b(B(x)))))) → a(a(a(a(c(c(c(b(b(b(b(b(B(x)))))))))))))
a(b(b(a(b(b(B(x))))))) → b(b(b(b(b(b(b(b(b(b(b(b(b(b(B(x)))))))))))))))
c(c(a(b(b(B(x)))))) → a(a(a(a(c(c(c(b(b(b(b(B(x))))))))))))
a(b(a(b(b(B(x)))))) → b(b(b(b(b(b(b(b(b(b(b(b(b(B(x))))))))))))))
c(c(b(a(b(b(B(x))))))) → a(a(a(a(c(c(c(b(b(B(x))))))))))
c(c(a(b(b(B(x)))))) → a(a(a(a(c(c(c(b(b(B(x))))))))))
c(c(b(B(x)))) → a(a(a(a(c(C(x))))))
a(a(b(b(B(x))))) → b(b(b(b(b(b(b(b(b(b(B(x)))))))))))
c(C(x)) → b(B(x))
c(c(a(b(b(B(x)))))) → a(a(a(a(c(c(c(b(B(x)))))))))
c(c(b(a(b(b(B(x))))))) → a(a(a(a(c(c(c(b(b(b(B(x)))))))))))
a(b(b(B(x)))) → b(B(x))
c(c(b(B(x)))) → a(a(a(a(c(c(C(x)))))))
c(C(x)) → b(b(B(x)))
a(a(b(b(B(x))))) → b(b(b(b(b(b(b(b(B(x)))))))))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule C1(c(a(b(b(B(x)))))) → C1(c(c(b(B(x))))) at position [0] we obtained the following new rules:
C1(c(a(b(b(B(y0)))))) → C1(b(b(b(a(a(a(b(B(y0)))))))))
C1(c(a(b(b(B(x0)))))) → C1(a(a(a(a(c(C(x0)))))))
C1(c(a(b(b(B(x0)))))) → C1(a(a(a(a(c(c(C(x0))))))))
↳ QTRS
↳ QTRS Reverse
↳ QTRS Reverse
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ SemLabProof
↳ SemLabProof2
↳ QDP
↳ QDPToSRSProof
↳ QTRS
↳ QTRS Reverse
↳ QTRS
↳ QTRS Reverse
↳ QTRS Reverse
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
Q DP problem:
The TRS P consists of the following rules:
C1(c(b(b(x)))) → C1(c(c(x)))
C1(c(b(B(x)))) → C1(c(C(x)))
C1(c(b(a(b(b(B(x))))))) → C1(c(c(b(b(b(B(x)))))))
C1(c(a(b(b(B(x)))))) → C1(c(b(b(b(B(x))))))
C1(c(b(b(x)))) → C1(x)
C1(c(a(b(b(B(x)))))) → C1(c(b(b(B(x)))))
C1(c(b(a(b(b(B(x))))))) → C1(c(b(b(b(B(x))))))
C1(c(a(b(b(B(x0)))))) → C1(a(a(a(a(c(C(x0)))))))
C1(c(b(a(b(b(B(x))))))) → C1(c(b(b(b(b(B(x)))))))
C1(c(b(b(x)))) → C1(c(x))
C1(c(a(b(b(B(x)))))) → C1(c(b(b(b(b(B(x)))))))
C1(c(b(a(b(b(B(x))))))) → C1(c(b(b(B(x)))))
C1(c(a(b(b(B(x0)))))) → C1(a(a(a(a(c(c(C(x0))))))))
C1(c(a(b(b(B(x)))))) → C1(c(b(B(x))))
C1(c(b(a(b(b(B(x))))))) → C1(c(b(b(b(b(b(B(x))))))))
C1(c(b(a(b(b(B(x))))))) → C1(c(b(b(b(b(b(b(B(x)))))))))
C1(c(a(b(b(B(x)))))) → C1(c(b(b(b(b(b(B(x))))))))
C1(c(a(b(b(B(y0)))))) → C1(b(b(b(a(a(a(b(B(y0)))))))))
The TRS R consists of the following rules:
c(c(x)) → b(b(b(a(a(a(x))))))
a(b(b(b(x)))) → b(b(b(b(b(b(b(b(x))))))))
c(c(b(b(x)))) → a(a(a(a(c(c(c(x)))))))
c(c(b(a(b(b(B(x))))))) → a(a(a(a(c(c(c(b(b(b(b(b(b(B(x))))))))))))))
a(a(b(b(B(x))))) → b(b(b(b(b(b(b(b(b(B(x))))))))))
a(b(a(b(b(B(x)))))) → b(b(b(b(b(b(b(b(b(B(x))))))))))
a(a(b(b(B(x))))) → b(b(b(b(b(b(b(b(b(b(b(B(x))))))))))))
a(b(a(b(b(B(x)))))) → b(b(b(b(b(b(b(b(b(b(b(B(x))))))))))))
a(b(b(a(b(b(B(x))))))) → b(b(b(b(b(b(b(b(b(b(b(B(x))))))))))))
a(b(b(a(b(b(B(x))))))) → b(b(b(b(b(b(b(b(b(b(B(x)))))))))))
a(a(b(b(B(x))))) → b(b(b(b(b(b(b(b(b(b(b(b(B(x)))))))))))))
c(c(a(b(b(B(x)))))) → a(a(a(a(c(c(c(b(b(b(B(x)))))))))))
a(b(b(B(x)))) → b(b(B(x)))
a(b(b(B(x)))) → B(x)
c(c(b(a(b(b(B(x))))))) → a(a(a(a(c(c(c(b(b(b(b(b(B(x)))))))))))))
a(b(b(a(b(b(B(x))))))) → b(b(b(b(b(b(b(b(b(b(b(b(b(B(x))))))))))))))
c(c(b(a(b(b(B(x))))))) → a(a(a(a(c(c(c(b(b(b(b(B(x))))))))))))
a(b(a(b(b(B(x)))))) → b(b(b(b(b(b(b(b(b(b(b(b(B(x)))))))))))))
a(b(a(b(b(B(x)))))) → b(b(b(b(b(b(b(b(b(b(B(x)))))))))))
a(b(b(a(b(b(B(x))))))) → b(b(b(b(b(b(b(b(b(b(b(b(B(x)))))))))))))
c(c(a(b(b(B(x)))))) → a(a(a(a(c(c(c(b(b(b(b(b(B(x)))))))))))))
a(b(b(a(b(b(B(x))))))) → b(b(b(b(b(b(b(b(b(b(b(b(b(b(B(x)))))))))))))))
c(c(a(b(b(B(x)))))) → a(a(a(a(c(c(c(b(b(b(b(B(x))))))))))))
a(b(a(b(b(B(x)))))) → b(b(b(b(b(b(b(b(b(b(b(b(b(B(x))))))))))))))
c(c(b(a(b(b(B(x))))))) → a(a(a(a(c(c(c(b(b(B(x))))))))))
c(c(a(b(b(B(x)))))) → a(a(a(a(c(c(c(b(b(B(x))))))))))
c(c(b(B(x)))) → a(a(a(a(c(C(x))))))
a(a(b(b(B(x))))) → b(b(b(b(b(b(b(b(b(b(B(x)))))))))))
c(C(x)) → b(B(x))
c(c(a(b(b(B(x)))))) → a(a(a(a(c(c(c(b(B(x)))))))))
c(c(b(a(b(b(B(x))))))) → a(a(a(a(c(c(c(b(b(b(B(x)))))))))))
a(b(b(B(x)))) → b(B(x))
c(c(b(B(x)))) → a(a(a(a(c(c(C(x)))))))
c(C(x)) → b(b(B(x)))
a(a(b(b(B(x))))) → b(b(b(b(b(b(b(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 3 less nodes.
↳ QTRS
↳ QTRS Reverse
↳ QTRS Reverse
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ SemLabProof
↳ SemLabProof2
↳ QDP
↳ QDPToSRSProof
↳ QTRS
↳ QTRS Reverse
↳ QTRS
↳ QTRS Reverse
↳ QTRS Reverse
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
Q DP problem:
The TRS P consists of the following rules:
C1(c(b(b(x)))) → C1(c(c(x)))
C1(c(b(a(b(b(B(x))))))) → C1(c(c(b(b(b(B(x)))))))
C1(c(b(B(x)))) → C1(c(C(x)))
C1(c(a(b(b(B(x)))))) → C1(c(b(b(b(B(x))))))
C1(c(b(b(x)))) → C1(x)
C1(c(a(b(b(B(x)))))) → C1(c(b(b(B(x)))))
C1(c(b(a(b(b(B(x))))))) → C1(c(b(b(b(B(x))))))
C1(c(b(a(b(b(B(x))))))) → C1(c(b(b(b(b(B(x)))))))
C1(c(b(b(x)))) → C1(c(x))
C1(c(a(b(b(B(x)))))) → C1(c(b(b(b(b(B(x)))))))
C1(c(b(a(b(b(B(x))))))) → C1(c(b(b(B(x)))))
C1(c(a(b(b(B(x)))))) → C1(c(b(B(x))))
C1(c(b(a(b(b(B(x))))))) → C1(c(b(b(b(b(b(b(B(x)))))))))
C1(c(b(a(b(b(B(x))))))) → C1(c(b(b(b(b(b(B(x))))))))
C1(c(a(b(b(B(x)))))) → C1(c(b(b(b(b(b(B(x))))))))
The TRS R consists of the following rules:
c(c(x)) → b(b(b(a(a(a(x))))))
a(b(b(b(x)))) → b(b(b(b(b(b(b(b(x))))))))
c(c(b(b(x)))) → a(a(a(a(c(c(c(x)))))))
c(c(b(a(b(b(B(x))))))) → a(a(a(a(c(c(c(b(b(b(b(b(b(B(x))))))))))))))
a(a(b(b(B(x))))) → b(b(b(b(b(b(b(b(b(B(x))))))))))
a(b(a(b(b(B(x)))))) → b(b(b(b(b(b(b(b(b(B(x))))))))))
a(a(b(b(B(x))))) → b(b(b(b(b(b(b(b(b(b(b(B(x))))))))))))
a(b(a(b(b(B(x)))))) → b(b(b(b(b(b(b(b(b(b(b(B(x))))))))))))
a(b(b(a(b(b(B(x))))))) → b(b(b(b(b(b(b(b(b(b(b(B(x))))))))))))
a(b(b(a(b(b(B(x))))))) → b(b(b(b(b(b(b(b(b(b(B(x)))))))))))
a(a(b(b(B(x))))) → b(b(b(b(b(b(b(b(b(b(b(b(B(x)))))))))))))
c(c(a(b(b(B(x)))))) → a(a(a(a(c(c(c(b(b(b(B(x)))))))))))
a(b(b(B(x)))) → b(b(B(x)))
a(b(b(B(x)))) → B(x)
c(c(b(a(b(b(B(x))))))) → a(a(a(a(c(c(c(b(b(b(b(b(B(x)))))))))))))
a(b(b(a(b(b(B(x))))))) → b(b(b(b(b(b(b(b(b(b(b(b(b(B(x))))))))))))))
c(c(b(a(b(b(B(x))))))) → a(a(a(a(c(c(c(b(b(b(b(B(x))))))))))))
a(b(a(b(b(B(x)))))) → b(b(b(b(b(b(b(b(b(b(b(b(B(x)))))))))))))
a(b(a(b(b(B(x)))))) → b(b(b(b(b(b(b(b(b(b(B(x)))))))))))
a(b(b(a(b(b(B(x))))))) → b(b(b(b(b(b(b(b(b(b(b(b(B(x)))))))))))))
c(c(a(b(b(B(x)))))) → a(a(a(a(c(c(c(b(b(b(b(b(B(x)))))))))))))
a(b(b(a(b(b(B(x))))))) → b(b(b(b(b(b(b(b(b(b(b(b(b(b(B(x)))))))))))))))
c(c(a(b(b(B(x)))))) → a(a(a(a(c(c(c(b(b(b(b(B(x))))))))))))
a(b(a(b(b(B(x)))))) → b(b(b(b(b(b(b(b(b(b(b(b(b(B(x))))))))))))))
c(c(b(a(b(b(B(x))))))) → a(a(a(a(c(c(c(b(b(B(x))))))))))
c(c(a(b(b(B(x)))))) → a(a(a(a(c(c(c(b(b(B(x))))))))))
c(c(b(B(x)))) → a(a(a(a(c(C(x))))))
a(a(b(b(B(x))))) → b(b(b(b(b(b(b(b(b(b(B(x)))))))))))
c(C(x)) → b(B(x))
c(c(a(b(b(B(x)))))) → a(a(a(a(c(c(c(b(B(x)))))))))
c(c(b(a(b(b(B(x))))))) → a(a(a(a(c(c(c(b(b(b(B(x)))))))))))
a(b(b(B(x)))) → b(B(x))
c(c(b(B(x)))) → a(a(a(a(c(c(C(x)))))))
c(C(x)) → b(b(B(x)))
a(a(b(b(B(x))))) → b(b(b(b(b(b(b(b(B(x)))))))))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule C1(c(b(B(x)))) → C1(c(C(x))) at position [0] we obtained the following new rules:
C1(c(b(B(x0)))) → C1(b(b(B(x0))))
C1(c(b(B(x0)))) → C1(b(B(x0)))
↳ QTRS
↳ QTRS Reverse
↳ QTRS Reverse
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ SemLabProof
↳ SemLabProof2
↳ QDP
↳ QDPToSRSProof
↳ QTRS
↳ QTRS Reverse
↳ QTRS
↳ QTRS Reverse
↳ QTRS Reverse
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
Q DP problem:
The TRS P consists of the following rules:
C1(c(b(b(x)))) → C1(c(c(x)))
C1(c(b(a(b(b(B(x))))))) → C1(c(c(b(b(b(B(x)))))))
C1(c(a(b(b(B(x)))))) → C1(c(b(b(b(B(x))))))
C1(c(b(b(x)))) → C1(x)
C1(c(a(b(b(B(x)))))) → C1(c(b(b(B(x)))))
C1(c(b(B(x0)))) → C1(b(b(B(x0))))
C1(c(b(a(b(b(B(x))))))) → C1(c(b(b(b(B(x))))))
C1(c(b(B(x0)))) → C1(b(B(x0)))
C1(c(b(a(b(b(B(x))))))) → C1(c(b(b(b(b(B(x)))))))
C1(c(b(b(x)))) → C1(c(x))
C1(c(a(b(b(B(x)))))) → C1(c(b(b(b(b(B(x)))))))
C1(c(b(a(b(b(B(x))))))) → C1(c(b(b(B(x)))))
C1(c(a(b(b(B(x)))))) → C1(c(b(B(x))))
C1(c(b(a(b(b(B(x))))))) → C1(c(b(b(b(b(b(B(x))))))))
C1(c(b(a(b(b(B(x))))))) → C1(c(b(b(b(b(b(b(B(x)))))))))
C1(c(a(b(b(B(x)))))) → C1(c(b(b(b(b(b(B(x))))))))
The TRS R consists of the following rules:
c(c(x)) → b(b(b(a(a(a(x))))))
a(b(b(b(x)))) → b(b(b(b(b(b(b(b(x))))))))
c(c(b(b(x)))) → a(a(a(a(c(c(c(x)))))))
c(c(b(a(b(b(B(x))))))) → a(a(a(a(c(c(c(b(b(b(b(b(b(B(x))))))))))))))
a(a(b(b(B(x))))) → b(b(b(b(b(b(b(b(b(B(x))))))))))
a(b(a(b(b(B(x)))))) → b(b(b(b(b(b(b(b(b(B(x))))))))))
a(a(b(b(B(x))))) → b(b(b(b(b(b(b(b(b(b(b(B(x))))))))))))
a(b(a(b(b(B(x)))))) → b(b(b(b(b(b(b(b(b(b(b(B(x))))))))))))
a(b(b(a(b(b(B(x))))))) → b(b(b(b(b(b(b(b(b(b(b(B(x))))))))))))
a(b(b(a(b(b(B(x))))))) → b(b(b(b(b(b(b(b(b(b(B(x)))))))))))
a(a(b(b(B(x))))) → b(b(b(b(b(b(b(b(b(b(b(b(B(x)))))))))))))
c(c(a(b(b(B(x)))))) → a(a(a(a(c(c(c(b(b(b(B(x)))))))))))
a(b(b(B(x)))) → b(b(B(x)))
a(b(b(B(x)))) → B(x)
c(c(b(a(b(b(B(x))))))) → a(a(a(a(c(c(c(b(b(b(b(b(B(x)))))))))))))
a(b(b(a(b(b(B(x))))))) → b(b(b(b(b(b(b(b(b(b(b(b(b(B(x))))))))))))))
c(c(b(a(b(b(B(x))))))) → a(a(a(a(c(c(c(b(b(b(b(B(x))))))))))))
a(b(a(b(b(B(x)))))) → b(b(b(b(b(b(b(b(b(b(b(b(B(x)))))))))))))
a(b(a(b(b(B(x)))))) → b(b(b(b(b(b(b(b(b(b(B(x)))))))))))
a(b(b(a(b(b(B(x))))))) → b(b(b(b(b(b(b(b(b(b(b(b(B(x)))))))))))))
c(c(a(b(b(B(x)))))) → a(a(a(a(c(c(c(b(b(b(b(b(B(x)))))))))))))
a(b(b(a(b(b(B(x))))))) → b(b(b(b(b(b(b(b(b(b(b(b(b(b(B(x)))))))))))))))
c(c(a(b(b(B(x)))))) → a(a(a(a(c(c(c(b(b(b(b(B(x))))))))))))
a(b(a(b(b(B(x)))))) → b(b(b(b(b(b(b(b(b(b(b(b(b(B(x))))))))))))))
c(c(b(a(b(b(B(x))))))) → a(a(a(a(c(c(c(b(b(B(x))))))))))
c(c(a(b(b(B(x)))))) → a(a(a(a(c(c(c(b(b(B(x))))))))))
c(c(b(B(x)))) → a(a(a(a(c(C(x))))))
a(a(b(b(B(x))))) → b(b(b(b(b(b(b(b(b(b(B(x)))))))))))
c(C(x)) → b(B(x))
c(c(a(b(b(B(x)))))) → a(a(a(a(c(c(c(b(B(x)))))))))
c(c(b(a(b(b(B(x))))))) → a(a(a(a(c(c(c(b(b(b(B(x)))))))))))
a(b(b(B(x)))) → b(B(x))
c(c(b(B(x)))) → a(a(a(a(c(c(C(x)))))))
c(C(x)) → b(b(B(x)))
a(a(b(b(B(x))))) → b(b(b(b(b(b(b(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 3 less nodes.
↳ QTRS
↳ QTRS Reverse
↳ QTRS Reverse
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ SemLabProof
↳ SemLabProof2
↳ QDP
↳ QDPToSRSProof
↳ QTRS
↳ QTRS Reverse
↳ QTRS
↳ QTRS Reverse
↳ QTRS Reverse
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
Q DP problem:
The TRS P consists of the following rules:
C1(c(b(b(x)))) → C1(c(c(x)))
C1(c(b(a(b(b(B(x))))))) → C1(c(c(b(b(b(B(x)))))))
C1(c(a(b(b(B(x)))))) → C1(c(b(b(b(B(x))))))
C1(c(b(b(x)))) → C1(x)
C1(c(a(b(b(B(x)))))) → C1(c(b(b(B(x)))))
C1(c(b(a(b(b(B(x))))))) → C1(c(b(b(b(B(x))))))
C1(c(b(a(b(b(B(x))))))) → C1(c(b(b(b(b(B(x)))))))
C1(c(b(b(x)))) → C1(c(x))
C1(c(a(b(b(B(x)))))) → C1(c(b(b(b(b(B(x)))))))
C1(c(b(a(b(b(B(x))))))) → C1(c(b(b(B(x)))))
C1(c(b(a(b(b(B(x))))))) → C1(c(b(b(b(b(b(b(B(x)))))))))
C1(c(b(a(b(b(B(x))))))) → C1(c(b(b(b(b(b(B(x))))))))
C1(c(a(b(b(B(x)))))) → C1(c(b(b(b(b(b(B(x))))))))
The TRS R consists of the following rules:
c(c(x)) → b(b(b(a(a(a(x))))))
a(b(b(b(x)))) → b(b(b(b(b(b(b(b(x))))))))
c(c(b(b(x)))) → a(a(a(a(c(c(c(x)))))))
c(c(b(a(b(b(B(x))))))) → a(a(a(a(c(c(c(b(b(b(b(b(b(B(x))))))))))))))
a(a(b(b(B(x))))) → b(b(b(b(b(b(b(b(b(B(x))))))))))
a(b(a(b(b(B(x)))))) → b(b(b(b(b(b(b(b(b(B(x))))))))))
a(a(b(b(B(x))))) → b(b(b(b(b(b(b(b(b(b(b(B(x))))))))))))
a(b(a(b(b(B(x)))))) → b(b(b(b(b(b(b(b(b(b(b(B(x))))))))))))
a(b(b(a(b(b(B(x))))))) → b(b(b(b(b(b(b(b(b(b(b(B(x))))))))))))
a(b(b(a(b(b(B(x))))))) → b(b(b(b(b(b(b(b(b(b(B(x)))))))))))
a(a(b(b(B(x))))) → b(b(b(b(b(b(b(b(b(b(b(b(B(x)))))))))))))
c(c(a(b(b(B(x)))))) → a(a(a(a(c(c(c(b(b(b(B(x)))))))))))
a(b(b(B(x)))) → b(b(B(x)))
a(b(b(B(x)))) → B(x)
c(c(b(a(b(b(B(x))))))) → a(a(a(a(c(c(c(b(b(b(b(b(B(x)))))))))))))
a(b(b(a(b(b(B(x))))))) → b(b(b(b(b(b(b(b(b(b(b(b(b(B(x))))))))))))))
c(c(b(a(b(b(B(x))))))) → a(a(a(a(c(c(c(b(b(b(b(B(x))))))))))))
a(b(a(b(b(B(x)))))) → b(b(b(b(b(b(b(b(b(b(b(b(B(x)))))))))))))
a(b(a(b(b(B(x)))))) → b(b(b(b(b(b(b(b(b(b(B(x)))))))))))
a(b(b(a(b(b(B(x))))))) → b(b(b(b(b(b(b(b(b(b(b(b(B(x)))))))))))))
c(c(a(b(b(B(x)))))) → a(a(a(a(c(c(c(b(b(b(b(b(B(x)))))))))))))
a(b(b(a(b(b(B(x))))))) → b(b(b(b(b(b(b(b(b(b(b(b(b(b(B(x)))))))))))))))
c(c(a(b(b(B(x)))))) → a(a(a(a(c(c(c(b(b(b(b(B(x))))))))))))
a(b(a(b(b(B(x)))))) → b(b(b(b(b(b(b(b(b(b(b(b(b(B(x))))))))))))))
c(c(b(a(b(b(B(x))))))) → a(a(a(a(c(c(c(b(b(B(x))))))))))
c(c(a(b(b(B(x)))))) → a(a(a(a(c(c(c(b(b(B(x))))))))))
c(c(b(B(x)))) → a(a(a(a(c(C(x))))))
a(a(b(b(B(x))))) → b(b(b(b(b(b(b(b(b(b(B(x)))))))))))
c(C(x)) → b(B(x))
c(c(a(b(b(B(x)))))) → a(a(a(a(c(c(c(b(B(x)))))))))
c(c(b(a(b(b(B(x))))))) → a(a(a(a(c(c(c(b(b(b(B(x)))))))))))
a(b(b(B(x)))) → b(B(x))
c(c(b(B(x)))) → a(a(a(a(c(c(C(x)))))))
c(C(x)) → b(b(B(x)))
a(a(b(b(B(x))))) → b(b(b(b(b(b(b(b(B(x)))))))))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule C1(c(b(a(b(b(B(x))))))) → C1(c(c(b(b(b(B(x))))))) at position [0] we obtained the following new rules:
C1(c(b(a(b(b(B(y0))))))) → C1(b(b(b(a(a(a(b(b(b(B(y0)))))))))))
C1(c(b(a(b(b(B(y0))))))) → C1(a(a(a(a(c(c(c(b(B(y0))))))))))
↳ QTRS
↳ QTRS Reverse
↳ QTRS Reverse
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ SemLabProof
↳ SemLabProof2
↳ QDP
↳ QDPToSRSProof
↳ QTRS
↳ QTRS Reverse
↳ QTRS
↳ QTRS Reverse
↳ QTRS Reverse
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
Q DP problem:
The TRS P consists of the following rules:
C1(c(b(b(x)))) → C1(c(c(x)))
C1(c(a(b(b(B(x)))))) → C1(c(b(b(b(B(x))))))
C1(c(b(b(x)))) → C1(x)
C1(c(a(b(b(B(x)))))) → C1(c(b(b(B(x)))))
C1(c(b(a(b(b(B(x))))))) → C1(c(b(b(b(B(x))))))
C1(c(b(a(b(b(B(x))))))) → C1(c(b(b(b(b(B(x)))))))
C1(c(b(b(x)))) → C1(c(x))
C1(c(b(a(b(b(B(y0))))))) → C1(b(b(b(a(a(a(b(b(b(B(y0)))))))))))
C1(c(a(b(b(B(x)))))) → C1(c(b(b(b(b(B(x)))))))
C1(c(b(a(b(b(B(y0))))))) → C1(a(a(a(a(c(c(c(b(B(y0))))))))))
C1(c(b(a(b(b(B(x))))))) → C1(c(b(b(B(x)))))
C1(c(b(a(b(b(B(x))))))) → C1(c(b(b(b(b(b(B(x))))))))
C1(c(b(a(b(b(B(x))))))) → C1(c(b(b(b(b(b(b(B(x)))))))))
C1(c(a(b(b(B(x)))))) → C1(c(b(b(b(b(b(B(x))))))))
The TRS R consists of the following rules:
c(c(x)) → b(b(b(a(a(a(x))))))
a(b(b(b(x)))) → b(b(b(b(b(b(b(b(x))))))))
c(c(b(b(x)))) → a(a(a(a(c(c(c(x)))))))
c(c(b(a(b(b(B(x))))))) → a(a(a(a(c(c(c(b(b(b(b(b(b(B(x))))))))))))))
a(a(b(b(B(x))))) → b(b(b(b(b(b(b(b(b(B(x))))))))))
a(b(a(b(b(B(x)))))) → b(b(b(b(b(b(b(b(b(B(x))))))))))
a(a(b(b(B(x))))) → b(b(b(b(b(b(b(b(b(b(b(B(x))))))))))))
a(b(a(b(b(B(x)))))) → b(b(b(b(b(b(b(b(b(b(b(B(x))))))))))))
a(b(b(a(b(b(B(x))))))) → b(b(b(b(b(b(b(b(b(b(b(B(x))))))))))))
a(b(b(a(b(b(B(x))))))) → b(b(b(b(b(b(b(b(b(b(B(x)))))))))))
a(a(b(b(B(x))))) → b(b(b(b(b(b(b(b(b(b(b(b(B(x)))))))))))))
c(c(a(b(b(B(x)))))) → a(a(a(a(c(c(c(b(b(b(B(x)))))))))))
a(b(b(B(x)))) → b(b(B(x)))
a(b(b(B(x)))) → B(x)
c(c(b(a(b(b(B(x))))))) → a(a(a(a(c(c(c(b(b(b(b(b(B(x)))))))))))))
a(b(b(a(b(b(B(x))))))) → b(b(b(b(b(b(b(b(b(b(b(b(b(B(x))))))))))))))
c(c(b(a(b(b(B(x))))))) → a(a(a(a(c(c(c(b(b(b(b(B(x))))))))))))
a(b(a(b(b(B(x)))))) → b(b(b(b(b(b(b(b(b(b(b(b(B(x)))))))))))))
a(b(a(b(b(B(x)))))) → b(b(b(b(b(b(b(b(b(b(B(x)))))))))))
a(b(b(a(b(b(B(x))))))) → b(b(b(b(b(b(b(b(b(b(b(b(B(x)))))))))))))
c(c(a(b(b(B(x)))))) → a(a(a(a(c(c(c(b(b(b(b(b(B(x)))))))))))))
a(b(b(a(b(b(B(x))))))) → b(b(b(b(b(b(b(b(b(b(b(b(b(b(B(x)))))))))))))))
c(c(a(b(b(B(x)))))) → a(a(a(a(c(c(c(b(b(b(b(B(x))))))))))))
a(b(a(b(b(B(x)))))) → b(b(b(b(b(b(b(b(b(b(b(b(b(B(x))))))))))))))
c(c(b(a(b(b(B(x))))))) → a(a(a(a(c(c(c(b(b(B(x))))))))))
c(c(a(b(b(B(x)))))) → a(a(a(a(c(c(c(b(b(B(x))))))))))
c(c(b(B(x)))) → a(a(a(a(c(C(x))))))
a(a(b(b(B(x))))) → b(b(b(b(b(b(b(b(b(b(B(x)))))))))))
c(C(x)) → b(B(x))
c(c(a(b(b(B(x)))))) → a(a(a(a(c(c(c(b(B(x)))))))))
c(c(b(a(b(b(B(x))))))) → a(a(a(a(c(c(c(b(b(b(B(x)))))))))))
a(b(b(B(x)))) → b(B(x))
c(c(b(B(x)))) → a(a(a(a(c(c(C(x)))))))
c(C(x)) → b(b(B(x)))
a(a(b(b(B(x))))) → b(b(b(b(b(b(b(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 2 less nodes.
↳ QTRS
↳ QTRS Reverse
↳ QTRS Reverse
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ SemLabProof
↳ SemLabProof2
↳ QDP
↳ QDPToSRSProof
↳ QTRS
↳ QTRS Reverse
↳ QTRS
↳ QTRS Reverse
↳ QTRS Reverse
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
Q DP problem:
The TRS P consists of the following rules:
C1(c(a(b(b(B(x)))))) → C1(c(b(b(b(b(B(x)))))))
C1(c(b(b(x)))) → C1(c(c(x)))
C1(c(b(a(b(b(B(x))))))) → C1(c(b(b(B(x)))))
C1(c(a(b(b(B(x)))))) → C1(c(b(b(b(B(x))))))
C1(c(b(b(x)))) → C1(x)
C1(c(a(b(b(B(x)))))) → C1(c(b(b(B(x)))))
C1(c(b(a(b(b(B(x))))))) → C1(c(b(b(b(b(b(B(x))))))))
C1(c(b(a(b(b(B(x))))))) → C1(c(b(b(b(b(b(b(B(x)))))))))
C1(c(a(b(b(B(x)))))) → C1(c(b(b(b(b(b(B(x))))))))
C1(c(b(a(b(b(B(x))))))) → C1(c(b(b(b(B(x))))))
C1(c(b(a(b(b(B(x))))))) → C1(c(b(b(b(b(B(x)))))))
C1(c(b(b(x)))) → C1(c(x))
The TRS R consists of the following rules:
c(c(x)) → b(b(b(a(a(a(x))))))
a(b(b(b(x)))) → b(b(b(b(b(b(b(b(x))))))))
c(c(b(b(x)))) → a(a(a(a(c(c(c(x)))))))
c(c(b(a(b(b(B(x))))))) → a(a(a(a(c(c(c(b(b(b(b(b(b(B(x))))))))))))))
a(a(b(b(B(x))))) → b(b(b(b(b(b(b(b(b(B(x))))))))))
a(b(a(b(b(B(x)))))) → b(b(b(b(b(b(b(b(b(B(x))))))))))
a(a(b(b(B(x))))) → b(b(b(b(b(b(b(b(b(b(b(B(x))))))))))))
a(b(a(b(b(B(x)))))) → b(b(b(b(b(b(b(b(b(b(b(B(x))))))))))))
a(b(b(a(b(b(B(x))))))) → b(b(b(b(b(b(b(b(b(b(b(B(x))))))))))))
a(b(b(a(b(b(B(x))))))) → b(b(b(b(b(b(b(b(b(b(B(x)))))))))))
a(a(b(b(B(x))))) → b(b(b(b(b(b(b(b(b(b(b(b(B(x)))))))))))))
c(c(a(b(b(B(x)))))) → a(a(a(a(c(c(c(b(b(b(B(x)))))))))))
a(b(b(B(x)))) → b(b(B(x)))
a(b(b(B(x)))) → B(x)
c(c(b(a(b(b(B(x))))))) → a(a(a(a(c(c(c(b(b(b(b(b(B(x)))))))))))))
a(b(b(a(b(b(B(x))))))) → b(b(b(b(b(b(b(b(b(b(b(b(b(B(x))))))))))))))
c(c(b(a(b(b(B(x))))))) → a(a(a(a(c(c(c(b(b(b(b(B(x))))))))))))
a(b(a(b(b(B(x)))))) → b(b(b(b(b(b(b(b(b(b(b(b(B(x)))))))))))))
a(b(a(b(b(B(x)))))) → b(b(b(b(b(b(b(b(b(b(B(x)))))))))))
a(b(b(a(b(b(B(x))))))) → b(b(b(b(b(b(b(b(b(b(b(b(B(x)))))))))))))
c(c(a(b(b(B(x)))))) → a(a(a(a(c(c(c(b(b(b(b(b(B(x)))))))))))))
a(b(b(a(b(b(B(x))))))) → b(b(b(b(b(b(b(b(b(b(b(b(b(b(B(x)))))))))))))))
c(c(a(b(b(B(x)))))) → a(a(a(a(c(c(c(b(b(b(b(B(x))))))))))))
a(b(a(b(b(B(x)))))) → b(b(b(b(b(b(b(b(b(b(b(b(b(B(x))))))))))))))
c(c(b(a(b(b(B(x))))))) → a(a(a(a(c(c(c(b(b(B(x))))))))))
c(c(a(b(b(B(x)))))) → a(a(a(a(c(c(c(b(b(B(x))))))))))
c(c(b(B(x)))) → a(a(a(a(c(C(x))))))
a(a(b(b(B(x))))) → b(b(b(b(b(b(b(b(b(b(B(x)))))))))))
c(C(x)) → b(B(x))
c(c(a(b(b(B(x)))))) → a(a(a(a(c(c(c(b(B(x)))))))))
c(c(b(a(b(b(B(x))))))) → a(a(a(a(c(c(c(b(b(b(B(x)))))))))))
a(b(b(B(x)))) → b(B(x))
c(c(b(B(x)))) → a(a(a(a(c(c(C(x)))))))
c(C(x)) → b(b(B(x)))
a(a(b(b(B(x))))) → b(b(b(b(b(b(b(b(B(x)))))))))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule C1(c(b(b(x)))) → C1(c(c(x))) at position [0] we obtained the following new rules:
C1(c(b(b(c(a(b(b(B(x0))))))))) → C1(c(a(a(a(a(c(c(c(b(b(b(b(B(x0))))))))))))))
C1(c(b(b(c(a(b(b(B(x0))))))))) → C1(c(a(a(a(a(c(c(c(b(b(b(B(x0)))))))))))))
C1(c(b(b(C(x0))))) → C1(c(b(B(x0))))
C1(c(b(b(b(a(b(b(B(x0))))))))) → C1(a(a(a(a(c(c(c(b(b(b(b(B(x0)))))))))))))
C1(c(b(b(b(a(b(b(B(x0))))))))) → C1(a(a(a(a(c(c(c(b(b(b(b(b(b(B(x0)))))))))))))))
C1(c(b(b(c(b(a(b(b(B(x0)))))))))) → C1(c(a(a(a(a(c(c(c(b(b(B(x0))))))))))))
C1(c(b(b(c(a(b(b(B(x0))))))))) → C1(c(a(a(a(a(c(c(c(b(B(x0)))))))))))
C1(c(b(b(c(b(a(b(b(B(x0)))))))))) → C1(c(a(a(a(a(c(c(c(b(b(b(b(b(B(x0)))))))))))))))
C1(c(b(b(c(a(b(b(B(x0))))))))) → C1(c(a(a(a(a(c(c(c(b(b(B(x0))))))))))))
C1(c(b(b(b(a(b(b(B(x0))))))))) → C1(a(a(a(a(c(c(c(b(b(b(b(b(B(x0))))))))))))))
C1(c(b(b(a(b(b(B(x0)))))))) → C1(a(a(a(a(c(c(c(b(b(b(b(B(x0)))))))))))))
C1(c(b(b(b(a(b(b(B(x0))))))))) → C1(a(a(a(a(c(c(c(b(b(B(x0)))))))))))
C1(c(b(b(a(b(b(B(x0)))))))) → C1(a(a(a(a(c(c(c(b(B(x0))))))))))
C1(c(b(b(c(x0))))) → C1(c(b(b(b(a(a(a(x0))))))))
C1(c(b(b(b(B(x0)))))) → C1(a(a(a(a(c(C(x0)))))))
C1(c(b(b(C(x0))))) → C1(c(b(b(B(x0)))))
C1(c(b(b(b(b(x0)))))) → C1(a(a(a(a(c(c(c(x0))))))))
C1(c(b(b(b(a(b(b(B(x0))))))))) → C1(a(a(a(a(c(c(c(b(b(b(B(x0))))))))))))
C1(c(b(b(a(b(b(B(x0)))))))) → C1(a(a(a(a(c(c(c(b(b(b(b(b(B(x0))))))))))))))
C1(c(b(b(c(b(a(b(b(B(x0)))))))))) → C1(c(a(a(a(a(c(c(c(b(b(b(B(x0)))))))))))))
C1(c(b(b(b(B(x0)))))) → C1(a(a(a(a(c(c(C(x0))))))))
C1(c(b(b(a(b(b(B(x0)))))))) → C1(a(a(a(a(c(c(c(b(b(B(x0)))))))))))
C1(c(b(b(c(b(B(x0))))))) → C1(c(a(a(a(a(c(C(x0))))))))
C1(c(b(b(c(b(a(b(b(B(x0)))))))))) → C1(c(a(a(a(a(c(c(c(b(b(b(b(B(x0))))))))))))))
C1(c(b(b(a(b(b(B(x0)))))))) → C1(a(a(a(a(c(c(c(b(b(b(B(x0))))))))))))
C1(c(b(b(x0)))) → C1(b(b(b(a(a(a(x0)))))))
C1(c(b(b(c(b(b(x0))))))) → C1(c(a(a(a(a(c(c(c(x0)))))))))
C1(c(b(b(c(b(a(b(b(B(x0)))))))))) → C1(c(a(a(a(a(c(c(c(b(b(b(b(b(b(B(x0))))))))))))))))
C1(c(b(b(c(b(B(x0))))))) → C1(c(a(a(a(a(c(c(C(x0)))))))))
C1(c(b(b(c(a(b(b(B(x0))))))))) → C1(c(a(a(a(a(c(c(c(b(b(b(b(b(B(x0)))))))))))))))
↳ QTRS
↳ QTRS Reverse
↳ QTRS Reverse
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ SemLabProof
↳ SemLabProof2
↳ QDP
↳ QDPToSRSProof
↳ QTRS
↳ QTRS Reverse
↳ QTRS
↳ QTRS Reverse
↳ QTRS Reverse
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
Q DP problem:
The TRS P consists of the following rules:
C1(c(b(b(c(a(b(b(B(x0))))))))) → C1(c(a(a(a(a(c(c(c(b(b(b(b(B(x0))))))))))))))
C1(c(b(b(c(a(b(b(B(x0))))))))) → C1(c(a(a(a(a(c(c(c(b(b(b(B(x0)))))))))))))
C1(c(a(b(b(B(x)))))) → C1(c(b(b(b(B(x))))))
C1(c(b(b(C(x0))))) → C1(c(b(B(x0))))
C1(c(b(b(b(a(b(b(B(x0))))))))) → C1(a(a(a(a(c(c(c(b(b(b(b(B(x0)))))))))))))
C1(c(b(b(x)))) → C1(x)
C1(c(b(b(b(a(b(b(B(x0))))))))) → C1(a(a(a(a(c(c(c(b(b(b(b(b(b(B(x0)))))))))))))))
C1(c(b(b(c(b(a(b(b(B(x0)))))))))) → C1(c(a(a(a(a(c(c(c(b(b(B(x0))))))))))))
C1(c(b(a(b(b(B(x))))))) → C1(c(b(b(b(B(x))))))
C1(c(b(a(b(b(B(x))))))) → C1(c(b(b(b(b(B(x)))))))
C1(c(b(b(x)))) → C1(c(x))
C1(c(b(b(c(a(b(b(B(x0))))))))) → C1(c(a(a(a(a(c(c(c(b(B(x0)))))))))))
C1(c(b(b(c(b(a(b(b(B(x0)))))))))) → C1(c(a(a(a(a(c(c(c(b(b(b(b(b(B(x0)))))))))))))))
C1(c(b(b(c(a(b(b(B(x0))))))))) → C1(c(a(a(a(a(c(c(c(b(b(B(x0))))))))))))
C1(c(b(b(b(a(b(b(B(x0))))))))) → C1(a(a(a(a(c(c(c(b(b(b(b(b(B(x0))))))))))))))
C1(c(b(a(b(b(B(x))))))) → C1(c(b(b(b(b(b(B(x))))))))
C1(c(b(b(a(b(b(B(x0)))))))) → C1(a(a(a(a(c(c(c(b(b(b(b(B(x0)))))))))))))
C1(c(b(b(b(a(b(b(B(x0))))))))) → C1(a(a(a(a(c(c(c(b(b(B(x0)))))))))))
C1(c(b(b(c(x0))))) → C1(c(b(b(b(a(a(a(x0))))))))
C1(c(b(b(a(b(b(B(x0)))))))) → C1(a(a(a(a(c(c(c(b(B(x0))))))))))
C1(c(b(b(b(B(x0)))))) → C1(a(a(a(a(c(C(x0)))))))
C1(c(b(b(C(x0))))) → C1(c(b(b(B(x0)))))
C1(c(a(b(b(B(x)))))) → C1(c(b(b(B(x)))))
C1(c(b(b(b(b(x0)))))) → C1(a(a(a(a(c(c(c(x0))))))))
C1(c(b(b(a(b(b(B(x0)))))))) → C1(a(a(a(a(c(c(c(b(b(b(b(b(B(x0))))))))))))))
C1(c(b(b(b(a(b(b(B(x0))))))))) → C1(a(a(a(a(c(c(c(b(b(b(B(x0))))))))))))
C1(c(b(b(c(b(a(b(b(B(x0)))))))))) → C1(c(a(a(a(a(c(c(c(b(b(b(B(x0)))))))))))))
C1(c(b(b(b(B(x0)))))) → C1(a(a(a(a(c(c(C(x0))))))))
C1(c(a(b(b(B(x)))))) → C1(c(b(b(b(b(B(x)))))))
C1(c(b(b(a(b(b(B(x0)))))))) → C1(a(a(a(a(c(c(c(b(b(B(x0)))))))))))
C1(c(b(a(b(b(B(x))))))) → C1(c(b(b(B(x)))))
C1(c(b(b(c(b(B(x0))))))) → C1(c(a(a(a(a(c(C(x0))))))))
C1(c(b(a(b(b(B(x))))))) → C1(c(b(b(b(b(b(b(B(x)))))))))
C1(c(a(b(b(B(x)))))) → C1(c(b(b(b(b(b(B(x))))))))
C1(c(b(b(c(b(a(b(b(B(x0)))))))))) → C1(c(a(a(a(a(c(c(c(b(b(b(b(B(x0))))))))))))))
C1(c(b(b(a(b(b(B(x0)))))))) → C1(a(a(a(a(c(c(c(b(b(b(B(x0))))))))))))
C1(c(b(b(c(b(b(x0))))))) → C1(c(a(a(a(a(c(c(c(x0)))))))))
C1(c(b(b(x0)))) → C1(b(b(b(a(a(a(x0)))))))
C1(c(b(b(c(b(a(b(b(B(x0)))))))))) → C1(c(a(a(a(a(c(c(c(b(b(b(b(b(b(B(x0))))))))))))))))
C1(c(b(b(c(b(B(x0))))))) → C1(c(a(a(a(a(c(c(C(x0)))))))))
C1(c(b(b(c(a(b(b(B(x0))))))))) → C1(c(a(a(a(a(c(c(c(b(b(b(b(b(B(x0)))))))))))))))
The TRS R consists of the following rules:
c(c(x)) → b(b(b(a(a(a(x))))))
a(b(b(b(x)))) → b(b(b(b(b(b(b(b(x))))))))
c(c(b(b(x)))) → a(a(a(a(c(c(c(x)))))))
c(c(b(a(b(b(B(x))))))) → a(a(a(a(c(c(c(b(b(b(b(b(b(B(x))))))))))))))
a(a(b(b(B(x))))) → b(b(b(b(b(b(b(b(b(B(x))))))))))
a(b(a(b(b(B(x)))))) → b(b(b(b(b(b(b(b(b(B(x))))))))))
a(a(b(b(B(x))))) → b(b(b(b(b(b(b(b(b(b(b(B(x))))))))))))
a(b(a(b(b(B(x)))))) → b(b(b(b(b(b(b(b(b(b(b(B(x))))))))))))
a(b(b(a(b(b(B(x))))))) → b(b(b(b(b(b(b(b(b(b(b(B(x))))))))))))
a(b(b(a(b(b(B(x))))))) → b(b(b(b(b(b(b(b(b(b(B(x)))))))))))
a(a(b(b(B(x))))) → b(b(b(b(b(b(b(b(b(b(b(b(B(x)))))))))))))
c(c(a(b(b(B(x)))))) → a(a(a(a(c(c(c(b(b(b(B(x)))))))))))
a(b(b(B(x)))) → b(b(B(x)))
a(b(b(B(x)))) → B(x)
c(c(b(a(b(b(B(x))))))) → a(a(a(a(c(c(c(b(b(b(b(b(B(x)))))))))))))
a(b(b(a(b(b(B(x))))))) → b(b(b(b(b(b(b(b(b(b(b(b(b(B(x))))))))))))))
c(c(b(a(b(b(B(x))))))) → a(a(a(a(c(c(c(b(b(b(b(B(x))))))))))))
a(b(a(b(b(B(x)))))) → b(b(b(b(b(b(b(b(b(b(b(b(B(x)))))))))))))
a(b(a(b(b(B(x)))))) → b(b(b(b(b(b(b(b(b(b(B(x)))))))))))
a(b(b(a(b(b(B(x))))))) → b(b(b(b(b(b(b(b(b(b(b(b(B(x)))))))))))))
c(c(a(b(b(B(x)))))) → a(a(a(a(c(c(c(b(b(b(b(b(B(x)))))))))))))
a(b(b(a(b(b(B(x))))))) → b(b(b(b(b(b(b(b(b(b(b(b(b(b(B(x)))))))))))))))
c(c(a(b(b(B(x)))))) → a(a(a(a(c(c(c(b(b(b(b(B(x))))))))))))
a(b(a(b(b(B(x)))))) → b(b(b(b(b(b(b(b(b(b(b(b(b(B(x))))))))))))))
c(c(b(a(b(b(B(x))))))) → a(a(a(a(c(c(c(b(b(B(x))))))))))
c(c(a(b(b(B(x)))))) → a(a(a(a(c(c(c(b(b(B(x))))))))))
c(c(b(B(x)))) → a(a(a(a(c(C(x))))))
a(a(b(b(B(x))))) → b(b(b(b(b(b(b(b(b(b(B(x)))))))))))
c(C(x)) → b(B(x))
c(c(a(b(b(B(x)))))) → a(a(a(a(c(c(c(b(B(x)))))))))
c(c(b(a(b(b(B(x))))))) → a(a(a(a(c(c(c(b(b(b(B(x)))))))))))
a(b(b(B(x)))) → b(B(x))
c(c(b(B(x)))) → a(a(a(a(c(c(C(x)))))))
c(C(x)) → b(b(B(x)))
a(a(b(b(B(x))))) → b(b(b(b(b(b(b(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 15 less nodes.
↳ QTRS
↳ QTRS Reverse
↳ QTRS Reverse
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ SemLabProof
↳ SemLabProof2
↳ QDP
↳ QDPToSRSProof
↳ QTRS
↳ QTRS Reverse
↳ QTRS
↳ QTRS Reverse
↳ QTRS Reverse
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ ForwardInstantiation
Q DP problem:
The TRS P consists of the following rules:
C1(c(b(b(c(a(b(b(B(x0))))))))) → C1(c(a(a(a(a(c(c(c(b(b(b(b(B(x0))))))))))))))
C1(c(b(b(c(a(b(b(B(x0))))))))) → C1(c(a(a(a(a(c(c(c(b(b(b(B(x0)))))))))))))
C1(c(a(b(b(B(x)))))) → C1(c(b(b(b(B(x))))))
C1(c(b(b(x)))) → C1(x)
C1(c(b(b(c(b(a(b(b(B(x0)))))))))) → C1(c(a(a(a(a(c(c(c(b(b(B(x0))))))))))))
C1(c(b(a(b(b(B(x))))))) → C1(c(b(b(b(B(x))))))
C1(c(b(a(b(b(B(x))))))) → C1(c(b(b(b(b(B(x)))))))
C1(c(b(b(x)))) → C1(c(x))
C1(c(b(b(c(a(b(b(B(x0))))))))) → C1(c(a(a(a(a(c(c(c(b(B(x0)))))))))))
C1(c(b(b(c(b(a(b(b(B(x0)))))))))) → C1(c(a(a(a(a(c(c(c(b(b(b(b(b(B(x0)))))))))))))))
C1(c(b(b(c(a(b(b(B(x0))))))))) → C1(c(a(a(a(a(c(c(c(b(b(B(x0))))))))))))
C1(c(b(a(b(b(B(x))))))) → C1(c(b(b(b(b(b(B(x))))))))
C1(c(b(b(c(x0))))) → C1(c(b(b(b(a(a(a(x0))))))))
C1(c(b(b(C(x0))))) → C1(c(b(b(B(x0)))))
C1(c(a(b(b(B(x)))))) → C1(c(b(b(B(x)))))
C1(c(b(b(c(b(a(b(b(B(x0)))))))))) → C1(c(a(a(a(a(c(c(c(b(b(b(B(x0)))))))))))))
C1(c(a(b(b(B(x)))))) → C1(c(b(b(b(b(B(x)))))))
C1(c(b(a(b(b(B(x))))))) → C1(c(b(b(B(x)))))
C1(c(b(b(c(b(B(x0))))))) → C1(c(a(a(a(a(c(C(x0))))))))
C1(c(b(a(b(b(B(x))))))) → C1(c(b(b(b(b(b(b(B(x)))))))))
C1(c(a(b(b(B(x)))))) → C1(c(b(b(b(b(b(B(x))))))))
C1(c(b(b(c(b(a(b(b(B(x0)))))))))) → C1(c(a(a(a(a(c(c(c(b(b(b(b(B(x0))))))))))))))
C1(c(b(b(c(b(b(x0))))))) → C1(c(a(a(a(a(c(c(c(x0)))))))))
C1(c(b(b(c(b(a(b(b(B(x0)))))))))) → C1(c(a(a(a(a(c(c(c(b(b(b(b(b(b(B(x0))))))))))))))))
C1(c(b(b(c(b(B(x0))))))) → C1(c(a(a(a(a(c(c(C(x0)))))))))
C1(c(b(b(c(a(b(b(B(x0))))))))) → C1(c(a(a(a(a(c(c(c(b(b(b(b(b(B(x0)))))))))))))))
The TRS R consists of the following rules:
c(c(x)) → b(b(b(a(a(a(x))))))
a(b(b(b(x)))) → b(b(b(b(b(b(b(b(x))))))))
c(c(b(b(x)))) → a(a(a(a(c(c(c(x)))))))
c(c(b(a(b(b(B(x))))))) → a(a(a(a(c(c(c(b(b(b(b(b(b(B(x))))))))))))))
a(a(b(b(B(x))))) → b(b(b(b(b(b(b(b(b(B(x))))))))))
a(b(a(b(b(B(x)))))) → b(b(b(b(b(b(b(b(b(B(x))))))))))
a(a(b(b(B(x))))) → b(b(b(b(b(b(b(b(b(b(b(B(x))))))))))))
a(b(a(b(b(B(x)))))) → b(b(b(b(b(b(b(b(b(b(b(B(x))))))))))))
a(b(b(a(b(b(B(x))))))) → b(b(b(b(b(b(b(b(b(b(b(B(x))))))))))))
a(b(b(a(b(b(B(x))))))) → b(b(b(b(b(b(b(b(b(b(B(x)))))))))))
a(a(b(b(B(x))))) → b(b(b(b(b(b(b(b(b(b(b(b(B(x)))))))))))))
c(c(a(b(b(B(x)))))) → a(a(a(a(c(c(c(b(b(b(B(x)))))))))))
a(b(b(B(x)))) → b(b(B(x)))
a(b(b(B(x)))) → B(x)
c(c(b(a(b(b(B(x))))))) → a(a(a(a(c(c(c(b(b(b(b(b(B(x)))))))))))))
a(b(b(a(b(b(B(x))))))) → b(b(b(b(b(b(b(b(b(b(b(b(b(B(x))))))))))))))
c(c(b(a(b(b(B(x))))))) → a(a(a(a(c(c(c(b(b(b(b(B(x))))))))))))
a(b(a(b(b(B(x)))))) → b(b(b(b(b(b(b(b(b(b(b(b(B(x)))))))))))))
a(b(a(b(b(B(x)))))) → b(b(b(b(b(b(b(b(b(b(B(x)))))))))))
a(b(b(a(b(b(B(x))))))) → b(b(b(b(b(b(b(b(b(b(b(b(B(x)))))))))))))
c(c(a(b(b(B(x)))))) → a(a(a(a(c(c(c(b(b(b(b(b(B(x)))))))))))))
a(b(b(a(b(b(B(x))))))) → b(b(b(b(b(b(b(b(b(b(b(b(b(b(B(x)))))))))))))))
c(c(a(b(b(B(x)))))) → a(a(a(a(c(c(c(b(b(b(b(B(x))))))))))))
a(b(a(b(b(B(x)))))) → b(b(b(b(b(b(b(b(b(b(b(b(b(B(x))))))))))))))
c(c(b(a(b(b(B(x))))))) → a(a(a(a(c(c(c(b(b(B(x))))))))))
c(c(a(b(b(B(x)))))) → a(a(a(a(c(c(c(b(b(B(x))))))))))
c(c(b(B(x)))) → a(a(a(a(c(C(x))))))
a(a(b(b(B(x))))) → b(b(b(b(b(b(b(b(b(b(B(x)))))))))))
c(C(x)) → b(B(x))
c(c(a(b(b(B(x)))))) → a(a(a(a(c(c(c(b(B(x)))))))))
c(c(b(a(b(b(B(x))))))) → a(a(a(a(c(c(c(b(b(b(B(x)))))))))))
a(b(b(B(x)))) → b(B(x))
c(c(b(B(x)))) → a(a(a(a(c(c(C(x)))))))
c(C(x)) → b(b(B(x)))
a(a(b(b(B(x))))) → b(b(b(b(b(b(b(b(B(x)))))))))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By forward instantiating [14] the rule C1(c(b(b(x)))) → C1(x) we obtained the following new rules:
C1(c(b(b(c(y_2))))) → C1(c(y_2))
C1(c(b(b(c(b(b(c(y_4)))))))) → C1(c(b(b(c(y_4)))))
C1(c(b(b(c(b(b(C(y_0)))))))) → C1(c(b(b(C(y_0)))))
↳ QTRS
↳ QTRS Reverse
↳ QTRS Reverse
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ SemLabProof
↳ SemLabProof2
↳ QDP
↳ QDPToSRSProof
↳ QTRS
↳ QTRS Reverse
↳ QTRS
↳ QTRS Reverse
↳ QTRS Reverse
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ SemLabProof
↳ SemLabProof2
Q DP problem:
The TRS P consists of the following rules:
C1(c(b(b(c(a(b(b(B(x0))))))))) → C1(c(a(a(a(a(c(c(c(b(b(b(b(B(x0))))))))))))))
C1(c(b(b(c(a(b(b(B(x0))))))))) → C1(c(a(a(a(a(c(c(c(b(b(b(B(x0)))))))))))))
C1(c(a(b(b(B(x)))))) → C1(c(b(b(b(B(x))))))
C1(c(b(b(c(b(a(b(b(B(x0)))))))))) → C1(c(a(a(a(a(c(c(c(b(b(B(x0))))))))))))
C1(c(b(a(b(b(B(x))))))) → C1(c(b(b(b(B(x))))))
C1(c(b(a(b(b(B(x))))))) → C1(c(b(b(b(b(B(x)))))))
C1(c(b(b(x)))) → C1(c(x))
C1(c(b(b(c(a(b(b(B(x0))))))))) → C1(c(a(a(a(a(c(c(c(b(b(B(x0))))))))))))
C1(c(b(b(c(b(a(b(b(B(x0)))))))))) → C1(c(a(a(a(a(c(c(c(b(b(b(b(b(B(x0)))))))))))))))
C1(c(b(b(c(a(b(b(B(x0))))))))) → C1(c(a(a(a(a(c(c(c(b(B(x0)))))))))))
C1(c(b(b(c(y_2))))) → C1(c(y_2))
C1(c(b(a(b(b(B(x))))))) → C1(c(b(b(b(b(b(B(x))))))))
C1(c(b(b(c(b(b(c(y_4)))))))) → C1(c(b(b(c(y_4)))))
C1(c(b(b(c(x0))))) → C1(c(b(b(b(a(a(a(x0))))))))
C1(c(b(b(C(x0))))) → C1(c(b(b(B(x0)))))
C1(c(a(b(b(B(x)))))) → C1(c(b(b(B(x)))))
C1(c(b(b(c(b(a(b(b(B(x0)))))))))) → C1(c(a(a(a(a(c(c(c(b(b(b(B(x0)))))))))))))
C1(c(a(b(b(B(x)))))) → C1(c(b(b(b(b(B(x)))))))
C1(c(b(a(b(b(B(x))))))) → C1(c(b(b(B(x)))))
C1(c(b(b(c(b(B(x0))))))) → C1(c(a(a(a(a(c(C(x0))))))))
C1(c(b(a(b(b(B(x))))))) → C1(c(b(b(b(b(b(b(B(x)))))))))
C1(c(a(b(b(B(x)))))) → C1(c(b(b(b(b(b(B(x))))))))
C1(c(b(b(c(b(a(b(b(B(x0)))))))))) → C1(c(a(a(a(a(c(c(c(b(b(b(b(B(x0))))))))))))))
C1(c(b(b(c(b(b(x0))))))) → C1(c(a(a(a(a(c(c(c(x0)))))))))
C1(c(b(b(c(b(a(b(b(B(x0)))))))))) → C1(c(a(a(a(a(c(c(c(b(b(b(b(b(b(B(x0))))))))))))))))
C1(c(b(b(c(b(B(x0))))))) → C1(c(a(a(a(a(c(c(C(x0)))))))))
C1(c(b(b(c(b(b(C(y_0)))))))) → C1(c(b(b(C(y_0)))))
C1(c(b(b(c(a(b(b(B(x0))))))))) → C1(c(a(a(a(a(c(c(c(b(b(b(b(b(B(x0)))))))))))))))
The TRS R consists of the following rules:
c(c(x)) → b(b(b(a(a(a(x))))))
a(b(b(b(x)))) → b(b(b(b(b(b(b(b(x))))))))
c(c(b(b(x)))) → a(a(a(a(c(c(c(x)))))))
c(c(b(a(b(b(B(x))))))) → a(a(a(a(c(c(c(b(b(b(b(b(b(B(x))))))))))))))
a(a(b(b(B(x))))) → b(b(b(b(b(b(b(b(b(B(x))))))))))
a(b(a(b(b(B(x)))))) → b(b(b(b(b(b(b(b(b(B(x))))))))))
a(a(b(b(B(x))))) → b(b(b(b(b(b(b(b(b(b(b(B(x))))))))))))
a(b(a(b(b(B(x)))))) → b(b(b(b(b(b(b(b(b(b(b(B(x))))))))))))
a(b(b(a(b(b(B(x))))))) → b(b(b(b(b(b(b(b(b(b(b(B(x))))))))))))
a(b(b(a(b(b(B(x))))))) → b(b(b(b(b(b(b(b(b(b(B(x)))))))))))
a(a(b(b(B(x))))) → b(b(b(b(b(b(b(b(b(b(b(b(B(x)))))))))))))
c(c(a(b(b(B(x)))))) → a(a(a(a(c(c(c(b(b(b(B(x)))))))))))
a(b(b(B(x)))) → b(b(B(x)))
a(b(b(B(x)))) → B(x)
c(c(b(a(b(b(B(x))))))) → a(a(a(a(c(c(c(b(b(b(b(b(B(x)))))))))))))
a(b(b(a(b(b(B(x))))))) → b(b(b(b(b(b(b(b(b(b(b(b(b(B(x))))))))))))))
c(c(b(a(b(b(B(x))))))) → a(a(a(a(c(c(c(b(b(b(b(B(x))))))))))))
a(b(a(b(b(B(x)))))) → b(b(b(b(b(b(b(b(b(b(b(b(B(x)))))))))))))
a(b(a(b(b(B(x)))))) → b(b(b(b(b(b(b(b(b(b(B(x)))))))))))
a(b(b(a(b(b(B(x))))))) → b(b(b(b(b(b(b(b(b(b(b(b(B(x)))))))))))))
c(c(a(b(b(B(x)))))) → a(a(a(a(c(c(c(b(b(b(b(b(B(x)))))))))))))
a(b(b(a(b(b(B(x))))))) → b(b(b(b(b(b(b(b(b(b(b(b(b(b(B(x)))))))))))))))
c(c(a(b(b(B(x)))))) → a(a(a(a(c(c(c(b(b(b(b(B(x))))))))))))
a(b(a(b(b(B(x)))))) → b(b(b(b(b(b(b(b(b(b(b(b(b(B(x))))))))))))))
c(c(b(a(b(b(B(x))))))) → a(a(a(a(c(c(c(b(b(B(x))))))))))
c(c(a(b(b(B(x)))))) → a(a(a(a(c(c(c(b(b(B(x))))))))))
c(c(b(B(x)))) → a(a(a(a(c(C(x))))))
a(a(b(b(B(x))))) → b(b(b(b(b(b(b(b(b(b(B(x)))))))))))
c(C(x)) → b(B(x))
c(c(a(b(b(B(x)))))) → a(a(a(a(c(c(c(b(B(x)))))))))
c(c(b(a(b(b(B(x))))))) → a(a(a(a(c(c(c(b(b(b(B(x)))))))))))
a(b(b(B(x)))) → b(B(x))
c(c(b(B(x)))) → a(a(a(a(c(c(C(x)))))))
c(C(x)) → b(b(B(x)))
a(a(b(b(B(x))))) → b(b(b(b(b(b(b(b(B(x)))))))))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We found the following model for the rules of the TRS R.
Interpretation over the domain with elements from 0 to 1.C: 1
c: 0
B: 0
a: 0
b: 0
C1: 0
By semantic labelling [33] we obtain the following labelled TRS:Q DP problem:
The TRS P consists of the following rules:
C1.0(c.0(b.0(a.0(b.0(b.0(B.1(x))))))) → C1.0(c.0(b.0(b.0(b.0(B.1(x))))))
C1.0(c.0(b.0(a.0(b.0(b.0(B.1(x))))))) → C1.0(c.0(b.0(b.0(b.0(b.0(b.0(b.0(B.1(x)))))))))
C1.0(c.0(b.0(b.0(c.0(b.0(a.0(b.0(b.0(B.0(x0)))))))))) → C1.0(c.0(a.0(a.0(a.0(a.0(c.0(c.0(c.0(b.0(b.0(b.0(b.0(B.0(x0))))))))))))))
C1.0(c.0(b.0(b.0(c.1(y_2))))) → C1.0(c.1(y_2))
C1.0(c.0(b.0(b.1(C.0(x0))))) → C1.0(c.0(b.0(b.0(B.0(x0)))))
C1.0(c.0(b.0(b.0(c.0(b.0(B.1(x0))))))) → C1.0(c.0(a.0(a.0(a.0(a.0(c.1(C.1(x0))))))))
C1.0(c.0(b.0(b.0(c.0(a.0(b.0(b.0(B.1(x0))))))))) → C1.0(c.0(a.0(a.0(a.0(a.0(c.0(c.0(c.0(b.0(b.0(b.0(b.0(B.1(x0))))))))))))))
C1.0(c.0(b.0(b.0(c.0(b.0(a.0(b.0(b.0(B.1(x0)))))))))) → C1.0(c.0(a.0(a.0(a.0(a.0(c.0(c.0(c.0(b.0(b.0(b.0(b.0(b.0(b.0(B.1(x0))))))))))))))))
C1.0(c.0(b.0(a.0(b.0(b.0(B.0(x))))))) → C1.0(c.0(b.0(b.0(B.0(x)))))
C1.0(c.0(b.0(b.0(c.0(b.0(b.1(x0))))))) → C1.0(c.0(a.0(a.0(a.0(a.0(c.0(c.0(c.1(x0)))))))))
C1.0(c.0(b.0(a.0(b.0(b.0(B.1(x))))))) → C1.0(c.0(b.0(b.0(b.0(b.0(b.0(B.1(x))))))))
C1.0(c.0(a.0(b.0(b.0(B.0(x)))))) → C1.0(c.0(b.0(b.0(b.0(b.0(b.0(B.0(x))))))))
C1.0(c.0(b.0(b.0(c.0(x0))))) → C1.0(c.0(b.0(b.0(b.0(a.0(a.0(a.0(x0))))))))
C1.0(c.0(b.0(b.0(c.0(a.0(b.0(b.0(B.0(x0))))))))) → C1.0(c.0(a.0(a.0(a.0(a.0(c.0(c.0(c.0(b.0(b.0(B.0(x0))))))))))))
C1.0(c.0(b.0(b.0(c.0(b.0(b.1(C.0(y_0)))))))) → C1.0(c.0(b.0(b.1(C.0(y_0)))))
C1.0(c.0(b.0(b.0(c.0(y_2))))) → C1.0(c.0(y_2))
C1.0(c.0(b.0(b.0(c.0(a.0(b.0(b.0(B.1(x0))))))))) → C1.0(c.0(a.0(a.0(a.0(a.0(c.0(c.0(c.0(b.0(b.0(b.0(B.1(x0)))))))))))))
C1.0(c.0(b.0(b.0(c.0(b.0(a.0(b.0(b.0(B.1(x0)))))))))) → C1.0(c.0(a.0(a.0(a.0(a.0(c.0(c.0(c.0(b.0(b.0(B.1(x0))))))))))))
C1.0(c.0(a.0(b.0(b.0(B.1(x)))))) → C1.0(c.0(b.0(b.0(b.0(B.1(x))))))
C1.0(c.0(a.0(b.0(b.0(B.0(x)))))) → C1.0(c.0(b.0(b.0(B.0(x)))))
C1.0(c.0(b.0(a.0(b.0(b.0(B.1(x))))))) → C1.0(c.0(b.0(b.0(B.1(x)))))
C1.0(c.0(b.0(b.0(c.0(b.0(B.0(x0))))))) → C1.0(c.0(a.0(a.0(a.0(a.0(c.0(c.1(C.0(x0)))))))))
C1.0(c.0(b.0(a.0(b.0(b.0(B.0(x))))))) → C1.0(c.0(b.0(b.0(b.0(b.0(b.0(B.0(x))))))))
C1.0(c.0(b.0(b.0(c.0(b.0(a.0(b.0(b.0(B.0(x0)))))))))) → C1.0(c.0(a.0(a.0(a.0(a.0(c.0(c.0(c.0(b.0(b.0(b.0(b.0(b.0(B.0(x0)))))))))))))))
C1.0(c.0(b.0(b.0(c.0(b.0(a.0(b.0(b.0(B.0(x0)))))))))) → C1.0(c.0(a.0(a.0(a.0(a.0(c.0(c.0(c.0(b.0(b.0(B.0(x0))))))))))))
C1.0(c.0(b.0(b.0(c.0(b.0(a.0(b.0(b.0(B.1(x0)))))))))) → C1.0(c.0(a.0(a.0(a.0(a.0(c.0(c.0(c.0(b.0(b.0(b.0(B.1(x0)))))))))))))
C1.0(c.0(b.0(b.0(c.0(a.0(b.0(b.0(B.0(x0))))))))) → C1.0(c.0(a.0(a.0(a.0(a.0(c.0(c.0(c.0(b.0(b.0(b.0(B.0(x0)))))))))))))
C1.0(c.0(b.0(a.0(b.0(b.0(B.0(x))))))) → C1.0(c.0(b.0(b.0(b.0(B.0(x))))))
C1.0(c.0(b.0(b.0(c.0(b.0(a.0(b.0(b.0(B.0(x0)))))))))) → C1.0(c.0(a.0(a.0(a.0(a.0(c.0(c.0(c.0(b.0(b.0(b.0(B.0(x0)))))))))))))
C1.0(c.0(a.0(b.0(b.0(B.1(x)))))) → C1.0(c.0(b.0(b.0(b.0(b.0(b.0(B.1(x))))))))
C1.0(c.0(b.0(b.0(c.0(b.0(a.0(b.0(b.0(B.0(x0)))))))))) → C1.0(c.0(a.0(a.0(a.0(a.0(c.0(c.0(c.0(b.0(b.0(b.0(b.0(b.0(b.0(B.0(x0))))))))))))))))
C1.0(c.0(b.0(b.1(x)))) → C1.0(c.1(x))
C1.0(c.0(b.0(b.0(c.0(a.0(b.0(b.0(B.1(x0))))))))) → C1.0(c.0(a.0(a.0(a.0(a.0(c.0(c.0(c.0(b.0(B.1(x0)))))))))))
C1.0(c.0(b.0(b.0(c.0(b.0(b.0(x0))))))) → C1.0(c.0(a.0(a.0(a.0(a.0(c.0(c.0(c.0(x0)))))))))
C1.0(c.0(b.0(b.0(c.0(b.0(a.0(b.0(b.0(B.1(x0)))))))))) → C1.0(c.0(a.0(a.0(a.0(a.0(c.0(c.0(c.0(b.0(b.0(b.0(b.0(B.1(x0))))))))))))))
C1.0(c.0(b.0(b.0(c.0(a.0(b.0(b.0(B.0(x0))))))))) → C1.0(c.0(a.0(a.0(a.0(a.0(c.0(c.0(c.0(b.0(b.0(b.0(b.0(B.0(x0))))))))))))))
C1.0(c.0(b.0(b.0(c.0(b.0(a.0(b.0(b.0(B.1(x0)))))))))) → C1.0(c.0(a.0(a.0(a.0(a.0(c.0(c.0(c.0(b.0(b.0(b.0(b.0(b.0(B.1(x0)))))))))))))))
C1.0(c.0(b.0(b.0(c.0(a.0(b.0(b.0(B.1(x0))))))))) → C1.0(c.0(a.0(a.0(a.0(a.0(c.0(c.0(c.0(b.0(b.0(b.0(b.0(b.0(B.1(x0)))))))))))))))
C1.0(c.0(b.0(b.0(c.0(b.0(B.0(x0))))))) → C1.0(c.0(a.0(a.0(a.0(a.0(c.1(C.0(x0))))))))
C1.0(c.0(b.0(a.0(b.0(b.0(B.0(x))))))) → C1.0(c.0(b.0(b.0(b.0(b.0(b.0(b.0(B.0(x)))))))))
C1.0(c.0(a.0(b.0(b.0(B.1(x)))))) → C1.0(c.0(b.0(b.0(B.1(x)))))
C1.0(c.0(b.0(b.0(c.1(x0))))) → C1.0(c.0(b.0(b.0(b.0(a.0(a.0(a.1(x0))))))))
C1.0(c.0(b.0(b.0(c.0(b.0(b.0(c.0(y_4)))))))) → C1.0(c.0(b.0(b.0(c.0(y_4)))))
C1.0(c.0(a.0(b.0(b.0(B.0(x)))))) → C1.0(c.0(b.0(b.0(b.0(B.0(x))))))
C1.0(c.0(b.0(b.0(c.0(b.0(b.0(c.1(y_4)))))))) → C1.0(c.0(b.0(b.0(c.1(y_4)))))
C1.0(c.0(b.0(b.1(C.1(x0))))) → C1.0(c.0(b.0(b.0(B.1(x0)))))
C1.0(c.0(b.0(b.0(c.0(a.0(b.0(b.0(B.1(x0))))))))) → C1.0(c.0(a.0(a.0(a.0(a.0(c.0(c.0(c.0(b.0(b.0(B.1(x0))))))))))))
C1.0(c.0(b.0(a.0(b.0(b.0(B.1(x))))))) → C1.0(c.0(b.0(b.0(b.0(b.0(B.1(x)))))))
C1.0(c.0(b.0(b.0(c.0(a.0(b.0(b.0(B.0(x0))))))))) → C1.0(c.0(a.0(a.0(a.0(a.0(c.0(c.0(c.0(b.0(b.0(b.0(b.0(b.0(B.0(x0)))))))))))))))
C1.0(c.0(b.0(b.0(c.0(a.0(b.0(b.0(B.0(x0))))))))) → C1.0(c.0(a.0(a.0(a.0(a.0(c.0(c.0(c.0(b.0(B.0(x0)))))))))))
C1.0(c.0(b.0(b.0(c.0(b.0(B.1(x0))))))) → C1.0(c.0(a.0(a.0(a.0(a.0(c.0(c.1(C.1(x0)))))))))
C1.0(c.0(b.0(b.0(c.0(b.0(b.1(C.1(y_0)))))))) → C1.0(c.0(b.0(b.1(C.1(y_0)))))
C1.0(c.0(a.0(b.0(b.0(B.1(x)))))) → C1.0(c.0(b.0(b.0(b.0(b.0(B.1(x)))))))
C1.0(c.0(b.0(a.0(b.0(b.0(B.0(x))))))) → C1.0(c.0(b.0(b.0(b.0(b.0(B.0(x)))))))
C1.0(c.0(b.0(b.0(x)))) → C1.0(c.0(x))
C1.0(c.0(a.0(b.0(b.0(B.0(x)))))) → C1.0(c.0(b.0(b.0(b.0(b.0(B.0(x)))))))
The TRS R consists of the following rules:
c.1(C.1(x)) → b.0(B.1(x))
c.0(c.0(b.0(B.0(x)))) → a.0(a.0(a.0(a.0(c.1(C.0(x))))))
a.0(b.0(b.0(b.1(x)))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.1(x))))))))
a.0(b.0(b.0(a.0(b.0(b.0(B.0(x))))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.0(x)))))))))))))))
a.0(b.0(b.0(B.0(x)))) → b.0(b.0(B.0(x)))
a.0(b.0(a.0(b.0(b.0(B.1(x)))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.1(x))))))))))))))
a.0(b.0(b.0(a.0(b.0(b.0(B.1(x))))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.1(x)))))))))))))))
a.0(a.0(b.0(b.0(B.1(x))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.1(x)))))))))))))
a.0(a.0(b.0(b.0(B.0(x))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.0(x)))))))))))))
a.0(b.0(b.0(B.0(x)))) → b.0(B.0(x))
a.0(b.0(a.0(b.0(b.0(B.0(x)))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.0(x))))))))))))))
a.0(b.0(b.0(B.1(x)))) → b.0(B.1(x))
a.0(b.0(b.0(B.1(x)))) → B.1(x)
c.0(c.0(b.0(a.0(b.0(b.0(B.0(x))))))) → a.0(a.0(a.0(a.0(c.0(c.0(c.0(b.0(b.0(b.0(B.0(x)))))))))))
c.0(c.0(b.0(a.0(b.0(b.0(B.1(x))))))) → a.0(a.0(a.0(a.0(c.0(c.0(c.0(b.0(b.0(b.0(B.1(x)))))))))))
a.0(b.0(b.0(a.0(b.0(b.0(B.1(x))))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.1(x))))))))))))))
c.0(c.0(b.0(b.0(x)))) → a.0(a.0(a.0(a.0(c.0(c.0(c.0(x)))))))
c.0(c.0(a.0(b.0(b.0(B.0(x)))))) → a.0(a.0(a.0(a.0(c.0(c.0(c.0(b.0(B.0(x)))))))))
a.0(a.0(b.0(b.0(B.0(x))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.0(x))))))))))
c.0(c.0(b.0(a.0(b.0(b.0(B.0(x))))))) → a.0(a.0(a.0(a.0(c.0(c.0(c.0(b.0(b.0(b.0(b.0(b.0(B.0(x)))))))))))))
a.0(a.0(b.0(b.0(B.0(x))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.0(x))))))))))))
c.0(c.0(a.0(b.0(b.0(B.1(x)))))) → a.0(a.0(a.0(a.0(c.0(c.0(c.0(b.0(b.0(b.0(B.1(x)))))))))))
c.0(c.0(b.0(B.1(x)))) → a.0(a.0(a.0(a.0(c.0(c.1(C.1(x)))))))
c.0(c.0(b.0(a.0(b.0(b.0(B.0(x))))))) → a.0(a.0(a.0(a.0(c.0(c.0(c.0(b.0(b.0(B.0(x))))))))))
a.0(b.0(b.0(a.0(b.0(b.0(B.1(x))))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.1(x)))))))))))))
c.1(C.0(x)) → b.0(b.0(B.0(x)))
a.0(b.0(a.0(b.0(b.0(B.0(x)))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.0(x))))))))))))
c.0(c.0(b.0(b.1(x)))) → a.0(a.0(a.0(a.0(c.0(c.0(c.1(x)))))))
a.0(b.0(a.0(b.0(b.0(B.1(x)))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.1(x)))))))))))))
a.0(a.0(b.0(b.0(B.0(x))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.0(x)))))))))))
c.0(c.0(b.0(a.0(b.0(b.0(B.0(x))))))) → a.0(a.0(a.0(a.0(c.0(c.0(c.0(b.0(b.0(b.0(b.0(B.0(x))))))))))))
c.0(c.0(a.0(b.0(b.0(B.0(x)))))) → a.0(a.0(a.0(a.0(c.0(c.0(c.0(b.0(b.0(b.0(B.0(x)))))))))))
c.0(c.0(a.0(b.0(b.0(B.1(x)))))) → a.0(a.0(a.0(a.0(c.0(c.0(c.0(b.0(b.0(B.1(x))))))))))
c.0(c.0(b.0(a.0(b.0(b.0(B.1(x))))))) → a.0(a.0(a.0(a.0(c.0(c.0(c.0(b.0(b.0(b.0(b.0(b.0(B.1(x)))))))))))))
c.0(c.0(a.0(b.0(b.0(B.1(x)))))) → a.0(a.0(a.0(a.0(c.0(c.0(c.0(b.0(b.0(b.0(b.0(b.0(B.1(x)))))))))))))
a.0(a.0(b.0(b.0(B.0(x))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.0(x)))))))))
a.0(b.0(a.0(b.0(b.0(B.1(x)))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.1(x))))))))))
a.0(a.0(b.0(b.0(B.1(x))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.1(x)))))))))
c.0(c.0(b.0(a.0(b.0(b.0(B.1(x))))))) → a.0(a.0(a.0(a.0(c.0(c.0(c.0(b.0(b.0(b.0(b.0(b.0(b.0(B.1(x))))))))))))))
c.0(c.0(b.0(B.1(x)))) → a.0(a.0(a.0(a.0(c.1(C.1(x))))))
c.0(c.0(a.0(b.0(b.0(B.0(x)))))) → a.0(a.0(a.0(a.0(c.0(c.0(c.0(b.0(b.0(b.0(b.0(B.0(x))))))))))))
a.0(b.0(b.0(B.1(x)))) → b.0(b.0(B.1(x)))
a.0(b.0(a.0(b.0(b.0(B.0(x)))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.0(x)))))))))))
a.0(b.0(b.0(a.0(b.0(b.0(B.1(x))))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.1(x)))))))))))
a.0(a.0(b.0(b.0(B.1(x))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.1(x)))))))))))
a.0(b.0(b.0(a.0(b.0(b.0(B.0(x))))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.0(x))))))))))))
a.0(a.0(b.0(b.0(B.1(x))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.1(x))))))))))))
c.1(C.1(x)) → b.0(b.0(B.1(x)))
a.0(b.0(a.0(b.0(b.0(B.1(x)))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.1(x))))))))))))
a.0(b.0(b.0(a.0(b.0(b.0(B.0(x))))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.0(x))))))))))))))
a.0(b.0(b.0(a.0(b.0(b.0(B.0(x))))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.0(x)))))))))))))
a.0(a.0(b.0(b.0(B.1(x))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.1(x))))))))))
a.0(b.0(b.0(B.0(x)))) → B.0(x)
c.0(c.0(b.0(B.0(x)))) → a.0(a.0(a.0(a.0(c.0(c.1(C.0(x)))))))
a.0(b.0(b.0(b.0(x)))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(x))))))))
a.0(b.0(a.0(b.0(b.0(B.0(x)))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.0(x))))))))))
c.0(c.1(x)) → b.0(b.0(b.0(a.0(a.0(a.1(x))))))
a.0(b.0(b.0(a.0(b.0(b.0(B.1(x))))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.1(x))))))))))))
c.0(c.0(a.0(b.0(b.0(B.1(x)))))) → a.0(a.0(a.0(a.0(c.0(c.0(c.0(b.0(b.0(b.0(b.0(B.1(x))))))))))))
c.0(c.0(b.0(a.0(b.0(b.0(B.0(x))))))) → a.0(a.0(a.0(a.0(c.0(c.0(c.0(b.0(b.0(b.0(b.0(b.0(b.0(B.0(x))))))))))))))
c.0(c.0(a.0(b.0(b.0(B.1(x)))))) → a.0(a.0(a.0(a.0(c.0(c.0(c.0(b.0(B.1(x)))))))))
c.0(c.0(b.0(a.0(b.0(b.0(B.1(x))))))) → a.0(a.0(a.0(a.0(c.0(c.0(c.0(b.0(b.0(B.1(x))))))))))
a.0(b.0(b.0(a.0(b.0(b.0(B.0(x))))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.0(x)))))))))))
c.0(c.0(a.0(b.0(b.0(B.0(x)))))) → a.0(a.0(a.0(a.0(c.0(c.0(c.0(b.0(b.0(B.0(x))))))))))
c.0(c.0(a.0(b.0(b.0(B.0(x)))))) → a.0(a.0(a.0(a.0(c.0(c.0(c.0(b.0(b.0(b.0(b.0(b.0(B.0(x)))))))))))))
c.0(c.0(x)) → b.0(b.0(b.0(a.0(a.0(a.0(x))))))
a.0(b.0(a.0(b.0(b.0(B.0(x)))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.0(x)))))))))))))
c.1(C.0(x)) → b.0(B.0(x))
a.0(b.0(a.0(b.0(b.0(B.1(x)))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.1(x)))))))))))
c.0(c.0(b.0(a.0(b.0(b.0(B.1(x))))))) → a.0(a.0(a.0(a.0(c.0(c.0(c.0(b.0(b.0(b.0(b.0(B.1(x))))))))))))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
↳ QTRS
↳ QTRS Reverse
↳ QTRS Reverse
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ SemLabProof
↳ SemLabProof2
↳ QDP
↳ QDPToSRSProof
↳ QTRS
↳ QTRS Reverse
↳ QTRS
↳ QTRS Reverse
↳ QTRS Reverse
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ SemLabProof
↳ QDP
↳ DependencyGraphProof
↳ SemLabProof2
Q DP problem:
The TRS P consists of the following rules:
C1.0(c.0(b.0(a.0(b.0(b.0(B.1(x))))))) → C1.0(c.0(b.0(b.0(b.0(B.1(x))))))
C1.0(c.0(b.0(a.0(b.0(b.0(B.1(x))))))) → C1.0(c.0(b.0(b.0(b.0(b.0(b.0(b.0(B.1(x)))))))))
C1.0(c.0(b.0(b.0(c.0(b.0(a.0(b.0(b.0(B.0(x0)))))))))) → C1.0(c.0(a.0(a.0(a.0(a.0(c.0(c.0(c.0(b.0(b.0(b.0(b.0(B.0(x0))))))))))))))
C1.0(c.0(b.0(b.0(c.1(y_2))))) → C1.0(c.1(y_2))
C1.0(c.0(b.0(b.1(C.0(x0))))) → C1.0(c.0(b.0(b.0(B.0(x0)))))
C1.0(c.0(b.0(b.0(c.0(b.0(B.1(x0))))))) → C1.0(c.0(a.0(a.0(a.0(a.0(c.1(C.1(x0))))))))
C1.0(c.0(b.0(b.0(c.0(a.0(b.0(b.0(B.1(x0))))))))) → C1.0(c.0(a.0(a.0(a.0(a.0(c.0(c.0(c.0(b.0(b.0(b.0(b.0(B.1(x0))))))))))))))
C1.0(c.0(b.0(b.0(c.0(b.0(a.0(b.0(b.0(B.1(x0)))))))))) → C1.0(c.0(a.0(a.0(a.0(a.0(c.0(c.0(c.0(b.0(b.0(b.0(b.0(b.0(b.0(B.1(x0))))))))))))))))
C1.0(c.0(b.0(a.0(b.0(b.0(B.0(x))))))) → C1.0(c.0(b.0(b.0(B.0(x)))))
C1.0(c.0(b.0(b.0(c.0(b.0(b.1(x0))))))) → C1.0(c.0(a.0(a.0(a.0(a.0(c.0(c.0(c.1(x0)))))))))
C1.0(c.0(b.0(a.0(b.0(b.0(B.1(x))))))) → C1.0(c.0(b.0(b.0(b.0(b.0(b.0(B.1(x))))))))
C1.0(c.0(a.0(b.0(b.0(B.0(x)))))) → C1.0(c.0(b.0(b.0(b.0(b.0(b.0(B.0(x))))))))
C1.0(c.0(b.0(b.0(c.0(x0))))) → C1.0(c.0(b.0(b.0(b.0(a.0(a.0(a.0(x0))))))))
C1.0(c.0(b.0(b.0(c.0(a.0(b.0(b.0(B.0(x0))))))))) → C1.0(c.0(a.0(a.0(a.0(a.0(c.0(c.0(c.0(b.0(b.0(B.0(x0))))))))))))
C1.0(c.0(b.0(b.0(c.0(b.0(b.1(C.0(y_0)))))))) → C1.0(c.0(b.0(b.1(C.0(y_0)))))
C1.0(c.0(b.0(b.0(c.0(y_2))))) → C1.0(c.0(y_2))
C1.0(c.0(b.0(b.0(c.0(a.0(b.0(b.0(B.1(x0))))))))) → C1.0(c.0(a.0(a.0(a.0(a.0(c.0(c.0(c.0(b.0(b.0(b.0(B.1(x0)))))))))))))
C1.0(c.0(b.0(b.0(c.0(b.0(a.0(b.0(b.0(B.1(x0)))))))))) → C1.0(c.0(a.0(a.0(a.0(a.0(c.0(c.0(c.0(b.0(b.0(B.1(x0))))))))))))
C1.0(c.0(a.0(b.0(b.0(B.1(x)))))) → C1.0(c.0(b.0(b.0(b.0(B.1(x))))))
C1.0(c.0(a.0(b.0(b.0(B.0(x)))))) → C1.0(c.0(b.0(b.0(B.0(x)))))
C1.0(c.0(b.0(a.0(b.0(b.0(B.1(x))))))) → C1.0(c.0(b.0(b.0(B.1(x)))))
C1.0(c.0(b.0(b.0(c.0(b.0(B.0(x0))))))) → C1.0(c.0(a.0(a.0(a.0(a.0(c.0(c.1(C.0(x0)))))))))
C1.0(c.0(b.0(a.0(b.0(b.0(B.0(x))))))) → C1.0(c.0(b.0(b.0(b.0(b.0(b.0(B.0(x))))))))
C1.0(c.0(b.0(b.0(c.0(b.0(a.0(b.0(b.0(B.0(x0)))))))))) → C1.0(c.0(a.0(a.0(a.0(a.0(c.0(c.0(c.0(b.0(b.0(b.0(b.0(b.0(B.0(x0)))))))))))))))
C1.0(c.0(b.0(b.0(c.0(b.0(a.0(b.0(b.0(B.0(x0)))))))))) → C1.0(c.0(a.0(a.0(a.0(a.0(c.0(c.0(c.0(b.0(b.0(B.0(x0))))))))))))
C1.0(c.0(b.0(b.0(c.0(b.0(a.0(b.0(b.0(B.1(x0)))))))))) → C1.0(c.0(a.0(a.0(a.0(a.0(c.0(c.0(c.0(b.0(b.0(b.0(B.1(x0)))))))))))))
C1.0(c.0(b.0(b.0(c.0(a.0(b.0(b.0(B.0(x0))))))))) → C1.0(c.0(a.0(a.0(a.0(a.0(c.0(c.0(c.0(b.0(b.0(b.0(B.0(x0)))))))))))))
C1.0(c.0(b.0(a.0(b.0(b.0(B.0(x))))))) → C1.0(c.0(b.0(b.0(b.0(B.0(x))))))
C1.0(c.0(b.0(b.0(c.0(b.0(a.0(b.0(b.0(B.0(x0)))))))))) → C1.0(c.0(a.0(a.0(a.0(a.0(c.0(c.0(c.0(b.0(b.0(b.0(B.0(x0)))))))))))))
C1.0(c.0(a.0(b.0(b.0(B.1(x)))))) → C1.0(c.0(b.0(b.0(b.0(b.0(b.0(B.1(x))))))))
C1.0(c.0(b.0(b.0(c.0(b.0(a.0(b.0(b.0(B.0(x0)))))))))) → C1.0(c.0(a.0(a.0(a.0(a.0(c.0(c.0(c.0(b.0(b.0(b.0(b.0(b.0(b.0(B.0(x0))))))))))))))))
C1.0(c.0(b.0(b.1(x)))) → C1.0(c.1(x))
C1.0(c.0(b.0(b.0(c.0(a.0(b.0(b.0(B.1(x0))))))))) → C1.0(c.0(a.0(a.0(a.0(a.0(c.0(c.0(c.0(b.0(B.1(x0)))))))))))
C1.0(c.0(b.0(b.0(c.0(b.0(b.0(x0))))))) → C1.0(c.0(a.0(a.0(a.0(a.0(c.0(c.0(c.0(x0)))))))))
C1.0(c.0(b.0(b.0(c.0(b.0(a.0(b.0(b.0(B.1(x0)))))))))) → C1.0(c.0(a.0(a.0(a.0(a.0(c.0(c.0(c.0(b.0(b.0(b.0(b.0(B.1(x0))))))))))))))
C1.0(c.0(b.0(b.0(c.0(a.0(b.0(b.0(B.0(x0))))))))) → C1.0(c.0(a.0(a.0(a.0(a.0(c.0(c.0(c.0(b.0(b.0(b.0(b.0(B.0(x0))))))))))))))
C1.0(c.0(b.0(b.0(c.0(b.0(a.0(b.0(b.0(B.1(x0)))))))))) → C1.0(c.0(a.0(a.0(a.0(a.0(c.0(c.0(c.0(b.0(b.0(b.0(b.0(b.0(B.1(x0)))))))))))))))
C1.0(c.0(b.0(b.0(c.0(a.0(b.0(b.0(B.1(x0))))))))) → C1.0(c.0(a.0(a.0(a.0(a.0(c.0(c.0(c.0(b.0(b.0(b.0(b.0(b.0(B.1(x0)))))))))))))))
C1.0(c.0(b.0(b.0(c.0(b.0(B.0(x0))))))) → C1.0(c.0(a.0(a.0(a.0(a.0(c.1(C.0(x0))))))))
C1.0(c.0(b.0(a.0(b.0(b.0(B.0(x))))))) → C1.0(c.0(b.0(b.0(b.0(b.0(b.0(b.0(B.0(x)))))))))
C1.0(c.0(a.0(b.0(b.0(B.1(x)))))) → C1.0(c.0(b.0(b.0(B.1(x)))))
C1.0(c.0(b.0(b.0(c.1(x0))))) → C1.0(c.0(b.0(b.0(b.0(a.0(a.0(a.1(x0))))))))
C1.0(c.0(b.0(b.0(c.0(b.0(b.0(c.0(y_4)))))))) → C1.0(c.0(b.0(b.0(c.0(y_4)))))
C1.0(c.0(a.0(b.0(b.0(B.0(x)))))) → C1.0(c.0(b.0(b.0(b.0(B.0(x))))))
C1.0(c.0(b.0(b.0(c.0(b.0(b.0(c.1(y_4)))))))) → C1.0(c.0(b.0(b.0(c.1(y_4)))))
C1.0(c.0(b.0(b.1(C.1(x0))))) → C1.0(c.0(b.0(b.0(B.1(x0)))))
C1.0(c.0(b.0(b.0(c.0(a.0(b.0(b.0(B.1(x0))))))))) → C1.0(c.0(a.0(a.0(a.0(a.0(c.0(c.0(c.0(b.0(b.0(B.1(x0))))))))))))
C1.0(c.0(b.0(a.0(b.0(b.0(B.1(x))))))) → C1.0(c.0(b.0(b.0(b.0(b.0(B.1(x)))))))
C1.0(c.0(b.0(b.0(c.0(a.0(b.0(b.0(B.0(x0))))))))) → C1.0(c.0(a.0(a.0(a.0(a.0(c.0(c.0(c.0(b.0(b.0(b.0(b.0(b.0(B.0(x0)))))))))))))))
C1.0(c.0(b.0(b.0(c.0(a.0(b.0(b.0(B.0(x0))))))))) → C1.0(c.0(a.0(a.0(a.0(a.0(c.0(c.0(c.0(b.0(B.0(x0)))))))))))
C1.0(c.0(b.0(b.0(c.0(b.0(B.1(x0))))))) → C1.0(c.0(a.0(a.0(a.0(a.0(c.0(c.1(C.1(x0)))))))))
C1.0(c.0(b.0(b.0(c.0(b.0(b.1(C.1(y_0)))))))) → C1.0(c.0(b.0(b.1(C.1(y_0)))))
C1.0(c.0(a.0(b.0(b.0(B.1(x)))))) → C1.0(c.0(b.0(b.0(b.0(b.0(B.1(x)))))))
C1.0(c.0(b.0(a.0(b.0(b.0(B.0(x))))))) → C1.0(c.0(b.0(b.0(b.0(b.0(B.0(x)))))))
C1.0(c.0(b.0(b.0(x)))) → C1.0(c.0(x))
C1.0(c.0(a.0(b.0(b.0(B.0(x)))))) → C1.0(c.0(b.0(b.0(b.0(b.0(B.0(x)))))))
The TRS R consists of the following rules:
c.1(C.1(x)) → b.0(B.1(x))
c.0(c.0(b.0(B.0(x)))) → a.0(a.0(a.0(a.0(c.1(C.0(x))))))
a.0(b.0(b.0(b.1(x)))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.1(x))))))))
a.0(b.0(b.0(a.0(b.0(b.0(B.0(x))))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.0(x)))))))))))))))
a.0(b.0(b.0(B.0(x)))) → b.0(b.0(B.0(x)))
a.0(b.0(a.0(b.0(b.0(B.1(x)))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.1(x))))))))))))))
a.0(b.0(b.0(a.0(b.0(b.0(B.1(x))))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.1(x)))))))))))))))
a.0(a.0(b.0(b.0(B.1(x))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.1(x)))))))))))))
a.0(a.0(b.0(b.0(B.0(x))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.0(x)))))))))))))
a.0(b.0(b.0(B.0(x)))) → b.0(B.0(x))
a.0(b.0(a.0(b.0(b.0(B.0(x)))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.0(x))))))))))))))
a.0(b.0(b.0(B.1(x)))) → b.0(B.1(x))
a.0(b.0(b.0(B.1(x)))) → B.1(x)
c.0(c.0(b.0(a.0(b.0(b.0(B.0(x))))))) → a.0(a.0(a.0(a.0(c.0(c.0(c.0(b.0(b.0(b.0(B.0(x)))))))))))
c.0(c.0(b.0(a.0(b.0(b.0(B.1(x))))))) → a.0(a.0(a.0(a.0(c.0(c.0(c.0(b.0(b.0(b.0(B.1(x)))))))))))
a.0(b.0(b.0(a.0(b.0(b.0(B.1(x))))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.1(x))))))))))))))
c.0(c.0(b.0(b.0(x)))) → a.0(a.0(a.0(a.0(c.0(c.0(c.0(x)))))))
c.0(c.0(a.0(b.0(b.0(B.0(x)))))) → a.0(a.0(a.0(a.0(c.0(c.0(c.0(b.0(B.0(x)))))))))
a.0(a.0(b.0(b.0(B.0(x))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.0(x))))))))))
c.0(c.0(b.0(a.0(b.0(b.0(B.0(x))))))) → a.0(a.0(a.0(a.0(c.0(c.0(c.0(b.0(b.0(b.0(b.0(b.0(B.0(x)))))))))))))
a.0(a.0(b.0(b.0(B.0(x))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.0(x))))))))))))
c.0(c.0(a.0(b.0(b.0(B.1(x)))))) → a.0(a.0(a.0(a.0(c.0(c.0(c.0(b.0(b.0(b.0(B.1(x)))))))))))
c.0(c.0(b.0(B.1(x)))) → a.0(a.0(a.0(a.0(c.0(c.1(C.1(x)))))))
c.0(c.0(b.0(a.0(b.0(b.0(B.0(x))))))) → a.0(a.0(a.0(a.0(c.0(c.0(c.0(b.0(b.0(B.0(x))))))))))
a.0(b.0(b.0(a.0(b.0(b.0(B.1(x))))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.1(x)))))))))))))
c.1(C.0(x)) → b.0(b.0(B.0(x)))
a.0(b.0(a.0(b.0(b.0(B.0(x)))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.0(x))))))))))))
c.0(c.0(b.0(b.1(x)))) → a.0(a.0(a.0(a.0(c.0(c.0(c.1(x)))))))
a.0(b.0(a.0(b.0(b.0(B.1(x)))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.1(x)))))))))))))
a.0(a.0(b.0(b.0(B.0(x))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.0(x)))))))))))
c.0(c.0(b.0(a.0(b.0(b.0(B.0(x))))))) → a.0(a.0(a.0(a.0(c.0(c.0(c.0(b.0(b.0(b.0(b.0(B.0(x))))))))))))
c.0(c.0(a.0(b.0(b.0(B.0(x)))))) → a.0(a.0(a.0(a.0(c.0(c.0(c.0(b.0(b.0(b.0(B.0(x)))))))))))
c.0(c.0(a.0(b.0(b.0(B.1(x)))))) → a.0(a.0(a.0(a.0(c.0(c.0(c.0(b.0(b.0(B.1(x))))))))))
c.0(c.0(b.0(a.0(b.0(b.0(B.1(x))))))) → a.0(a.0(a.0(a.0(c.0(c.0(c.0(b.0(b.0(b.0(b.0(b.0(B.1(x)))))))))))))
c.0(c.0(a.0(b.0(b.0(B.1(x)))))) → a.0(a.0(a.0(a.0(c.0(c.0(c.0(b.0(b.0(b.0(b.0(b.0(B.1(x)))))))))))))
a.0(a.0(b.0(b.0(B.0(x))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.0(x)))))))))
a.0(b.0(a.0(b.0(b.0(B.1(x)))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.1(x))))))))))
a.0(a.0(b.0(b.0(B.1(x))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.1(x)))))))))
c.0(c.0(b.0(a.0(b.0(b.0(B.1(x))))))) → a.0(a.0(a.0(a.0(c.0(c.0(c.0(b.0(b.0(b.0(b.0(b.0(b.0(B.1(x))))))))))))))
c.0(c.0(b.0(B.1(x)))) → a.0(a.0(a.0(a.0(c.1(C.1(x))))))
c.0(c.0(a.0(b.0(b.0(B.0(x)))))) → a.0(a.0(a.0(a.0(c.0(c.0(c.0(b.0(b.0(b.0(b.0(B.0(x))))))))))))
a.0(b.0(b.0(B.1(x)))) → b.0(b.0(B.1(x)))
a.0(b.0(a.0(b.0(b.0(B.0(x)))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.0(x)))))))))))
a.0(b.0(b.0(a.0(b.0(b.0(B.1(x))))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.1(x)))))))))))
a.0(a.0(b.0(b.0(B.1(x))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.1(x)))))))))))
a.0(b.0(b.0(a.0(b.0(b.0(B.0(x))))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.0(x))))))))))))
a.0(a.0(b.0(b.0(B.1(x))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.1(x))))))))))))
c.1(C.1(x)) → b.0(b.0(B.1(x)))
a.0(b.0(a.0(b.0(b.0(B.1(x)))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.1(x))))))))))))
a.0(b.0(b.0(a.0(b.0(b.0(B.0(x))))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.0(x))))))))))))))
a.0(b.0(b.0(a.0(b.0(b.0(B.0(x))))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.0(x)))))))))))))
a.0(a.0(b.0(b.0(B.1(x))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.1(x))))))))))
a.0(b.0(b.0(B.0(x)))) → B.0(x)
c.0(c.0(b.0(B.0(x)))) → a.0(a.0(a.0(a.0(c.0(c.1(C.0(x)))))))
a.0(b.0(b.0(b.0(x)))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(x))))))))
a.0(b.0(a.0(b.0(b.0(B.0(x)))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.0(x))))))))))
c.0(c.1(x)) → b.0(b.0(b.0(a.0(a.0(a.1(x))))))
a.0(b.0(b.0(a.0(b.0(b.0(B.1(x))))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.1(x))))))))))))
c.0(c.0(a.0(b.0(b.0(B.1(x)))))) → a.0(a.0(a.0(a.0(c.0(c.0(c.0(b.0(b.0(b.0(b.0(B.1(x))))))))))))
c.0(c.0(b.0(a.0(b.0(b.0(B.0(x))))))) → a.0(a.0(a.0(a.0(c.0(c.0(c.0(b.0(b.0(b.0(b.0(b.0(b.0(B.0(x))))))))))))))
c.0(c.0(a.0(b.0(b.0(B.1(x)))))) → a.0(a.0(a.0(a.0(c.0(c.0(c.0(b.0(B.1(x)))))))))
c.0(c.0(b.0(a.0(b.0(b.0(B.1(x))))))) → a.0(a.0(a.0(a.0(c.0(c.0(c.0(b.0(b.0(B.1(x))))))))))
a.0(b.0(b.0(a.0(b.0(b.0(B.0(x))))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.0(x)))))))))))
c.0(c.0(a.0(b.0(b.0(B.0(x)))))) → a.0(a.0(a.0(a.0(c.0(c.0(c.0(b.0(b.0(B.0(x))))))))))
c.0(c.0(a.0(b.0(b.0(B.0(x)))))) → a.0(a.0(a.0(a.0(c.0(c.0(c.0(b.0(b.0(b.0(b.0(b.0(B.0(x)))))))))))))
c.0(c.0(x)) → b.0(b.0(b.0(a.0(a.0(a.0(x))))))
a.0(b.0(a.0(b.0(b.0(B.0(x)))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.0(x)))))))))))))
c.1(C.0(x)) → b.0(B.0(x))
a.0(b.0(a.0(b.0(b.0(B.1(x)))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.1(x)))))))))))
c.0(c.0(b.0(a.0(b.0(b.0(B.1(x))))))) → a.0(a.0(a.0(a.0(c.0(c.0(c.0(b.0(b.0(b.0(b.0(B.1(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
↳ QTRS Reverse
↳ QTRS Reverse
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ SemLabProof
↳ SemLabProof2
↳ QDP
↳ QDPToSRSProof
↳ QTRS
↳ QTRS Reverse
↳ QTRS
↳ QTRS Reverse
↳ QTRS Reverse
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ SemLabProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ RuleRemovalProof
↳ SemLabProof2
Q DP problem:
The TRS P consists of the following rules:
C1.0(c.0(b.0(a.0(b.0(b.0(B.1(x))))))) → C1.0(c.0(b.0(b.0(b.0(B.1(x))))))
C1.0(c.0(b.0(a.0(b.0(b.0(B.1(x))))))) → C1.0(c.0(b.0(b.0(b.0(b.0(b.0(b.0(B.1(x)))))))))
C1.0(c.0(b.0(b.0(c.0(b.0(a.0(b.0(b.0(B.0(x0)))))))))) → C1.0(c.0(a.0(a.0(a.0(a.0(c.0(c.0(c.0(b.0(b.0(b.0(b.0(B.0(x0))))))))))))))
C1.0(c.0(b.0(b.1(C.0(x0))))) → C1.0(c.0(b.0(b.0(B.0(x0)))))
C1.0(c.0(b.0(b.0(c.0(b.0(B.1(x0))))))) → C1.0(c.0(a.0(a.0(a.0(a.0(c.1(C.1(x0))))))))
C1.0(c.0(b.0(b.0(c.0(a.0(b.0(b.0(B.1(x0))))))))) → C1.0(c.0(a.0(a.0(a.0(a.0(c.0(c.0(c.0(b.0(b.0(b.0(b.0(B.1(x0))))))))))))))
C1.0(c.0(b.0(b.0(c.0(b.0(a.0(b.0(b.0(B.1(x0)))))))))) → C1.0(c.0(a.0(a.0(a.0(a.0(c.0(c.0(c.0(b.0(b.0(b.0(b.0(b.0(b.0(B.1(x0))))))))))))))))
C1.0(c.0(b.0(a.0(b.0(b.0(B.0(x))))))) → C1.0(c.0(b.0(b.0(B.0(x)))))
C1.0(c.0(b.0(b.0(c.0(b.0(b.1(x0))))))) → C1.0(c.0(a.0(a.0(a.0(a.0(c.0(c.0(c.1(x0)))))))))
C1.0(c.0(b.0(a.0(b.0(b.0(B.1(x))))))) → C1.0(c.0(b.0(b.0(b.0(b.0(b.0(B.1(x))))))))
C1.0(c.0(a.0(b.0(b.0(B.0(x)))))) → C1.0(c.0(b.0(b.0(b.0(b.0(b.0(B.0(x))))))))
C1.0(c.0(b.0(b.0(c.0(x0))))) → C1.0(c.0(b.0(b.0(b.0(a.0(a.0(a.0(x0))))))))
C1.0(c.0(b.0(b.0(c.0(a.0(b.0(b.0(B.0(x0))))))))) → C1.0(c.0(a.0(a.0(a.0(a.0(c.0(c.0(c.0(b.0(b.0(B.0(x0))))))))))))
C1.0(c.0(b.0(b.0(c.0(b.0(b.1(C.0(y_0)))))))) → C1.0(c.0(b.0(b.1(C.0(y_0)))))
C1.0(c.0(b.0(b.0(c.0(y_2))))) → C1.0(c.0(y_2))
C1.0(c.0(b.0(b.0(c.0(b.0(a.0(b.0(b.0(B.1(x0)))))))))) → C1.0(c.0(a.0(a.0(a.0(a.0(c.0(c.0(c.0(b.0(b.0(B.1(x0))))))))))))
C1.0(c.0(b.0(b.0(c.0(a.0(b.0(b.0(B.1(x0))))))))) → C1.0(c.0(a.0(a.0(a.0(a.0(c.0(c.0(c.0(b.0(b.0(b.0(B.1(x0)))))))))))))
C1.0(c.0(a.0(b.0(b.0(B.1(x)))))) → C1.0(c.0(b.0(b.0(b.0(B.1(x))))))
C1.0(c.0(a.0(b.0(b.0(B.0(x)))))) → C1.0(c.0(b.0(b.0(B.0(x)))))
C1.0(c.0(b.0(a.0(b.0(b.0(B.1(x))))))) → C1.0(c.0(b.0(b.0(B.1(x)))))
C1.0(c.0(b.0(b.0(c.0(b.0(B.0(x0))))))) → C1.0(c.0(a.0(a.0(a.0(a.0(c.0(c.1(C.0(x0)))))))))
C1.0(c.0(b.0(a.0(b.0(b.0(B.0(x))))))) → C1.0(c.0(b.0(b.0(b.0(b.0(b.0(B.0(x))))))))
C1.0(c.0(b.0(b.0(c.0(b.0(a.0(b.0(b.0(B.0(x0)))))))))) → C1.0(c.0(a.0(a.0(a.0(a.0(c.0(c.0(c.0(b.0(b.0(b.0(b.0(b.0(B.0(x0)))))))))))))))
C1.0(c.0(b.0(b.0(c.0(b.0(a.0(b.0(b.0(B.0(x0)))))))))) → C1.0(c.0(a.0(a.0(a.0(a.0(c.0(c.0(c.0(b.0(b.0(B.0(x0))))))))))))
C1.0(c.0(b.0(b.0(c.0(b.0(a.0(b.0(b.0(B.1(x0)))))))))) → C1.0(c.0(a.0(a.0(a.0(a.0(c.0(c.0(c.0(b.0(b.0(b.0(B.1(x0)))))))))))))
C1.0(c.0(b.0(b.0(c.0(a.0(b.0(b.0(B.0(x0))))))))) → C1.0(c.0(a.0(a.0(a.0(a.0(c.0(c.0(c.0(b.0(b.0(b.0(B.0(x0)))))))))))))
C1.0(c.0(b.0(a.0(b.0(b.0(B.0(x))))))) → C1.0(c.0(b.0(b.0(b.0(B.0(x))))))
C1.0(c.0(b.0(b.0(c.0(b.0(a.0(b.0(b.0(B.0(x0)))))))))) → C1.0(c.0(a.0(a.0(a.0(a.0(c.0(c.0(c.0(b.0(b.0(b.0(B.0(x0)))))))))))))
C1.0(c.0(a.0(b.0(b.0(B.1(x)))))) → C1.0(c.0(b.0(b.0(b.0(b.0(b.0(B.1(x))))))))
C1.0(c.0(b.0(b.0(c.0(b.0(a.0(b.0(b.0(B.0(x0)))))))))) → C1.0(c.0(a.0(a.0(a.0(a.0(c.0(c.0(c.0(b.0(b.0(b.0(b.0(b.0(b.0(B.0(x0))))))))))))))))
C1.0(c.0(b.0(b.0(c.0(a.0(b.0(b.0(B.1(x0))))))))) → C1.0(c.0(a.0(a.0(a.0(a.0(c.0(c.0(c.0(b.0(B.1(x0)))))))))))
C1.0(c.0(b.0(b.0(c.0(b.0(b.0(x0))))))) → C1.0(c.0(a.0(a.0(a.0(a.0(c.0(c.0(c.0(x0)))))))))
C1.0(c.0(b.0(b.0(c.0(b.0(a.0(b.0(b.0(B.1(x0)))))))))) → C1.0(c.0(a.0(a.0(a.0(a.0(c.0(c.0(c.0(b.0(b.0(b.0(b.0(B.1(x0))))))))))))))
C1.0(c.0(b.0(b.0(c.0(a.0(b.0(b.0(B.0(x0))))))))) → C1.0(c.0(a.0(a.0(a.0(a.0(c.0(c.0(c.0(b.0(b.0(b.0(b.0(B.0(x0))))))))))))))
C1.0(c.0(b.0(b.0(c.0(b.0(a.0(b.0(b.0(B.1(x0)))))))))) → C1.0(c.0(a.0(a.0(a.0(a.0(c.0(c.0(c.0(b.0(b.0(b.0(b.0(b.0(B.1(x0)))))))))))))))
C1.0(c.0(b.0(b.0(c.0(a.0(b.0(b.0(B.1(x0))))))))) → C1.0(c.0(a.0(a.0(a.0(a.0(c.0(c.0(c.0(b.0(b.0(b.0(b.0(b.0(B.1(x0)))))))))))))))
C1.0(c.0(b.0(b.0(c.0(b.0(B.0(x0))))))) → C1.0(c.0(a.0(a.0(a.0(a.0(c.1(C.0(x0))))))))
C1.0(c.0(b.0(a.0(b.0(b.0(B.0(x))))))) → C1.0(c.0(b.0(b.0(b.0(b.0(b.0(b.0(B.0(x)))))))))
C1.0(c.0(a.0(b.0(b.0(B.1(x)))))) → C1.0(c.0(b.0(b.0(B.1(x)))))
C1.0(c.0(b.0(b.0(c.1(x0))))) → C1.0(c.0(b.0(b.0(b.0(a.0(a.0(a.1(x0))))))))
C1.0(c.0(b.0(b.0(c.0(b.0(b.0(c.0(y_4)))))))) → C1.0(c.0(b.0(b.0(c.0(y_4)))))
C1.0(c.0(a.0(b.0(b.0(B.0(x)))))) → C1.0(c.0(b.0(b.0(b.0(B.0(x))))))
C1.0(c.0(b.0(b.0(c.0(b.0(b.0(c.1(y_4)))))))) → C1.0(c.0(b.0(b.0(c.1(y_4)))))
C1.0(c.0(b.0(b.1(C.1(x0))))) → C1.0(c.0(b.0(b.0(B.1(x0)))))
C1.0(c.0(b.0(b.0(c.0(a.0(b.0(b.0(B.1(x0))))))))) → C1.0(c.0(a.0(a.0(a.0(a.0(c.0(c.0(c.0(b.0(b.0(B.1(x0))))))))))))
C1.0(c.0(b.0(a.0(b.0(b.0(B.1(x))))))) → C1.0(c.0(b.0(b.0(b.0(b.0(B.1(x)))))))
C1.0(c.0(b.0(b.0(c.0(a.0(b.0(b.0(B.0(x0))))))))) → C1.0(c.0(a.0(a.0(a.0(a.0(c.0(c.0(c.0(b.0(b.0(b.0(b.0(b.0(B.0(x0)))))))))))))))
C1.0(c.0(b.0(b.0(c.0(a.0(b.0(b.0(B.0(x0))))))))) → C1.0(c.0(a.0(a.0(a.0(a.0(c.0(c.0(c.0(b.0(B.0(x0)))))))))))
C1.0(c.0(b.0(b.0(c.0(b.0(B.1(x0))))))) → C1.0(c.0(a.0(a.0(a.0(a.0(c.0(c.1(C.1(x0)))))))))
C1.0(c.0(b.0(b.0(c.0(b.0(b.1(C.1(y_0)))))))) → C1.0(c.0(b.0(b.1(C.1(y_0)))))
C1.0(c.0(b.0(a.0(b.0(b.0(B.0(x))))))) → C1.0(c.0(b.0(b.0(b.0(b.0(B.0(x)))))))
C1.0(c.0(a.0(b.0(b.0(B.1(x)))))) → C1.0(c.0(b.0(b.0(b.0(b.0(B.1(x)))))))
C1.0(c.0(b.0(b.0(x)))) → C1.0(c.0(x))
C1.0(c.0(a.0(b.0(b.0(B.0(x)))))) → C1.0(c.0(b.0(b.0(b.0(b.0(B.0(x)))))))
The TRS R consists of the following rules:
c.1(C.1(x)) → b.0(B.1(x))
c.0(c.0(b.0(B.0(x)))) → a.0(a.0(a.0(a.0(c.1(C.0(x))))))
a.0(b.0(b.0(b.1(x)))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.1(x))))))))
a.0(b.0(b.0(a.0(b.0(b.0(B.0(x))))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.0(x)))))))))))))))
a.0(b.0(b.0(B.0(x)))) → b.0(b.0(B.0(x)))
a.0(b.0(a.0(b.0(b.0(B.1(x)))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.1(x))))))))))))))
a.0(b.0(b.0(a.0(b.0(b.0(B.1(x))))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.1(x)))))))))))))))
a.0(a.0(b.0(b.0(B.1(x))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.1(x)))))))))))))
a.0(a.0(b.0(b.0(B.0(x))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.0(x)))))))))))))
a.0(b.0(b.0(B.0(x)))) → b.0(B.0(x))
a.0(b.0(a.0(b.0(b.0(B.0(x)))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.0(x))))))))))))))
a.0(b.0(b.0(B.1(x)))) → b.0(B.1(x))
a.0(b.0(b.0(B.1(x)))) → B.1(x)
c.0(c.0(b.0(a.0(b.0(b.0(B.0(x))))))) → a.0(a.0(a.0(a.0(c.0(c.0(c.0(b.0(b.0(b.0(B.0(x)))))))))))
c.0(c.0(b.0(a.0(b.0(b.0(B.1(x))))))) → a.0(a.0(a.0(a.0(c.0(c.0(c.0(b.0(b.0(b.0(B.1(x)))))))))))
a.0(b.0(b.0(a.0(b.0(b.0(B.1(x))))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.1(x))))))))))))))
c.0(c.0(b.0(b.0(x)))) → a.0(a.0(a.0(a.0(c.0(c.0(c.0(x)))))))
c.0(c.0(a.0(b.0(b.0(B.0(x)))))) → a.0(a.0(a.0(a.0(c.0(c.0(c.0(b.0(B.0(x)))))))))
a.0(a.0(b.0(b.0(B.0(x))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.0(x))))))))))
c.0(c.0(b.0(a.0(b.0(b.0(B.0(x))))))) → a.0(a.0(a.0(a.0(c.0(c.0(c.0(b.0(b.0(b.0(b.0(b.0(B.0(x)))))))))))))
a.0(a.0(b.0(b.0(B.0(x))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.0(x))))))))))))
c.0(c.0(a.0(b.0(b.0(B.1(x)))))) → a.0(a.0(a.0(a.0(c.0(c.0(c.0(b.0(b.0(b.0(B.1(x)))))))))))
c.0(c.0(b.0(B.1(x)))) → a.0(a.0(a.0(a.0(c.0(c.1(C.1(x)))))))
c.0(c.0(b.0(a.0(b.0(b.0(B.0(x))))))) → a.0(a.0(a.0(a.0(c.0(c.0(c.0(b.0(b.0(B.0(x))))))))))
a.0(b.0(b.0(a.0(b.0(b.0(B.1(x))))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.1(x)))))))))))))
c.1(C.0(x)) → b.0(b.0(B.0(x)))
a.0(b.0(a.0(b.0(b.0(B.0(x)))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.0(x))))))))))))
c.0(c.0(b.0(b.1(x)))) → a.0(a.0(a.0(a.0(c.0(c.0(c.1(x)))))))
a.0(b.0(a.0(b.0(b.0(B.1(x)))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.1(x)))))))))))))
a.0(a.0(b.0(b.0(B.0(x))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.0(x)))))))))))
c.0(c.0(b.0(a.0(b.0(b.0(B.0(x))))))) → a.0(a.0(a.0(a.0(c.0(c.0(c.0(b.0(b.0(b.0(b.0(B.0(x))))))))))))
c.0(c.0(a.0(b.0(b.0(B.0(x)))))) → a.0(a.0(a.0(a.0(c.0(c.0(c.0(b.0(b.0(b.0(B.0(x)))))))))))
c.0(c.0(a.0(b.0(b.0(B.1(x)))))) → a.0(a.0(a.0(a.0(c.0(c.0(c.0(b.0(b.0(B.1(x))))))))))
c.0(c.0(b.0(a.0(b.0(b.0(B.1(x))))))) → a.0(a.0(a.0(a.0(c.0(c.0(c.0(b.0(b.0(b.0(b.0(b.0(B.1(x)))))))))))))
c.0(c.0(a.0(b.0(b.0(B.1(x)))))) → a.0(a.0(a.0(a.0(c.0(c.0(c.0(b.0(b.0(b.0(b.0(b.0(B.1(x)))))))))))))
a.0(a.0(b.0(b.0(B.0(x))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.0(x)))))))))
a.0(b.0(a.0(b.0(b.0(B.1(x)))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.1(x))))))))))
a.0(a.0(b.0(b.0(B.1(x))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.1(x)))))))))
c.0(c.0(b.0(a.0(b.0(b.0(B.1(x))))))) → a.0(a.0(a.0(a.0(c.0(c.0(c.0(b.0(b.0(b.0(b.0(b.0(b.0(B.1(x))))))))))))))
c.0(c.0(b.0(B.1(x)))) → a.0(a.0(a.0(a.0(c.1(C.1(x))))))
c.0(c.0(a.0(b.0(b.0(B.0(x)))))) → a.0(a.0(a.0(a.0(c.0(c.0(c.0(b.0(b.0(b.0(b.0(B.0(x))))))))))))
a.0(b.0(b.0(B.1(x)))) → b.0(b.0(B.1(x)))
a.0(b.0(a.0(b.0(b.0(B.0(x)))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.0(x)))))))))))
a.0(b.0(b.0(a.0(b.0(b.0(B.1(x))))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.1(x)))))))))))
a.0(a.0(b.0(b.0(B.1(x))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.1(x)))))))))))
a.0(b.0(b.0(a.0(b.0(b.0(B.0(x))))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.0(x))))))))))))
a.0(a.0(b.0(b.0(B.1(x))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.1(x))))))))))))
c.1(C.1(x)) → b.0(b.0(B.1(x)))
a.0(b.0(a.0(b.0(b.0(B.1(x)))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.1(x))))))))))))
a.0(b.0(b.0(a.0(b.0(b.0(B.0(x))))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.0(x))))))))))))))
a.0(b.0(b.0(a.0(b.0(b.0(B.0(x))))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.0(x)))))))))))))
a.0(a.0(b.0(b.0(B.1(x))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.1(x))))))))))
a.0(b.0(b.0(B.0(x)))) → B.0(x)
c.0(c.0(b.0(B.0(x)))) → a.0(a.0(a.0(a.0(c.0(c.1(C.0(x)))))))
a.0(b.0(b.0(b.0(x)))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(x))))))))
a.0(b.0(a.0(b.0(b.0(B.0(x)))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.0(x))))))))))
c.0(c.1(x)) → b.0(b.0(b.0(a.0(a.0(a.1(x))))))
a.0(b.0(b.0(a.0(b.0(b.0(B.1(x))))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.1(x))))))))))))
c.0(c.0(a.0(b.0(b.0(B.1(x)))))) → a.0(a.0(a.0(a.0(c.0(c.0(c.0(b.0(b.0(b.0(b.0(B.1(x))))))))))))
c.0(c.0(b.0(a.0(b.0(b.0(B.0(x))))))) → a.0(a.0(a.0(a.0(c.0(c.0(c.0(b.0(b.0(b.0(b.0(b.0(b.0(B.0(x))))))))))))))
c.0(c.0(a.0(b.0(b.0(B.1(x)))))) → a.0(a.0(a.0(a.0(c.0(c.0(c.0(b.0(B.1(x)))))))))
c.0(c.0(b.0(a.0(b.0(b.0(B.1(x))))))) → a.0(a.0(a.0(a.0(c.0(c.0(c.0(b.0(b.0(B.1(x))))))))))
a.0(b.0(b.0(a.0(b.0(b.0(B.0(x))))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.0(x)))))))))))
c.0(c.0(a.0(b.0(b.0(B.0(x)))))) → a.0(a.0(a.0(a.0(c.0(c.0(c.0(b.0(b.0(B.0(x))))))))))
c.0(c.0(a.0(b.0(b.0(B.0(x)))))) → a.0(a.0(a.0(a.0(c.0(c.0(c.0(b.0(b.0(b.0(b.0(b.0(B.0(x)))))))))))))
c.0(c.0(x)) → b.0(b.0(b.0(a.0(a.0(a.0(x))))))
a.0(b.0(a.0(b.0(b.0(B.0(x)))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.0(x)))))))))))))
c.1(C.0(x)) → b.0(B.0(x))
a.0(b.0(a.0(b.0(b.0(B.1(x)))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.1(x)))))))))))
c.0(c.0(b.0(a.0(b.0(b.0(B.1(x))))))) → a.0(a.0(a.0(a.0(c.0(c.0(c.0(b.0(b.0(b.0(b.0(B.1(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:
C1.0(c.0(b.0(b.1(C.0(x0))))) → C1.0(c.0(b.0(b.0(B.0(x0)))))
C1.0(c.0(b.0(b.0(c.0(b.0(b.1(x0))))))) → C1.0(c.0(a.0(a.0(a.0(a.0(c.0(c.0(c.1(x0)))))))))
C1.0(c.0(b.0(b.1(C.1(x0))))) → C1.0(c.0(b.0(b.0(B.1(x0)))))
Strictly oriented rules of the TRS R:
c.0(c.0(b.0(b.1(x)))) → a.0(a.0(a.0(a.0(c.0(c.0(c.1(x)))))))
Used ordering: POLO with Polynomial interpretation [25]:
POL(B.0(x1)) = 1 + x1
POL(B.1(x1)) = 1 + x1
POL(C.0(x1)) = 1 + x1
POL(C.1(x1)) = 1 + x1
POL(C1.0(x1)) = x1
POL(a.0(x1)) = x1
POL(a.1(x1)) = x1
POL(b.0(x1)) = x1
POL(b.1(x1)) = 1 + x1
POL(c.0(x1)) = x1
POL(c.1(x1)) = x1
↳ QTRS
↳ QTRS Reverse
↳ QTRS Reverse
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ SemLabProof
↳ SemLabProof2
↳ QDP
↳ QDPToSRSProof
↳ QTRS
↳ QTRS Reverse
↳ QTRS
↳ QTRS Reverse
↳ QTRS Reverse
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ SemLabProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ RuleRemovalProof
↳ QDP
↳ DependencyGraphProof
↳ SemLabProof2
Q DP problem:
The TRS P consists of the following rules:
C1.0(c.0(b.0(a.0(b.0(b.0(B.1(x))))))) → C1.0(c.0(b.0(b.0(b.0(B.1(x))))))
C1.0(c.0(b.0(a.0(b.0(b.0(B.1(x))))))) → C1.0(c.0(b.0(b.0(b.0(b.0(b.0(b.0(B.1(x)))))))))
C1.0(c.0(b.0(b.0(c.0(b.0(a.0(b.0(b.0(B.0(x0)))))))))) → C1.0(c.0(a.0(a.0(a.0(a.0(c.0(c.0(c.0(b.0(b.0(b.0(b.0(B.0(x0))))))))))))))
C1.0(c.0(b.0(b.0(c.0(b.0(B.1(x0))))))) → C1.0(c.0(a.0(a.0(a.0(a.0(c.1(C.1(x0))))))))
C1.0(c.0(b.0(b.0(c.0(a.0(b.0(b.0(B.1(x0))))))))) → C1.0(c.0(a.0(a.0(a.0(a.0(c.0(c.0(c.0(b.0(b.0(b.0(b.0(B.1(x0))))))))))))))
C1.0(c.0(b.0(b.0(c.0(b.0(a.0(b.0(b.0(B.1(x0)))))))))) → C1.0(c.0(a.0(a.0(a.0(a.0(c.0(c.0(c.0(b.0(b.0(b.0(b.0(b.0(b.0(B.1(x0))))))))))))))))
C1.0(c.0(b.0(a.0(b.0(b.0(B.0(x))))))) → C1.0(c.0(b.0(b.0(B.0(x)))))
C1.0(c.0(b.0(a.0(b.0(b.0(B.1(x))))))) → C1.0(c.0(b.0(b.0(b.0(b.0(b.0(B.1(x))))))))
C1.0(c.0(a.0(b.0(b.0(B.0(x)))))) → C1.0(c.0(b.0(b.0(b.0(b.0(b.0(B.0(x))))))))
C1.0(c.0(b.0(b.0(c.0(x0))))) → C1.0(c.0(b.0(b.0(b.0(a.0(a.0(a.0(x0))))))))
C1.0(c.0(b.0(b.0(c.0(a.0(b.0(b.0(B.0(x0))))))))) → C1.0(c.0(a.0(a.0(a.0(a.0(c.0(c.0(c.0(b.0(b.0(B.0(x0))))))))))))
C1.0(c.0(b.0(b.0(c.0(b.0(b.1(C.0(y_0)))))))) → C1.0(c.0(b.0(b.1(C.0(y_0)))))
C1.0(c.0(b.0(b.0(c.0(y_2))))) → C1.0(c.0(y_2))
C1.0(c.0(b.0(b.0(c.0(a.0(b.0(b.0(B.1(x0))))))))) → C1.0(c.0(a.0(a.0(a.0(a.0(c.0(c.0(c.0(b.0(b.0(b.0(B.1(x0)))))))))))))
C1.0(c.0(b.0(b.0(c.0(b.0(a.0(b.0(b.0(B.1(x0)))))))))) → C1.0(c.0(a.0(a.0(a.0(a.0(c.0(c.0(c.0(b.0(b.0(B.1(x0))))))))))))
C1.0(c.0(a.0(b.0(b.0(B.1(x)))))) → C1.0(c.0(b.0(b.0(b.0(B.1(x))))))
C1.0(c.0(a.0(b.0(b.0(B.0(x)))))) → C1.0(c.0(b.0(b.0(B.0(x)))))
C1.0(c.0(b.0(a.0(b.0(b.0(B.1(x))))))) → C1.0(c.0(b.0(b.0(B.1(x)))))
C1.0(c.0(b.0(b.0(c.0(b.0(B.0(x0))))))) → C1.0(c.0(a.0(a.0(a.0(a.0(c.0(c.1(C.0(x0)))))))))
C1.0(c.0(b.0(b.0(c.0(b.0(a.0(b.0(b.0(B.0(x0)))))))))) → C1.0(c.0(a.0(a.0(a.0(a.0(c.0(c.0(c.0(b.0(b.0(b.0(b.0(b.0(B.0(x0)))))))))))))))
C1.0(c.0(b.0(a.0(b.0(b.0(B.0(x))))))) → C1.0(c.0(b.0(b.0(b.0(b.0(b.0(B.0(x))))))))
C1.0(c.0(b.0(b.0(c.0(b.0(a.0(b.0(b.0(B.0(x0)))))))))) → C1.0(c.0(a.0(a.0(a.0(a.0(c.0(c.0(c.0(b.0(b.0(B.0(x0))))))))))))
C1.0(c.0(b.0(b.0(c.0(b.0(a.0(b.0(b.0(B.1(x0)))))))))) → C1.0(c.0(a.0(a.0(a.0(a.0(c.0(c.0(c.0(b.0(b.0(b.0(B.1(x0)))))))))))))
C1.0(c.0(b.0(b.0(c.0(a.0(b.0(b.0(B.0(x0))))))))) → C1.0(c.0(a.0(a.0(a.0(a.0(c.0(c.0(c.0(b.0(b.0(b.0(B.0(x0)))))))))))))
C1.0(c.0(b.0(a.0(b.0(b.0(B.0(x))))))) → C1.0(c.0(b.0(b.0(b.0(B.0(x))))))
C1.0(c.0(b.0(b.0(c.0(b.0(a.0(b.0(b.0(B.0(x0)))))))))) → C1.0(c.0(a.0(a.0(a.0(a.0(c.0(c.0(c.0(b.0(b.0(b.0(B.0(x0)))))))))))))
C1.0(c.0(a.0(b.0(b.0(B.1(x)))))) → C1.0(c.0(b.0(b.0(b.0(b.0(b.0(B.1(x))))))))
C1.0(c.0(b.0(b.0(c.0(b.0(a.0(b.0(b.0(B.0(x0)))))))))) → C1.0(c.0(a.0(a.0(a.0(a.0(c.0(c.0(c.0(b.0(b.0(b.0(b.0(b.0(b.0(B.0(x0))))))))))))))))
C1.0(c.0(b.0(b.0(c.0(a.0(b.0(b.0(B.1(x0))))))))) → C1.0(c.0(a.0(a.0(a.0(a.0(c.0(c.0(c.0(b.0(B.1(x0)))))))))))
C1.0(c.0(b.0(b.0(c.0(b.0(b.0(x0))))))) → C1.0(c.0(a.0(a.0(a.0(a.0(c.0(c.0(c.0(x0)))))))))
C1.0(c.0(b.0(b.0(c.0(b.0(a.0(b.0(b.0(B.1(x0)))))))))) → C1.0(c.0(a.0(a.0(a.0(a.0(c.0(c.0(c.0(b.0(b.0(b.0(b.0(B.1(x0))))))))))))))
C1.0(c.0(b.0(b.0(c.0(a.0(b.0(b.0(B.0(x0))))))))) → C1.0(c.0(a.0(a.0(a.0(a.0(c.0(c.0(c.0(b.0(b.0(b.0(b.0(B.0(x0))))))))))))))
C1.0(c.0(b.0(b.0(c.0(b.0(a.0(b.0(b.0(B.1(x0)))))))))) → C1.0(c.0(a.0(a.0(a.0(a.0(c.0(c.0(c.0(b.0(b.0(b.0(b.0(b.0(B.1(x0)))))))))))))))
C1.0(c.0(b.0(b.0(c.0(a.0(b.0(b.0(B.1(x0))))))))) → C1.0(c.0(a.0(a.0(a.0(a.0(c.0(c.0(c.0(b.0(b.0(b.0(b.0(b.0(B.1(x0)))))))))))))))
C1.0(c.0(b.0(b.0(c.0(b.0(B.0(x0))))))) → C1.0(c.0(a.0(a.0(a.0(a.0(c.1(C.0(x0))))))))
C1.0(c.0(b.0(a.0(b.0(b.0(B.0(x))))))) → C1.0(c.0(b.0(b.0(b.0(b.0(b.0(b.0(B.0(x)))))))))
C1.0(c.0(a.0(b.0(b.0(B.1(x)))))) → C1.0(c.0(b.0(b.0(B.1(x)))))
C1.0(c.0(b.0(b.0(c.1(x0))))) → C1.0(c.0(b.0(b.0(b.0(a.0(a.0(a.1(x0))))))))
C1.0(c.0(b.0(b.0(c.0(b.0(b.0(c.0(y_4)))))))) → C1.0(c.0(b.0(b.0(c.0(y_4)))))
C1.0(c.0(a.0(b.0(b.0(B.0(x)))))) → C1.0(c.0(b.0(b.0(b.0(B.0(x))))))
C1.0(c.0(b.0(b.0(c.0(b.0(b.0(c.1(y_4)))))))) → C1.0(c.0(b.0(b.0(c.1(y_4)))))
C1.0(c.0(b.0(b.0(c.0(a.0(b.0(b.0(B.1(x0))))))))) → C1.0(c.0(a.0(a.0(a.0(a.0(c.0(c.0(c.0(b.0(b.0(B.1(x0))))))))))))
C1.0(c.0(b.0(a.0(b.0(b.0(B.1(x))))))) → C1.0(c.0(b.0(b.0(b.0(b.0(B.1(x)))))))
C1.0(c.0(b.0(b.0(c.0(a.0(b.0(b.0(B.0(x0))))))))) → C1.0(c.0(a.0(a.0(a.0(a.0(c.0(c.0(c.0(b.0(b.0(b.0(b.0(b.0(B.0(x0)))))))))))))))
C1.0(c.0(b.0(b.0(c.0(a.0(b.0(b.0(B.0(x0))))))))) → C1.0(c.0(a.0(a.0(a.0(a.0(c.0(c.0(c.0(b.0(B.0(x0)))))))))))
C1.0(c.0(b.0(b.0(c.0(b.0(B.1(x0))))))) → C1.0(c.0(a.0(a.0(a.0(a.0(c.0(c.1(C.1(x0)))))))))
C1.0(c.0(b.0(b.0(c.0(b.0(b.1(C.1(y_0)))))))) → C1.0(c.0(b.0(b.1(C.1(y_0)))))
C1.0(c.0(b.0(a.0(b.0(b.0(B.0(x))))))) → C1.0(c.0(b.0(b.0(b.0(b.0(B.0(x)))))))
C1.0(c.0(a.0(b.0(b.0(B.1(x)))))) → C1.0(c.0(b.0(b.0(b.0(b.0(B.1(x)))))))
C1.0(c.0(b.0(b.0(x)))) → C1.0(c.0(x))
C1.0(c.0(a.0(b.0(b.0(B.0(x)))))) → C1.0(c.0(b.0(b.0(b.0(b.0(B.0(x)))))))
The TRS R consists of the following rules:
c.1(C.1(x)) → b.0(B.1(x))
c.0(c.0(b.0(B.0(x)))) → a.0(a.0(a.0(a.0(c.1(C.0(x))))))
a.0(b.0(b.0(b.1(x)))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.1(x))))))))
a.0(b.0(b.0(a.0(b.0(b.0(B.0(x))))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.0(x)))))))))))))))
a.0(b.0(b.0(B.0(x)))) → b.0(b.0(B.0(x)))
a.0(b.0(a.0(b.0(b.0(B.1(x)))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.1(x))))))))))))))
a.0(b.0(b.0(a.0(b.0(b.0(B.1(x))))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.1(x)))))))))))))))
a.0(a.0(b.0(b.0(B.1(x))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.1(x)))))))))))))
a.0(a.0(b.0(b.0(B.0(x))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.0(x)))))))))))))
a.0(b.0(b.0(B.0(x)))) → b.0(B.0(x))
a.0(b.0(a.0(b.0(b.0(B.0(x)))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.0(x))))))))))))))
a.0(b.0(b.0(B.1(x)))) → b.0(B.1(x))
a.0(b.0(b.0(B.1(x)))) → B.1(x)
c.0(c.0(b.0(a.0(b.0(b.0(B.0(x))))))) → a.0(a.0(a.0(a.0(c.0(c.0(c.0(b.0(b.0(b.0(B.0(x)))))))))))
c.0(c.0(b.0(a.0(b.0(b.0(B.1(x))))))) → a.0(a.0(a.0(a.0(c.0(c.0(c.0(b.0(b.0(b.0(B.1(x)))))))))))
a.0(b.0(b.0(a.0(b.0(b.0(B.1(x))))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.1(x))))))))))))))
c.0(c.0(b.0(b.0(x)))) → a.0(a.0(a.0(a.0(c.0(c.0(c.0(x)))))))
c.0(c.0(a.0(b.0(b.0(B.0(x)))))) → a.0(a.0(a.0(a.0(c.0(c.0(c.0(b.0(B.0(x)))))))))
a.0(a.0(b.0(b.0(B.0(x))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.0(x))))))))))
c.0(c.0(b.0(a.0(b.0(b.0(B.0(x))))))) → a.0(a.0(a.0(a.0(c.0(c.0(c.0(b.0(b.0(b.0(b.0(b.0(B.0(x)))))))))))))
a.0(a.0(b.0(b.0(B.0(x))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.0(x))))))))))))
c.0(c.0(a.0(b.0(b.0(B.1(x)))))) → a.0(a.0(a.0(a.0(c.0(c.0(c.0(b.0(b.0(b.0(B.1(x)))))))))))
c.0(c.0(b.0(B.1(x)))) → a.0(a.0(a.0(a.0(c.0(c.1(C.1(x)))))))
c.0(c.0(b.0(a.0(b.0(b.0(B.0(x))))))) → a.0(a.0(a.0(a.0(c.0(c.0(c.0(b.0(b.0(B.0(x))))))))))
a.0(b.0(b.0(a.0(b.0(b.0(B.1(x))))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.1(x)))))))))))))
c.1(C.0(x)) → b.0(b.0(B.0(x)))
a.0(b.0(a.0(b.0(b.0(B.0(x)))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.0(x))))))))))))
a.0(b.0(a.0(b.0(b.0(B.1(x)))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.1(x)))))))))))))
a.0(a.0(b.0(b.0(B.0(x))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.0(x)))))))))))
c.0(c.0(b.0(a.0(b.0(b.0(B.0(x))))))) → a.0(a.0(a.0(a.0(c.0(c.0(c.0(b.0(b.0(b.0(b.0(B.0(x))))))))))))
c.0(c.0(a.0(b.0(b.0(B.0(x)))))) → a.0(a.0(a.0(a.0(c.0(c.0(c.0(b.0(b.0(b.0(B.0(x)))))))))))
c.0(c.0(a.0(b.0(b.0(B.1(x)))))) → a.0(a.0(a.0(a.0(c.0(c.0(c.0(b.0(b.0(B.1(x))))))))))
c.0(c.0(b.0(a.0(b.0(b.0(B.1(x))))))) → a.0(a.0(a.0(a.0(c.0(c.0(c.0(b.0(b.0(b.0(b.0(b.0(B.1(x)))))))))))))
c.0(c.0(a.0(b.0(b.0(B.1(x)))))) → a.0(a.0(a.0(a.0(c.0(c.0(c.0(b.0(b.0(b.0(b.0(b.0(B.1(x)))))))))))))
a.0(a.0(b.0(b.0(B.0(x))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.0(x)))))))))
a.0(b.0(a.0(b.0(b.0(B.1(x)))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.1(x))))))))))
a.0(a.0(b.0(b.0(B.1(x))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.1(x)))))))))
c.0(c.0(b.0(a.0(b.0(b.0(B.1(x))))))) → a.0(a.0(a.0(a.0(c.0(c.0(c.0(b.0(b.0(b.0(b.0(b.0(b.0(B.1(x))))))))))))))
c.0(c.0(b.0(B.1(x)))) → a.0(a.0(a.0(a.0(c.1(C.1(x))))))
c.0(c.0(a.0(b.0(b.0(B.0(x)))))) → a.0(a.0(a.0(a.0(c.0(c.0(c.0(b.0(b.0(b.0(b.0(B.0(x))))))))))))
a.0(b.0(b.0(B.1(x)))) → b.0(b.0(B.1(x)))
a.0(b.0(a.0(b.0(b.0(B.0(x)))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.0(x)))))))))))
a.0(b.0(b.0(a.0(b.0(b.0(B.1(x))))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.1(x)))))))))))
a.0(a.0(b.0(b.0(B.1(x))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.1(x)))))))))))
a.0(b.0(b.0(a.0(b.0(b.0(B.0(x))))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.0(x))))))))))))
a.0(a.0(b.0(b.0(B.1(x))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.1(x))))))))))))
c.1(C.1(x)) → b.0(b.0(B.1(x)))
a.0(b.0(a.0(b.0(b.0(B.1(x)))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.1(x))))))))))))
a.0(b.0(b.0(a.0(b.0(b.0(B.0(x))))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.0(x))))))))))))))
a.0(b.0(b.0(a.0(b.0(b.0(B.0(x))))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.0(x)))))))))))))
a.0(a.0(b.0(b.0(B.1(x))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.1(x))))))))))
a.0(b.0(b.0(B.0(x)))) → B.0(x)
c.0(c.0(b.0(B.0(x)))) → a.0(a.0(a.0(a.0(c.0(c.1(C.0(x)))))))
a.0(b.0(b.0(b.0(x)))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(x))))))))
a.0(b.0(a.0(b.0(b.0(B.0(x)))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.0(x))))))))))
c.0(c.1(x)) → b.0(b.0(b.0(a.0(a.0(a.1(x))))))
a.0(b.0(b.0(a.0(b.0(b.0(B.1(x))))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.1(x))))))))))))
c.0(c.0(a.0(b.0(b.0(B.1(x)))))) → a.0(a.0(a.0(a.0(c.0(c.0(c.0(b.0(b.0(b.0(b.0(B.1(x))))))))))))
c.0(c.0(b.0(a.0(b.0(b.0(B.0(x))))))) → a.0(a.0(a.0(a.0(c.0(c.0(c.0(b.0(b.0(b.0(b.0(b.0(b.0(B.0(x))))))))))))))
c.0(c.0(a.0(b.0(b.0(B.1(x)))))) → a.0(a.0(a.0(a.0(c.0(c.0(c.0(b.0(B.1(x)))))))))
c.0(c.0(b.0(a.0(b.0(b.0(B.1(x))))))) → a.0(a.0(a.0(a.0(c.0(c.0(c.0(b.0(b.0(B.1(x))))))))))
a.0(b.0(b.0(a.0(b.0(b.0(B.0(x))))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.0(x)))))))))))
c.0(c.0(a.0(b.0(b.0(B.0(x)))))) → a.0(a.0(a.0(a.0(c.0(c.0(c.0(b.0(b.0(B.0(x))))))))))
c.0(c.0(a.0(b.0(b.0(B.0(x)))))) → a.0(a.0(a.0(a.0(c.0(c.0(c.0(b.0(b.0(b.0(b.0(b.0(B.0(x)))))))))))))
c.0(c.0(x)) → b.0(b.0(b.0(a.0(a.0(a.0(x))))))
a.0(b.0(a.0(b.0(b.0(B.0(x)))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.0(x)))))))))))))
c.1(C.0(x)) → b.0(B.0(x))
a.0(b.0(a.0(b.0(b.0(B.1(x)))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.1(x)))))))))))
c.0(c.0(b.0(a.0(b.0(b.0(B.1(x))))))) → a.0(a.0(a.0(a.0(c.0(c.0(c.0(b.0(b.0(b.0(b.0(B.1(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
↳ QTRS Reverse
↳ QTRS Reverse
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ SemLabProof
↳ SemLabProof2
↳ QDP
↳ QDPToSRSProof
↳ QTRS
↳ QTRS Reverse
↳ QTRS
↳ QTRS Reverse
↳ QTRS Reverse
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ SemLabProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ RuleRemovalProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ RuleRemovalProof
↳ SemLabProof2
Q DP problem:
The TRS P consists of the following rules:
C1.0(c.0(b.0(a.0(b.0(b.0(B.1(x))))))) → C1.0(c.0(b.0(b.0(b.0(B.1(x))))))
C1.0(c.0(b.0(a.0(b.0(b.0(B.1(x))))))) → C1.0(c.0(b.0(b.0(b.0(b.0(b.0(b.0(B.1(x)))))))))
C1.0(c.0(b.0(b.0(c.0(b.0(a.0(b.0(b.0(B.0(x0)))))))))) → C1.0(c.0(a.0(a.0(a.0(a.0(c.0(c.0(c.0(b.0(b.0(b.0(b.0(B.0(x0))))))))))))))
C1.0(c.0(b.0(b.0(c.0(b.0(B.1(x0))))))) → C1.0(c.0(a.0(a.0(a.0(a.0(c.1(C.1(x0))))))))
C1.0(c.0(b.0(b.0(c.0(a.0(b.0(b.0(B.1(x0))))))))) → C1.0(c.0(a.0(a.0(a.0(a.0(c.0(c.0(c.0(b.0(b.0(b.0(b.0(B.1(x0))))))))))))))
C1.0(c.0(b.0(b.0(c.0(b.0(a.0(b.0(b.0(B.1(x0)))))))))) → C1.0(c.0(a.0(a.0(a.0(a.0(c.0(c.0(c.0(b.0(b.0(b.0(b.0(b.0(b.0(B.1(x0))))))))))))))))
C1.0(c.0(b.0(a.0(b.0(b.0(B.0(x))))))) → C1.0(c.0(b.0(b.0(B.0(x)))))
C1.0(c.0(b.0(a.0(b.0(b.0(B.1(x))))))) → C1.0(c.0(b.0(b.0(b.0(b.0(b.0(B.1(x))))))))
C1.0(c.0(a.0(b.0(b.0(B.0(x)))))) → C1.0(c.0(b.0(b.0(b.0(b.0(b.0(B.0(x))))))))
C1.0(c.0(b.0(b.0(c.0(x0))))) → C1.0(c.0(b.0(b.0(b.0(a.0(a.0(a.0(x0))))))))
C1.0(c.0(b.0(b.0(c.0(a.0(b.0(b.0(B.0(x0))))))))) → C1.0(c.0(a.0(a.0(a.0(a.0(c.0(c.0(c.0(b.0(b.0(B.0(x0))))))))))))
C1.0(c.0(b.0(b.0(c.0(y_2))))) → C1.0(c.0(y_2))
C1.0(c.0(b.0(b.0(c.0(b.0(a.0(b.0(b.0(B.1(x0)))))))))) → C1.0(c.0(a.0(a.0(a.0(a.0(c.0(c.0(c.0(b.0(b.0(B.1(x0))))))))))))
C1.0(c.0(b.0(b.0(c.0(a.0(b.0(b.0(B.1(x0))))))))) → C1.0(c.0(a.0(a.0(a.0(a.0(c.0(c.0(c.0(b.0(b.0(b.0(B.1(x0)))))))))))))
C1.0(c.0(a.0(b.0(b.0(B.1(x)))))) → C1.0(c.0(b.0(b.0(b.0(B.1(x))))))
C1.0(c.0(a.0(b.0(b.0(B.0(x)))))) → C1.0(c.0(b.0(b.0(B.0(x)))))
C1.0(c.0(b.0(a.0(b.0(b.0(B.1(x))))))) → C1.0(c.0(b.0(b.0(B.1(x)))))
C1.0(c.0(b.0(b.0(c.0(b.0(B.0(x0))))))) → C1.0(c.0(a.0(a.0(a.0(a.0(c.0(c.1(C.0(x0)))))))))
C1.0(c.0(b.0(a.0(b.0(b.0(B.0(x))))))) → C1.0(c.0(b.0(b.0(b.0(b.0(b.0(B.0(x))))))))
C1.0(c.0(b.0(b.0(c.0(b.0(a.0(b.0(b.0(B.0(x0)))))))))) → C1.0(c.0(a.0(a.0(a.0(a.0(c.0(c.0(c.0(b.0(b.0(b.0(b.0(b.0(B.0(x0)))))))))))))))
C1.0(c.0(b.0(b.0(c.0(b.0(a.0(b.0(b.0(B.0(x0)))))))))) → C1.0(c.0(a.0(a.0(a.0(a.0(c.0(c.0(c.0(b.0(b.0(B.0(x0))))))))))))
C1.0(c.0(b.0(b.0(c.0(b.0(a.0(b.0(b.0(B.1(x0)))))))))) → C1.0(c.0(a.0(a.0(a.0(a.0(c.0(c.0(c.0(b.0(b.0(b.0(B.1(x0)))))))))))))
C1.0(c.0(b.0(b.0(c.0(a.0(b.0(b.0(B.0(x0))))))))) → C1.0(c.0(a.0(a.0(a.0(a.0(c.0(c.0(c.0(b.0(b.0(b.0(B.0(x0)))))))))))))
C1.0(c.0(b.0(a.0(b.0(b.0(B.0(x))))))) → C1.0(c.0(b.0(b.0(b.0(B.0(x))))))
C1.0(c.0(b.0(b.0(c.0(b.0(a.0(b.0(b.0(B.0(x0)))))))))) → C1.0(c.0(a.0(a.0(a.0(a.0(c.0(c.0(c.0(b.0(b.0(b.0(B.0(x0)))))))))))))
C1.0(c.0(a.0(b.0(b.0(B.1(x)))))) → C1.0(c.0(b.0(b.0(b.0(b.0(b.0(B.1(x))))))))
C1.0(c.0(b.0(b.0(c.0(b.0(a.0(b.0(b.0(B.0(x0)))))))))) → C1.0(c.0(a.0(a.0(a.0(a.0(c.0(c.0(c.0(b.0(b.0(b.0(b.0(b.0(b.0(B.0(x0))))))))))))))))
C1.0(c.0(b.0(b.0(c.0(a.0(b.0(b.0(B.1(x0))))))))) → C1.0(c.0(a.0(a.0(a.0(a.0(c.0(c.0(c.0(b.0(B.1(x0)))))))))))
C1.0(c.0(b.0(b.0(c.0(b.0(b.0(x0))))))) → C1.0(c.0(a.0(a.0(a.0(a.0(c.0(c.0(c.0(x0)))))))))
C1.0(c.0(b.0(b.0(c.0(b.0(a.0(b.0(b.0(B.1(x0)))))))))) → C1.0(c.0(a.0(a.0(a.0(a.0(c.0(c.0(c.0(b.0(b.0(b.0(b.0(B.1(x0))))))))))))))
C1.0(c.0(b.0(b.0(c.0(a.0(b.0(b.0(B.0(x0))))))))) → C1.0(c.0(a.0(a.0(a.0(a.0(c.0(c.0(c.0(b.0(b.0(b.0(b.0(B.0(x0))))))))))))))
C1.0(c.0(b.0(b.0(c.0(b.0(a.0(b.0(b.0(B.1(x0)))))))))) → C1.0(c.0(a.0(a.0(a.0(a.0(c.0(c.0(c.0(b.0(b.0(b.0(b.0(b.0(B.1(x0)))))))))))))))
C1.0(c.0(b.0(b.0(c.0(a.0(b.0(b.0(B.1(x0))))))))) → C1.0(c.0(a.0(a.0(a.0(a.0(c.0(c.0(c.0(b.0(b.0(b.0(b.0(b.0(B.1(x0)))))))))))))))
C1.0(c.0(b.0(b.0(c.0(b.0(B.0(x0))))))) → C1.0(c.0(a.0(a.0(a.0(a.0(c.1(C.0(x0))))))))
C1.0(c.0(b.0(a.0(b.0(b.0(B.0(x))))))) → C1.0(c.0(b.0(b.0(b.0(b.0(b.0(b.0(B.0(x)))))))))
C1.0(c.0(a.0(b.0(b.0(B.1(x)))))) → C1.0(c.0(b.0(b.0(B.1(x)))))
C1.0(c.0(b.0(b.0(c.1(x0))))) → C1.0(c.0(b.0(b.0(b.0(a.0(a.0(a.1(x0))))))))
C1.0(c.0(b.0(b.0(c.0(b.0(b.0(c.0(y_4)))))))) → C1.0(c.0(b.0(b.0(c.0(y_4)))))
C1.0(c.0(a.0(b.0(b.0(B.0(x)))))) → C1.0(c.0(b.0(b.0(b.0(B.0(x))))))
C1.0(c.0(b.0(b.0(c.0(b.0(b.0(c.1(y_4)))))))) → C1.0(c.0(b.0(b.0(c.1(y_4)))))
C1.0(c.0(b.0(b.0(c.0(a.0(b.0(b.0(B.1(x0))))))))) → C1.0(c.0(a.0(a.0(a.0(a.0(c.0(c.0(c.0(b.0(b.0(B.1(x0))))))))))))
C1.0(c.0(b.0(a.0(b.0(b.0(B.1(x))))))) → C1.0(c.0(b.0(b.0(b.0(b.0(B.1(x)))))))
C1.0(c.0(b.0(b.0(c.0(a.0(b.0(b.0(B.0(x0))))))))) → C1.0(c.0(a.0(a.0(a.0(a.0(c.0(c.0(c.0(b.0(b.0(b.0(b.0(b.0(B.0(x0)))))))))))))))
C1.0(c.0(b.0(b.0(c.0(a.0(b.0(b.0(B.0(x0))))))))) → C1.0(c.0(a.0(a.0(a.0(a.0(c.0(c.0(c.0(b.0(B.0(x0)))))))))))
C1.0(c.0(b.0(b.0(c.0(b.0(B.1(x0))))))) → C1.0(c.0(a.0(a.0(a.0(a.0(c.0(c.1(C.1(x0)))))))))
C1.0(c.0(b.0(a.0(b.0(b.0(B.0(x))))))) → C1.0(c.0(b.0(b.0(b.0(b.0(B.0(x)))))))
C1.0(c.0(a.0(b.0(b.0(B.1(x)))))) → C1.0(c.0(b.0(b.0(b.0(b.0(B.1(x)))))))
C1.0(c.0(b.0(b.0(x)))) → C1.0(c.0(x))
C1.0(c.0(a.0(b.0(b.0(B.0(x)))))) → C1.0(c.0(b.0(b.0(b.0(b.0(B.0(x)))))))
The TRS R consists of the following rules:
c.1(C.1(x)) → b.0(B.1(x))
c.0(c.0(b.0(B.0(x)))) → a.0(a.0(a.0(a.0(c.1(C.0(x))))))
a.0(b.0(b.0(b.1(x)))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.1(x))))))))
a.0(b.0(b.0(a.0(b.0(b.0(B.0(x))))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.0(x)))))))))))))))
a.0(b.0(b.0(B.0(x)))) → b.0(b.0(B.0(x)))
a.0(b.0(a.0(b.0(b.0(B.1(x)))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.1(x))))))))))))))
a.0(b.0(b.0(a.0(b.0(b.0(B.1(x))))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.1(x)))))))))))))))
a.0(a.0(b.0(b.0(B.1(x))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.1(x)))))))))))))
a.0(a.0(b.0(b.0(B.0(x))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.0(x)))))))))))))
a.0(b.0(b.0(B.0(x)))) → b.0(B.0(x))
a.0(b.0(a.0(b.0(b.0(B.0(x)))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.0(x))))))))))))))
a.0(b.0(b.0(B.1(x)))) → b.0(B.1(x))
a.0(b.0(b.0(B.1(x)))) → B.1(x)
c.0(c.0(b.0(a.0(b.0(b.0(B.0(x))))))) → a.0(a.0(a.0(a.0(c.0(c.0(c.0(b.0(b.0(b.0(B.0(x)))))))))))
c.0(c.0(b.0(a.0(b.0(b.0(B.1(x))))))) → a.0(a.0(a.0(a.0(c.0(c.0(c.0(b.0(b.0(b.0(B.1(x)))))))))))
a.0(b.0(b.0(a.0(b.0(b.0(B.1(x))))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.1(x))))))))))))))
c.0(c.0(b.0(b.0(x)))) → a.0(a.0(a.0(a.0(c.0(c.0(c.0(x)))))))
c.0(c.0(a.0(b.0(b.0(B.0(x)))))) → a.0(a.0(a.0(a.0(c.0(c.0(c.0(b.0(B.0(x)))))))))
a.0(a.0(b.0(b.0(B.0(x))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.0(x))))))))))
c.0(c.0(b.0(a.0(b.0(b.0(B.0(x))))))) → a.0(a.0(a.0(a.0(c.0(c.0(c.0(b.0(b.0(b.0(b.0(b.0(B.0(x)))))))))))))
a.0(a.0(b.0(b.0(B.0(x))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.0(x))))))))))))
c.0(c.0(a.0(b.0(b.0(B.1(x)))))) → a.0(a.0(a.0(a.0(c.0(c.0(c.0(b.0(b.0(b.0(B.1(x)))))))))))
c.0(c.0(b.0(B.1(x)))) → a.0(a.0(a.0(a.0(c.0(c.1(C.1(x)))))))
c.0(c.0(b.0(a.0(b.0(b.0(B.0(x))))))) → a.0(a.0(a.0(a.0(c.0(c.0(c.0(b.0(b.0(B.0(x))))))))))
a.0(b.0(b.0(a.0(b.0(b.0(B.1(x))))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.1(x)))))))))))))
c.1(C.0(x)) → b.0(b.0(B.0(x)))
a.0(b.0(a.0(b.0(b.0(B.0(x)))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.0(x))))))))))))
a.0(b.0(a.0(b.0(b.0(B.1(x)))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.1(x)))))))))))))
a.0(a.0(b.0(b.0(B.0(x))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.0(x)))))))))))
c.0(c.0(b.0(a.0(b.0(b.0(B.0(x))))))) → a.0(a.0(a.0(a.0(c.0(c.0(c.0(b.0(b.0(b.0(b.0(B.0(x))))))))))))
c.0(c.0(a.0(b.0(b.0(B.0(x)))))) → a.0(a.0(a.0(a.0(c.0(c.0(c.0(b.0(b.0(b.0(B.0(x)))))))))))
c.0(c.0(a.0(b.0(b.0(B.1(x)))))) → a.0(a.0(a.0(a.0(c.0(c.0(c.0(b.0(b.0(B.1(x))))))))))
c.0(c.0(b.0(a.0(b.0(b.0(B.1(x))))))) → a.0(a.0(a.0(a.0(c.0(c.0(c.0(b.0(b.0(b.0(b.0(b.0(B.1(x)))))))))))))
c.0(c.0(a.0(b.0(b.0(B.1(x)))))) → a.0(a.0(a.0(a.0(c.0(c.0(c.0(b.0(b.0(b.0(b.0(b.0(B.1(x)))))))))))))
a.0(a.0(b.0(b.0(B.0(x))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.0(x)))))))))
a.0(b.0(a.0(b.0(b.0(B.1(x)))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.1(x))))))))))
a.0(a.0(b.0(b.0(B.1(x))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.1(x)))))))))
c.0(c.0(b.0(a.0(b.0(b.0(B.1(x))))))) → a.0(a.0(a.0(a.0(c.0(c.0(c.0(b.0(b.0(b.0(b.0(b.0(b.0(B.1(x))))))))))))))
c.0(c.0(b.0(B.1(x)))) → a.0(a.0(a.0(a.0(c.1(C.1(x))))))
c.0(c.0(a.0(b.0(b.0(B.0(x)))))) → a.0(a.0(a.0(a.0(c.0(c.0(c.0(b.0(b.0(b.0(b.0(B.0(x))))))))))))
a.0(b.0(b.0(B.1(x)))) → b.0(b.0(B.1(x)))
a.0(b.0(a.0(b.0(b.0(B.0(x)))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.0(x)))))))))))
a.0(b.0(b.0(a.0(b.0(b.0(B.1(x))))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.1(x)))))))))))
a.0(a.0(b.0(b.0(B.1(x))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.1(x)))))))))))
a.0(b.0(b.0(a.0(b.0(b.0(B.0(x))))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.0(x))))))))))))
a.0(a.0(b.0(b.0(B.1(x))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.1(x))))))))))))
c.1(C.1(x)) → b.0(b.0(B.1(x)))
a.0(b.0(a.0(b.0(b.0(B.1(x)))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.1(x))))))))))))
a.0(b.0(b.0(a.0(b.0(b.0(B.0(x))))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.0(x))))))))))))))
a.0(b.0(b.0(a.0(b.0(b.0(B.0(x))))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.0(x)))))))))))))
a.0(a.0(b.0(b.0(B.1(x))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.1(x))))))))))
a.0(b.0(b.0(B.0(x)))) → B.0(x)
c.0(c.0(b.0(B.0(x)))) → a.0(a.0(a.0(a.0(c.0(c.1(C.0(x)))))))
a.0(b.0(b.0(b.0(x)))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(x))))))))
a.0(b.0(a.0(b.0(b.0(B.0(x)))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.0(x))))))))))
c.0(c.1(x)) → b.0(b.0(b.0(a.0(a.0(a.1(x))))))
a.0(b.0(b.0(a.0(b.0(b.0(B.1(x))))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.1(x))))))))))))
c.0(c.0(a.0(b.0(b.0(B.1(x)))))) → a.0(a.0(a.0(a.0(c.0(c.0(c.0(b.0(b.0(b.0(b.0(B.1(x))))))))))))
c.0(c.0(b.0(a.0(b.0(b.0(B.0(x))))))) → a.0(a.0(a.0(a.0(c.0(c.0(c.0(b.0(b.0(b.0(b.0(b.0(b.0(B.0(x))))))))))))))
c.0(c.0(a.0(b.0(b.0(B.1(x)))))) → a.0(a.0(a.0(a.0(c.0(c.0(c.0(b.0(B.1(x)))))))))
c.0(c.0(b.0(a.0(b.0(b.0(B.1(x))))))) → a.0(a.0(a.0(a.0(c.0(c.0(c.0(b.0(b.0(B.1(x))))))))))
a.0(b.0(b.0(a.0(b.0(b.0(B.0(x))))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.0(x)))))))))))
c.0(c.0(a.0(b.0(b.0(B.0(x)))))) → a.0(a.0(a.0(a.0(c.0(c.0(c.0(b.0(b.0(B.0(x))))))))))
c.0(c.0(a.0(b.0(b.0(B.0(x)))))) → a.0(a.0(a.0(a.0(c.0(c.0(c.0(b.0(b.0(b.0(b.0(b.0(B.0(x)))))))))))))
c.0(c.0(x)) → b.0(b.0(b.0(a.0(a.0(a.0(x))))))
a.0(b.0(a.0(b.0(b.0(B.0(x)))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.0(x)))))))))))))
c.1(C.0(x)) → b.0(B.0(x))
a.0(b.0(a.0(b.0(b.0(B.1(x)))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.1(x)))))))))))
c.0(c.0(b.0(a.0(b.0(b.0(B.1(x))))))) → a.0(a.0(a.0(a.0(c.0(c.0(c.0(b.0(b.0(b.0(b.0(B.1(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:
C1.0(c.0(b.0(b.0(c.1(x0))))) → C1.0(c.0(b.0(b.0(b.0(a.0(a.0(a.1(x0))))))))
Strictly oriented rules of the TRS R:
c.0(c.1(x)) → b.0(b.0(b.0(a.0(a.0(a.1(x))))))
Used ordering: POLO with Polynomial interpretation [25]:
POL(B.0(x1)) = 1 + x1
POL(B.1(x1)) = 1 + x1
POL(C.0(x1)) = x1
POL(C.1(x1)) = x1
POL(C1.0(x1)) = x1
POL(a.0(x1)) = x1
POL(a.1(x1)) = x1
POL(b.0(x1)) = x1
POL(b.1(x1)) = x1
POL(c.0(x1)) = x1
POL(c.1(x1)) = 1 + x1
↳ QTRS
↳ QTRS Reverse
↳ QTRS Reverse
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ SemLabProof
↳ SemLabProof2
↳ QDP
↳ QDPToSRSProof
↳ QTRS
↳ QTRS Reverse
↳ QTRS
↳ QTRS Reverse
↳ QTRS Reverse
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ SemLabProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ RuleRemovalProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ RuleRemovalProof
↳ QDP
↳ SemLabProof2
Q DP problem:
The TRS P consists of the following rules:
C1.0(c.0(b.0(a.0(b.0(b.0(B.1(x))))))) → C1.0(c.0(b.0(b.0(b.0(B.1(x))))))
C1.0(c.0(b.0(a.0(b.0(b.0(B.1(x))))))) → C1.0(c.0(b.0(b.0(b.0(b.0(b.0(b.0(B.1(x)))))))))
C1.0(c.0(b.0(b.0(c.0(b.0(a.0(b.0(b.0(B.0(x0)))))))))) → C1.0(c.0(a.0(a.0(a.0(a.0(c.0(c.0(c.0(b.0(b.0(b.0(b.0(b.0(b.0(B.0(x0))))))))))))))))
C1.0(c.0(b.0(b.0(c.0(b.0(a.0(b.0(b.0(B.0(x0)))))))))) → C1.0(c.0(a.0(a.0(a.0(a.0(c.0(c.0(c.0(b.0(b.0(b.0(b.0(B.0(x0))))))))))))))
C1.0(c.0(b.0(b.0(c.0(a.0(b.0(b.0(B.1(x0))))))))) → C1.0(c.0(a.0(a.0(a.0(a.0(c.0(c.0(c.0(b.0(B.1(x0)))))))))))
C1.0(c.0(b.0(b.0(c.0(b.0(B.1(x0))))))) → C1.0(c.0(a.0(a.0(a.0(a.0(c.1(C.1(x0))))))))
C1.0(c.0(b.0(b.0(c.0(b.0(b.0(x0))))))) → C1.0(c.0(a.0(a.0(a.0(a.0(c.0(c.0(c.0(x0)))))))))
C1.0(c.0(b.0(b.0(c.0(a.0(b.0(b.0(B.1(x0))))))))) → C1.0(c.0(a.0(a.0(a.0(a.0(c.0(c.0(c.0(b.0(b.0(b.0(b.0(B.1(x0))))))))))))))
C1.0(c.0(b.0(b.0(c.0(b.0(a.0(b.0(b.0(B.1(x0)))))))))) → C1.0(c.0(a.0(a.0(a.0(a.0(c.0(c.0(c.0(b.0(b.0(b.0(b.0(B.1(x0))))))))))))))
C1.0(c.0(b.0(b.0(c.0(b.0(a.0(b.0(b.0(B.1(x0)))))))))) → C1.0(c.0(a.0(a.0(a.0(a.0(c.0(c.0(c.0(b.0(b.0(b.0(b.0(b.0(B.1(x0)))))))))))))))
C1.0(c.0(b.0(b.0(c.0(a.0(b.0(b.0(B.0(x0))))))))) → C1.0(c.0(a.0(a.0(a.0(a.0(c.0(c.0(c.0(b.0(b.0(b.0(b.0(B.0(x0))))))))))))))
C1.0(c.0(b.0(b.0(c.0(b.0(a.0(b.0(b.0(B.1(x0)))))))))) → C1.0(c.0(a.0(a.0(a.0(a.0(c.0(c.0(c.0(b.0(b.0(b.0(b.0(b.0(b.0(B.1(x0))))))))))))))))
C1.0(c.0(b.0(b.0(c.0(a.0(b.0(b.0(B.1(x0))))))))) → C1.0(c.0(a.0(a.0(a.0(a.0(c.0(c.0(c.0(b.0(b.0(b.0(b.0(b.0(B.1(x0)))))))))))))))
C1.0(c.0(b.0(a.0(b.0(b.0(B.0(x))))))) → C1.0(c.0(b.0(b.0(B.0(x)))))
C1.0(c.0(b.0(b.0(c.0(b.0(B.0(x0))))))) → C1.0(c.0(a.0(a.0(a.0(a.0(c.1(C.0(x0))))))))
C1.0(c.0(b.0(a.0(b.0(b.0(B.0(x))))))) → C1.0(c.0(b.0(b.0(b.0(b.0(b.0(b.0(B.0(x)))))))))
C1.0(c.0(b.0(a.0(b.0(b.0(B.1(x))))))) → C1.0(c.0(b.0(b.0(b.0(b.0(b.0(B.1(x))))))))
C1.0(c.0(a.0(b.0(b.0(B.1(x)))))) → C1.0(c.0(b.0(b.0(B.1(x)))))
C1.0(c.0(a.0(b.0(b.0(B.0(x)))))) → C1.0(c.0(b.0(b.0(b.0(b.0(b.0(B.0(x))))))))
C1.0(c.0(b.0(b.0(c.0(x0))))) → C1.0(c.0(b.0(b.0(b.0(a.0(a.0(a.0(x0))))))))
C1.0(c.0(b.0(b.0(c.0(b.0(b.0(c.0(y_4)))))))) → C1.0(c.0(b.0(b.0(c.0(y_4)))))
C1.0(c.0(b.0(b.0(c.0(a.0(b.0(b.0(B.0(x0))))))))) → C1.0(c.0(a.0(a.0(a.0(a.0(c.0(c.0(c.0(b.0(b.0(B.0(x0))))))))))))
C1.0(c.0(a.0(b.0(b.0(B.0(x)))))) → C1.0(c.0(b.0(b.0(b.0(B.0(x))))))
C1.0(c.0(b.0(b.0(c.0(y_2))))) → C1.0(c.0(y_2))
C1.0(c.0(b.0(b.0(c.0(b.0(b.0(c.1(y_4)))))))) → C1.0(c.0(b.0(b.0(c.1(y_4)))))
C1.0(c.0(b.0(b.0(c.0(b.0(a.0(b.0(b.0(B.1(x0)))))))))) → C1.0(c.0(a.0(a.0(a.0(a.0(c.0(c.0(c.0(b.0(b.0(B.1(x0))))))))))))
C1.0(c.0(b.0(b.0(c.0(a.0(b.0(b.0(B.1(x0))))))))) → C1.0(c.0(a.0(a.0(a.0(a.0(c.0(c.0(c.0(b.0(b.0(b.0(B.1(x0)))))))))))))
C1.0(c.0(a.0(b.0(b.0(B.1(x)))))) → C1.0(c.0(b.0(b.0(b.0(B.1(x))))))
C1.0(c.0(b.0(b.0(c.0(a.0(b.0(b.0(B.1(x0))))))))) → C1.0(c.0(a.0(a.0(a.0(a.0(c.0(c.0(c.0(b.0(b.0(B.1(x0))))))))))))
C1.0(c.0(a.0(b.0(b.0(B.0(x)))))) → C1.0(c.0(b.0(b.0(B.0(x)))))
C1.0(c.0(b.0(a.0(b.0(b.0(B.1(x))))))) → C1.0(c.0(b.0(b.0(b.0(b.0(B.1(x)))))))
C1.0(c.0(b.0(b.0(c.0(a.0(b.0(b.0(B.0(x0))))))))) → C1.0(c.0(a.0(a.0(a.0(a.0(c.0(c.0(c.0(b.0(b.0(b.0(b.0(b.0(B.0(x0)))))))))))))))
C1.0(c.0(b.0(a.0(b.0(b.0(B.1(x))))))) → C1.0(c.0(b.0(b.0(B.1(x)))))
C1.0(c.0(b.0(b.0(c.0(a.0(b.0(b.0(B.0(x0))))))))) → C1.0(c.0(a.0(a.0(a.0(a.0(c.0(c.0(c.0(b.0(B.0(x0)))))))))))
C1.0(c.0(b.0(b.0(c.0(b.0(B.0(x0))))))) → C1.0(c.0(a.0(a.0(a.0(a.0(c.0(c.1(C.0(x0)))))))))
C1.0(c.0(b.0(a.0(b.0(b.0(B.0(x))))))) → C1.0(c.0(b.0(b.0(b.0(b.0(b.0(B.0(x))))))))
C1.0(c.0(b.0(b.0(c.0(b.0(a.0(b.0(b.0(B.0(x0)))))))))) → C1.0(c.0(a.0(a.0(a.0(a.0(c.0(c.0(c.0(b.0(b.0(b.0(b.0(b.0(B.0(x0)))))))))))))))
C1.0(c.0(b.0(b.0(c.0(b.0(a.0(b.0(b.0(B.0(x0)))))))))) → C1.0(c.0(a.0(a.0(a.0(a.0(c.0(c.0(c.0(b.0(b.0(B.0(x0))))))))))))
C1.0(c.0(b.0(b.0(c.0(b.0(B.1(x0))))))) → C1.0(c.0(a.0(a.0(a.0(a.0(c.0(c.1(C.1(x0)))))))))
C1.0(c.0(b.0(b.0(c.0(b.0(a.0(b.0(b.0(B.1(x0)))))))))) → C1.0(c.0(a.0(a.0(a.0(a.0(c.0(c.0(c.0(b.0(b.0(b.0(B.1(x0)))))))))))))
C1.0(c.0(b.0(b.0(c.0(a.0(b.0(b.0(B.0(x0))))))))) → C1.0(c.0(a.0(a.0(a.0(a.0(c.0(c.0(c.0(b.0(b.0(b.0(B.0(x0)))))))))))))
C1.0(c.0(a.0(b.0(b.0(B.1(x)))))) → C1.0(c.0(b.0(b.0(b.0(b.0(B.1(x)))))))
C1.0(c.0(b.0(a.0(b.0(b.0(B.0(x))))))) → C1.0(c.0(b.0(b.0(b.0(b.0(B.0(x)))))))
C1.0(c.0(b.0(a.0(b.0(b.0(B.0(x))))))) → C1.0(c.0(b.0(b.0(b.0(B.0(x))))))
C1.0(c.0(b.0(b.0(c.0(b.0(a.0(b.0(b.0(B.0(x0)))))))))) → C1.0(c.0(a.0(a.0(a.0(a.0(c.0(c.0(c.0(b.0(b.0(b.0(B.0(x0)))))))))))))
C1.0(c.0(b.0(b.0(x)))) → C1.0(c.0(x))
C1.0(c.0(a.0(b.0(b.0(B.1(x)))))) → C1.0(c.0(b.0(b.0(b.0(b.0(b.0(B.1(x))))))))
C1.0(c.0(a.0(b.0(b.0(B.0(x)))))) → C1.0(c.0(b.0(b.0(b.0(b.0(B.0(x)))))))
The TRS R consists of the following rules:
c.1(C.1(x)) → b.0(B.1(x))
c.0(c.0(b.0(B.0(x)))) → a.0(a.0(a.0(a.0(c.1(C.0(x))))))
a.0(b.0(b.0(b.1(x)))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.1(x))))))))
a.0(b.0(b.0(a.0(b.0(b.0(B.0(x))))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.0(x)))))))))))))))
a.0(b.0(b.0(B.0(x)))) → b.0(b.0(B.0(x)))
a.0(b.0(a.0(b.0(b.0(B.1(x)))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.1(x))))))))))))))
a.0(b.0(b.0(a.0(b.0(b.0(B.1(x))))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.1(x)))))))))))))))
a.0(a.0(b.0(b.0(B.1(x))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.1(x)))))))))))))
a.0(a.0(b.0(b.0(B.0(x))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.0(x)))))))))))))
a.0(b.0(b.0(B.0(x)))) → b.0(B.0(x))
a.0(b.0(a.0(b.0(b.0(B.0(x)))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.0(x))))))))))))))
a.0(b.0(b.0(B.1(x)))) → b.0(B.1(x))
a.0(b.0(b.0(B.1(x)))) → B.1(x)
c.0(c.0(b.0(a.0(b.0(b.0(B.0(x))))))) → a.0(a.0(a.0(a.0(c.0(c.0(c.0(b.0(b.0(b.0(B.0(x)))))))))))
c.0(c.0(b.0(a.0(b.0(b.0(B.1(x))))))) → a.0(a.0(a.0(a.0(c.0(c.0(c.0(b.0(b.0(b.0(B.1(x)))))))))))
a.0(b.0(b.0(a.0(b.0(b.0(B.1(x))))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.1(x))))))))))))))
c.0(c.0(b.0(b.0(x)))) → a.0(a.0(a.0(a.0(c.0(c.0(c.0(x)))))))
c.0(c.0(a.0(b.0(b.0(B.0(x)))))) → a.0(a.0(a.0(a.0(c.0(c.0(c.0(b.0(B.0(x)))))))))
a.0(a.0(b.0(b.0(B.0(x))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.0(x))))))))))
c.0(c.0(b.0(a.0(b.0(b.0(B.0(x))))))) → a.0(a.0(a.0(a.0(c.0(c.0(c.0(b.0(b.0(b.0(b.0(b.0(B.0(x)))))))))))))
a.0(a.0(b.0(b.0(B.0(x))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.0(x))))))))))))
c.0(c.0(a.0(b.0(b.0(B.1(x)))))) → a.0(a.0(a.0(a.0(c.0(c.0(c.0(b.0(b.0(b.0(B.1(x)))))))))))
c.0(c.0(b.0(B.1(x)))) → a.0(a.0(a.0(a.0(c.0(c.1(C.1(x)))))))
c.0(c.0(b.0(a.0(b.0(b.0(B.0(x))))))) → a.0(a.0(a.0(a.0(c.0(c.0(c.0(b.0(b.0(B.0(x))))))))))
a.0(b.0(b.0(a.0(b.0(b.0(B.1(x))))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.1(x)))))))))))))
c.1(C.0(x)) → b.0(b.0(B.0(x)))
a.0(b.0(a.0(b.0(b.0(B.0(x)))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.0(x))))))))))))
a.0(b.0(a.0(b.0(b.0(B.1(x)))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.1(x)))))))))))))
a.0(a.0(b.0(b.0(B.0(x))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.0(x)))))))))))
c.0(c.0(b.0(a.0(b.0(b.0(B.0(x))))))) → a.0(a.0(a.0(a.0(c.0(c.0(c.0(b.0(b.0(b.0(b.0(B.0(x))))))))))))
c.0(c.0(a.0(b.0(b.0(B.0(x)))))) → a.0(a.0(a.0(a.0(c.0(c.0(c.0(b.0(b.0(b.0(B.0(x)))))))))))
c.0(c.0(a.0(b.0(b.0(B.1(x)))))) → a.0(a.0(a.0(a.0(c.0(c.0(c.0(b.0(b.0(B.1(x))))))))))
c.0(c.0(b.0(a.0(b.0(b.0(B.1(x))))))) → a.0(a.0(a.0(a.0(c.0(c.0(c.0(b.0(b.0(b.0(b.0(b.0(B.1(x)))))))))))))
c.0(c.0(a.0(b.0(b.0(B.1(x)))))) → a.0(a.0(a.0(a.0(c.0(c.0(c.0(b.0(b.0(b.0(b.0(b.0(B.1(x)))))))))))))
a.0(a.0(b.0(b.0(B.0(x))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.0(x)))))))))
a.0(b.0(a.0(b.0(b.0(B.1(x)))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.1(x))))))))))
a.0(a.0(b.0(b.0(B.1(x))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.1(x)))))))))
c.0(c.0(b.0(a.0(b.0(b.0(B.1(x))))))) → a.0(a.0(a.0(a.0(c.0(c.0(c.0(b.0(b.0(b.0(b.0(b.0(b.0(B.1(x))))))))))))))
c.0(c.0(b.0(B.1(x)))) → a.0(a.0(a.0(a.0(c.1(C.1(x))))))
c.0(c.0(a.0(b.0(b.0(B.0(x)))))) → a.0(a.0(a.0(a.0(c.0(c.0(c.0(b.0(b.0(b.0(b.0(B.0(x))))))))))))
a.0(b.0(b.0(B.1(x)))) → b.0(b.0(B.1(x)))
a.0(b.0(a.0(b.0(b.0(B.0(x)))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.0(x)))))))))))
a.0(b.0(b.0(a.0(b.0(b.0(B.1(x))))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.1(x)))))))))))
a.0(a.0(b.0(b.0(B.1(x))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.1(x)))))))))))
a.0(b.0(b.0(a.0(b.0(b.0(B.0(x))))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.0(x))))))))))))
a.0(a.0(b.0(b.0(B.1(x))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.1(x))))))))))))
c.1(C.1(x)) → b.0(b.0(B.1(x)))
a.0(b.0(a.0(b.0(b.0(B.1(x)))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.1(x))))))))))))
a.0(b.0(b.0(a.0(b.0(b.0(B.0(x))))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.0(x))))))))))))))
a.0(b.0(b.0(a.0(b.0(b.0(B.0(x))))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.0(x)))))))))))))
a.0(a.0(b.0(b.0(B.1(x))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.1(x))))))))))
a.0(b.0(b.0(B.0(x)))) → B.0(x)
c.0(c.0(b.0(B.0(x)))) → a.0(a.0(a.0(a.0(c.0(c.1(C.0(x)))))))
a.0(b.0(b.0(b.0(x)))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(x))))))))
a.0(b.0(a.0(b.0(b.0(B.0(x)))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.0(x))))))))))
a.0(b.0(b.0(a.0(b.0(b.0(B.1(x))))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.1(x))))))))))))
c.0(c.0(a.0(b.0(b.0(B.1(x)))))) → a.0(a.0(a.0(a.0(c.0(c.0(c.0(b.0(b.0(b.0(b.0(B.1(x))))))))))))
c.0(c.0(b.0(a.0(b.0(b.0(B.0(x))))))) → a.0(a.0(a.0(a.0(c.0(c.0(c.0(b.0(b.0(b.0(b.0(b.0(b.0(B.0(x))))))))))))))
c.0(c.0(a.0(b.0(b.0(B.1(x)))))) → a.0(a.0(a.0(a.0(c.0(c.0(c.0(b.0(B.1(x)))))))))
c.0(c.0(b.0(a.0(b.0(b.0(B.1(x))))))) → a.0(a.0(a.0(a.0(c.0(c.0(c.0(b.0(b.0(B.1(x))))))))))
a.0(b.0(b.0(a.0(b.0(b.0(B.0(x))))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.0(x)))))))))))
c.0(c.0(a.0(b.0(b.0(B.0(x)))))) → a.0(a.0(a.0(a.0(c.0(c.0(c.0(b.0(b.0(B.0(x))))))))))
c.0(c.0(a.0(b.0(b.0(B.0(x)))))) → a.0(a.0(a.0(a.0(c.0(c.0(c.0(b.0(b.0(b.0(b.0(b.0(B.0(x)))))))))))))
c.0(c.0(x)) → b.0(b.0(b.0(a.0(a.0(a.0(x))))))
a.0(b.0(a.0(b.0(b.0(B.0(x)))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.0(x)))))))))))))
c.1(C.0(x)) → b.0(B.0(x))
a.0(b.0(a.0(b.0(b.0(B.1(x)))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.1(x)))))))))))
c.0(c.0(b.0(a.0(b.0(b.0(B.1(x))))))) → a.0(a.0(a.0(a.0(c.0(c.0(c.0(b.0(b.0(b.0(b.0(B.1(x))))))))))))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
As can be seen after transforming the QDP problem by semantic labelling [33] and then some rule deleting processors, only certain labelled rules and pairs can be used.
Hence, we only have to consider all unlabelled pairs and rules (without the decreasing rules for quasi-models).
↳ QTRS
↳ QTRS Reverse
↳ QTRS Reverse
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ SemLabProof
↳ SemLabProof2
↳ QDP
↳ QDPToSRSProof
↳ QTRS
↳ QTRS Reverse
↳ QTRS
↳ QTRS Reverse
↳ QTRS Reverse
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ SemLabProof
↳ SemLabProof2
↳ QDP
↳ SemLabProof
Q DP problem:
The TRS P consists of the following rules:
C1(c(b(b(c(a(b(b(B(x0))))))))) → C1(c(a(a(a(a(c(c(c(b(b(b(b(B(x0))))))))))))))
C1(c(b(b(c(a(b(b(B(x0))))))))) → C1(c(a(a(a(a(c(c(c(b(b(b(B(x0)))))))))))))
C1(c(a(b(b(B(x)))))) → C1(c(b(b(b(B(x))))))
C1(c(b(b(c(b(a(b(b(B(x0)))))))))) → C1(c(a(a(a(a(c(c(c(b(b(B(x0))))))))))))
C1(c(b(a(b(b(B(x))))))) → C1(c(b(b(b(B(x))))))
C1(c(b(a(b(b(B(x))))))) → C1(c(b(b(b(b(B(x)))))))
C1(c(b(b(x)))) → C1(c(x))
C1(c(b(b(c(b(a(b(b(B(x0)))))))))) → C1(c(a(a(a(a(c(c(c(b(b(b(b(b(B(x0)))))))))))))))
C1(c(b(b(c(a(b(b(B(x0))))))))) → C1(c(a(a(a(a(c(c(c(b(B(x0)))))))))))
C1(c(b(b(c(a(b(b(B(x0))))))))) → C1(c(a(a(a(a(c(c(c(b(b(B(x0))))))))))))
C1(c(b(b(c(y_2))))) → C1(c(y_2))
C1(c(b(a(b(b(B(x))))))) → C1(c(b(b(b(b(b(B(x))))))))
C1(c(b(b(c(b(b(c(y_4)))))))) → C1(c(b(b(c(y_4)))))
C1(c(b(b(c(x0))))) → C1(c(b(b(b(a(a(a(x0))))))))
C1(c(a(b(b(B(x)))))) → C1(c(b(b(B(x)))))
C1(c(b(b(c(b(a(b(b(B(x0)))))))))) → C1(c(a(a(a(a(c(c(c(b(b(b(B(x0)))))))))))))
C1(c(a(b(b(B(x)))))) → C1(c(b(b(b(b(B(x)))))))
C1(c(b(a(b(b(B(x))))))) → C1(c(b(b(B(x)))))
C1(c(b(b(c(b(B(x0))))))) → C1(c(a(a(a(a(c(C(x0))))))))
C1(c(b(a(b(b(B(x))))))) → C1(c(b(b(b(b(b(b(B(x)))))))))
C1(c(a(b(b(B(x)))))) → C1(c(b(b(b(b(b(B(x))))))))
C1(c(b(b(c(b(a(b(b(B(x0)))))))))) → C1(c(a(a(a(a(c(c(c(b(b(b(b(B(x0))))))))))))))
C1(c(b(b(c(b(b(x0))))))) → C1(c(a(a(a(a(c(c(c(x0)))))))))
C1(c(b(b(c(b(a(b(b(B(x0)))))))))) → C1(c(a(a(a(a(c(c(c(b(b(b(b(b(b(B(x0))))))))))))))))
C1(c(b(b(c(b(B(x0))))))) → C1(c(a(a(a(a(c(c(C(x0)))))))))
C1(c(b(b(c(a(b(b(B(x0))))))))) → C1(c(a(a(a(a(c(c(c(b(b(b(b(b(B(x0)))))))))))))))
The TRS R consists of the following rules:
c(c(x)) → b(b(b(a(a(a(x))))))
a(b(b(b(x)))) → b(b(b(b(b(b(b(b(x))))))))
c(c(b(b(x)))) → a(a(a(a(c(c(c(x)))))))
c(c(b(a(b(b(B(x))))))) → a(a(a(a(c(c(c(b(b(b(b(b(b(B(x))))))))))))))
a(a(b(b(B(x))))) → b(b(b(b(b(b(b(b(b(B(x))))))))))
a(b(a(b(b(B(x)))))) → b(b(b(b(b(b(b(b(b(B(x))))))))))
a(a(b(b(B(x))))) → b(b(b(b(b(b(b(b(b(b(b(B(x))))))))))))
a(b(a(b(b(B(x)))))) → b(b(b(b(b(b(b(b(b(b(b(B(x))))))))))))
a(b(b(a(b(b(B(x))))))) → b(b(b(b(b(b(b(b(b(b(b(B(x))))))))))))
a(b(b(a(b(b(B(x))))))) → b(b(b(b(b(b(b(b(b(b(B(x)))))))))))
a(a(b(b(B(x))))) → b(b(b(b(b(b(b(b(b(b(b(b(B(x)))))))))))))
c(c(a(b(b(B(x)))))) → a(a(a(a(c(c(c(b(b(b(B(x)))))))))))
a(b(b(B(x)))) → b(b(B(x)))
a(b(b(B(x)))) → B(x)
c(c(b(a(b(b(B(x))))))) → a(a(a(a(c(c(c(b(b(b(b(b(B(x)))))))))))))
a(b(b(a(b(b(B(x))))))) → b(b(b(b(b(b(b(b(b(b(b(b(b(B(x))))))))))))))
c(c(b(a(b(b(B(x))))))) → a(a(a(a(c(c(c(b(b(b(b(B(x))))))))))))
a(b(a(b(b(B(x)))))) → b(b(b(b(b(b(b(b(b(b(b(b(B(x)))))))))))))
a(b(a(b(b(B(x)))))) → b(b(b(b(b(b(b(b(b(b(B(x)))))))))))
a(b(b(a(b(b(B(x))))))) → b(b(b(b(b(b(b(b(b(b(b(b(B(x)))))))))))))
c(c(a(b(b(B(x)))))) → a(a(a(a(c(c(c(b(b(b(b(b(B(x)))))))))))))
a(b(b(a(b(b(B(x))))))) → b(b(b(b(b(b(b(b(b(b(b(b(b(b(B(x)))))))))))))))
c(c(a(b(b(B(x)))))) → a(a(a(a(c(c(c(b(b(b(b(B(x))))))))))))
a(b(a(b(b(B(x)))))) → b(b(b(b(b(b(b(b(b(b(b(b(b(B(x))))))))))))))
c(c(b(a(b(b(B(x))))))) → a(a(a(a(c(c(c(b(b(B(x))))))))))
c(c(a(b(b(B(x)))))) → a(a(a(a(c(c(c(b(b(B(x))))))))))
c(c(b(B(x)))) → a(a(a(a(c(C(x))))))
a(a(b(b(B(x))))) → b(b(b(b(b(b(b(b(b(b(B(x)))))))))))
c(C(x)) → b(B(x))
c(c(a(b(b(B(x)))))) → a(a(a(a(c(c(c(b(B(x)))))))))
c(c(b(a(b(b(B(x))))))) → a(a(a(a(c(c(c(b(b(b(B(x)))))))))))
a(b(b(B(x)))) → b(B(x))
c(c(b(B(x)))) → a(a(a(a(c(c(C(x)))))))
c(C(x)) → b(b(B(x)))
a(a(b(b(B(x))))) → b(b(b(b(b(b(b(b(B(x)))))))))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We found the following quasi-model for the rules of the TRS R.
Interpretation over the domain with elements from 0 to 1.C: 0
c: 1
B: 0
a: 0
b: 0
C1: 0
By semantic labelling [33] we obtain the following labelled TRS:Q DP problem:
The TRS P consists of the following rules:
C1.1(c.0(b.0(a.0(b.0(b.0(B.1(x))))))) → C1.0(c.0(b.0(b.0(b.0(B.1(x))))))
C1.1(c.0(b.0(b.1(c.0(b.0(a.0(b.0(b.0(B.1(x0)))))))))) → C1.0(c.0(a.0(a.0(a.0(a.1(c.1(c.1(c.0(b.0(b.0(b.0(b.0(b.0(b.0(B.1(x0))))))))))))))))
C1.1(c.0(b.0(b.1(c.0(a.0(b.0(b.0(B.0(x0))))))))) → C1.0(c.0(a.0(a.0(a.0(a.1(c.1(c.1(c.0(b.0(b.0(b.0(b.0(b.0(B.0(x0)))))))))))))))
C1.1(c.0(b.0(b.1(c.0(b.0(B.1(x0))))))) → C1.0(c.0(a.0(a.0(a.0(a.1(c.1(c.0(C.1(x0)))))))))
C1.1(c.0(b.0(a.0(b.0(b.0(B.0(x))))))) → C1.1(c.0(b.0(b.0(B.0(x)))))
C1.1(c.0(b.0(b.1(c.0(b.0(B.0(x0))))))) → C1.0(c.0(a.0(a.0(a.0(a.1(c.0(C.0(x0))))))))
C1.1(c.0(b.0(a.0(b.0(b.0(B.0(x))))))) → C1.1(c.0(b.0(b.0(b.0(b.0(B.0(x)))))))
C1.1(c.0(b.0(b.1(c.0(b.0(b.0(x0))))))) → C1.1(c.0(a.0(a.0(a.0(a.1(c.1(c.1(c.0(x0)))))))))
C1.1(c.0(b.0(b.1(c.0(a.0(b.0(b.0(B.0(x0))))))))) → C1.0(c.0(a.0(a.0(a.0(a.1(c.1(c.1(c.0(b.0(b.0(b.0(B.0(x0)))))))))))))
C1.1(c.0(b.0(b.1(c.0(b.0(b.0(x0))))))) → C1.0(c.0(a.0(a.0(a.0(a.1(c.1(c.1(c.0(x0)))))))))
C1.1(c.0(b.0(b.1(c.0(b.0(b.1(c.0(y_4)))))))) → C1.0(c.0(b.0(b.1(c.0(y_4)))))
C1.1(c.0(a.0(b.0(b.0(B.0(x)))))) → C1.0(c.0(b.0(b.0(b.0(b.0(b.0(B.0(x))))))))
C1.1(c.0(b.0(b.1(x)))) → C1.0(c.1(x))
C1.1(c.0(b.0(a.0(b.0(b.0(B.1(x))))))) → C1.1(c.0(b.0(b.0(b.0(b.0(b.0(B.1(x))))))))
C1.1(c.0(b.0(b.1(c.0(b.0(b.1(x0))))))) → C1.1(c.0(a.0(a.0(a.0(a.1(c.1(c.1(c.1(x0)))))))))
C1.1(c.0(b.0(b.1(c.0(b.0(a.0(b.0(b.0(B.0(x0)))))))))) → C1.1(c.0(a.0(a.0(a.0(a.1(c.1(c.1(c.0(b.0(b.0(b.0(b.0(B.0(x0))))))))))))))
C1.1(c.0(b.0(b.1(c.0(a.0(b.0(b.0(B.0(x0))))))))) → C1.0(c.0(a.0(a.0(a.0(a.1(c.1(c.1(c.0(b.0(b.0(B.0(x0))))))))))))
C1.1(c.0(a.0(b.0(b.0(B.1(x)))))) → C1.0(c.0(b.0(b.0(b.0(b.0(B.1(x)))))))
C1.1(c.0(b.0(b.1(c.0(b.0(b.1(x0))))))) → C1.0(c.0(a.0(a.0(a.0(a.1(c.1(c.1(c.1(x0)))))))))
C1.1(c.0(b.0(b.1(c.0(a.0(b.0(b.0(B.1(x0))))))))) → C1.1(c.0(a.0(a.0(a.0(a.1(c.1(c.1(c.0(b.0(B.1(x0)))))))))))
C1.1(c.0(a.0(b.0(b.0(B.1(x)))))) → C1.0(c.0(b.0(b.0(B.1(x)))))
C1.1(c.0(b.0(b.1(c.0(b.0(a.0(b.0(b.0(B.0(x0)))))))))) → C1.0(c.0(a.0(a.0(a.0(a.1(c.1(c.1(c.0(b.0(b.0(B.0(x0))))))))))))
C1.1(c.0(b.0(a.0(b.0(b.0(B.0(x))))))) → C1.0(c.0(b.0(b.0(b.0(B.0(x))))))
C1.1(c.0(b.0(b.1(c.0(b.0(a.0(b.0(b.0(B.1(x0)))))))))) → C1.1(c.0(a.0(a.0(a.0(a.1(c.1(c.1(c.0(b.0(b.0(b.0(b.0(b.0(b.0(B.1(x0))))))))))))))))
C1.1(c.0(b.0(b.1(c.0(b.0(b.1(c.1(y_4)))))))) → C1.1(c.0(b.0(b.1(c.1(y_4)))))
C1.1(c.0(a.0(b.0(b.0(B.0(x)))))) → C1.1(c.0(b.0(b.0(b.0(B.0(x))))))
C1.1(c.0(b.0(b.1(c.1(y_2))))) → C1.0(c.1(y_2))
C1.1(c.0(b.0(b.1(c.0(b.0(a.0(b.0(b.0(B.0(x0)))))))))) → C1.0(c.0(a.0(a.0(a.0(a.1(c.1(c.1(c.0(b.0(b.0(b.0(b.0(b.0(b.0(B.0(x0))))))))))))))))
C1.1(c.0(b.0(b.1(c.0(y_2))))) → C1.1(c.0(y_2))
C1.1(c.0(b.0(b.1(c.0(b.0(a.0(b.0(b.0(B.1(x0)))))))))) → C1.0(c.0(a.0(a.0(a.0(a.1(c.1(c.1(c.0(b.0(b.0(b.0(b.0(B.1(x0))))))))))))))
C1.1(c.0(a.0(b.0(b.0(B.1(x)))))) → C1.1(c.0(b.0(b.0(B.1(x)))))
C1.1(c.0(b.0(b.1(c.0(a.0(b.0(b.0(B.1(x0))))))))) → C1.0(c.0(a.0(a.0(a.0(a.1(c.1(c.1(c.0(b.0(b.0(b.0(B.1(x0)))))))))))))
C1.1(c.0(b.0(a.0(b.0(b.0(B.1(x))))))) → C1.0(c.0(b.0(b.0(B.1(x)))))
C1.1(c.0(b.0(b.1(c.0(b.0(a.0(b.0(b.0(B.0(x0)))))))))) → C1.0(c.0(a.0(a.0(a.0(a.1(c.1(c.1(c.0(b.0(b.0(b.0(B.0(x0)))))))))))))
C1.1(c.0(b.0(b.1(c.0(b.0(a.0(b.0(b.0(B.1(x0)))))))))) → C1.1(c.0(a.0(a.0(a.0(a.1(c.1(c.1(c.0(b.0(b.0(b.0(b.0(b.0(B.1(x0)))))))))))))))
C1.1(c.0(b.0(b.1(c.0(y_2))))) → C1.0(c.0(y_2))
C1.1(c.0(b.0(b.1(c.0(b.0(a.0(b.0(b.0(B.0(x0)))))))))) → C1.1(c.0(a.0(a.0(a.0(a.1(c.1(c.1(c.0(b.0(b.0(b.0(B.0(x0)))))))))))))
C1.1(c.0(b.0(b.1(c.0(a.0(b.0(b.0(B.1(x0))))))))) → C1.0(c.0(a.0(a.0(a.0(a.1(c.1(c.1(c.0(b.0(B.1(x0)))))))))))
C1.1(c.0(b.0(b.1(c.0(b.0(a.0(b.0(b.0(B.0(x0)))))))))) → C1.0(c.0(a.0(a.0(a.0(a.1(c.1(c.1(c.0(b.0(b.0(b.0(b.0(B.0(x0))))))))))))))
C1.1(c.0(b.0(b.1(c.0(b.0(a.0(b.0(b.0(B.1(x0)))))))))) → C1.1(c.0(a.0(a.0(a.0(a.1(c.1(c.1(c.0(b.0(b.0(b.0(B.1(x0)))))))))))))
C1.1(c.0(a.0(b.0(b.0(B.0(x)))))) → C1.1(c.0(b.0(b.0(B.0(x)))))
C1.1(c.0(b.0(b.1(c.0(b.0(a.0(b.0(b.0(B.0(x0)))))))))) → C1.0(c.0(a.0(a.0(a.0(a.1(c.1(c.1(c.0(b.0(b.0(b.0(b.0(b.0(B.0(x0)))))))))))))))
C1.1(c.0(b.0(a.0(b.0(b.0(B.1(x))))))) → C1.1(c.0(b.0(b.0(B.1(x)))))
C1.1(c.0(b.0(b.1(c.0(b.0(b.1(c.0(y_4)))))))) → C1.1(c.0(b.0(b.1(c.0(y_4)))))
C1.1(c.0(b.0(b.1(c.0(b.0(a.0(b.0(b.0(B.1(x0)))))))))) → C1.0(c.0(a.0(a.0(a.0(a.1(c.1(c.1(c.0(b.0(b.0(b.0(b.0(b.0(B.1(x0)))))))))))))))
C1.1(c.0(b.0(b.0(x)))) → C1.0(c.0(x))
C1.1(c.0(b.0(b.1(c.0(a.0(b.0(b.0(B.0(x0))))))))) → C1.1(c.0(a.0(a.0(a.0(a.1(c.1(c.1(c.0(b.0(b.0(b.0(b.0(b.0(B.0(x0)))))))))))))))
C1.1(c.0(b.0(a.0(b.0(b.0(B.0(x))))))) → C1.1(c.0(b.0(b.0(b.0(B.0(x))))))
C1.1(c.0(b.0(b.1(c.0(a.0(b.0(b.0(B.0(x0))))))))) → C1.0(c.0(a.0(a.0(a.0(a.1(c.1(c.1(c.0(b.0(B.0(x0)))))))))))
C1.1(c.0(a.0(b.0(b.0(B.0(x)))))) → C1.1(c.0(b.0(b.0(b.0(b.0(B.0(x)))))))
C1.1(c.0(b.0(b.1(c.0(b.0(B.1(x0))))))) → C1.0(c.0(a.0(a.0(a.0(a.1(c.0(C.1(x0))))))))
C1.1(c.0(b.0(a.0(b.0(b.0(B.1(x))))))) → C1.1(c.0(b.0(b.0(b.0(b.0(b.0(b.0(B.1(x)))))))))
C1.1(c.0(b.0(b.1(c.0(a.0(b.0(b.0(B.0(x0))))))))) → C1.0(c.0(a.0(a.0(a.0(a.1(c.1(c.1(c.0(b.0(b.0(b.0(b.0(B.0(x0))))))))))))))
C1.1(c.0(b.0(b.1(c.0(b.0(a.0(b.0(b.0(B.1(x0)))))))))) → C1.1(c.0(a.0(a.0(a.0(a.1(c.1(c.1(c.0(b.0(b.0(b.0(b.0(B.1(x0))))))))))))))
C1.1(c.0(b.0(b.1(c.0(b.0(B.0(x0))))))) → C1.1(c.0(a.0(a.0(a.0(a.1(c.0(C.0(x0))))))))
C1.1(c.0(a.0(b.0(b.0(B.1(x)))))) → C1.1(c.0(b.0(b.0(b.0(b.0(B.1(x)))))))
C1.1(c.0(b.0(b.1(c.0(a.0(b.0(b.0(B.1(x0))))))))) → C1.1(c.0(a.0(a.0(a.0(a.1(c.1(c.1(c.0(b.0(b.0(b.0(b.0(B.1(x0))))))))))))))
C1.1(c.0(b.0(b.1(c.0(a.0(b.0(b.0(B.1(x0))))))))) → C1.0(c.0(a.0(a.0(a.0(a.1(c.1(c.1(c.0(b.0(b.0(B.1(x0))))))))))))
C1.1(c.0(b.0(b.1(c.0(b.0(b.1(c.1(y_4)))))))) → C1.0(c.0(b.0(b.1(c.1(y_4)))))
C1.1(c.0(b.0(a.0(b.0(b.0(B.0(x))))))) → C1.1(c.0(b.0(b.0(b.0(b.0(b.0(B.0(x))))))))
C1.1(c.0(a.0(b.0(b.0(B.1(x)))))) → C1.1(c.0(b.0(b.0(b.0(b.0(b.0(B.1(x))))))))
C1.1(c.0(b.0(b.1(c.0(a.0(b.0(b.0(B.0(x0))))))))) → C1.1(c.0(a.0(a.0(a.0(a.1(c.1(c.1(c.0(b.0(b.0(B.0(x0))))))))))))
C1.1(c.0(b.0(a.0(b.0(b.0(B.0(x))))))) → C1.0(c.0(b.0(b.0(b.0(b.0(B.0(x)))))))
C1.1(c.0(b.0(b.1(c.0(a.0(b.0(b.0(B.1(x0))))))))) → C1.1(c.0(a.0(a.0(a.0(a.1(c.1(c.1(c.0(b.0(b.0(b.0(B.1(x0)))))))))))))
C1.1(c.0(b.0(a.0(b.0(b.0(B.0(x))))))) → C1.0(c.0(b.0(b.0(b.0(b.0(b.0(b.0(B.0(x)))))))))
C1.1(c.0(b.0(b.1(c.0(a.0(b.0(b.0(B.0(x0))))))))) → C1.1(c.0(a.0(a.0(a.0(a.1(c.1(c.1(c.0(b.0(b.0(b.0(B.0(x0)))))))))))))
C1.1(c.0(a.0(b.0(b.0(B.0(x)))))) → C1.0(c.0(b.0(b.0(B.0(x)))))
C1.1(c.0(b.0(b.1(c.0(a.0(b.0(b.0(B.0(x0))))))))) → C1.1(c.0(a.0(a.0(a.0(a.1(c.1(c.1(c.0(b.0(b.0(b.0(b.0(B.0(x0))))))))))))))
C1.1(c.0(b.0(b.1(c.1(y_2))))) → C1.1(c.1(y_2))
C1.1(c.0(b.0(b.1(c.1(x0))))) → C1.1(c.0(b.0(b.0(b.0(a.0(a.0(a.1(x0))))))))
C1.1(c.0(b.0(b.1(c.0(a.0(b.0(b.0(B.0(x0))))))))) → C1.1(c.0(a.0(a.0(a.0(a.1(c.1(c.1(c.0(b.0(B.0(x0)))))))))))
C1.1(c.0(b.0(a.0(b.0(b.0(B.1(x))))))) → C1.1(c.0(b.0(b.0(b.0(b.0(B.1(x)))))))
C1.1(c.0(a.0(b.0(b.0(B.1(x)))))) → C1.1(c.0(b.0(b.0(b.0(B.1(x))))))
C1.1(c.0(b.0(b.1(c.0(b.0(B.0(x0))))))) → C1.0(c.0(a.0(a.0(a.0(a.1(c.1(c.0(C.0(x0)))))))))
C1.1(c.0(b.0(a.0(b.0(b.0(B.1(x))))))) → C1.0(c.0(b.0(b.0(b.0(b.0(b.0(B.1(x))))))))
C1.1(c.0(b.0(a.0(b.0(b.0(B.1(x))))))) → C1.0(c.0(b.0(b.0(b.0(b.0(b.0(b.0(B.1(x)))))))))
C1.1(c.0(b.0(a.0(b.0(b.0(B.1(x))))))) → C1.1(c.0(b.0(b.0(b.0(B.1(x))))))
C1.1(c.0(b.0(b.1(c.0(a.0(b.0(b.0(B.1(x0))))))))) → C1.0(c.0(a.0(a.0(a.0(a.1(c.1(c.1(c.0(b.0(b.0(b.0(b.0(B.1(x0))))))))))))))
C1.1(c.0(b.0(b.1(c.0(b.0(B.0(x0))))))) → C1.1(c.0(a.0(a.0(a.0(a.1(c.1(c.0(C.0(x0)))))))))
C1.1(c.0(b.0(b.1(c.0(b.0(a.0(b.0(b.0(B.1(x0)))))))))) → C1.1(c.0(a.0(a.0(a.0(a.1(c.1(c.1(c.0(b.0(b.0(B.1(x0))))))))))))
C1.1(c.0(a.0(b.0(b.0(B.0(x)))))) → C1.0(c.0(b.0(b.0(b.0(B.0(x))))))
C1.1(c.0(b.0(b.1(x)))) → C1.1(c.1(x))
C1.1(c.0(b.0(b.1(c.0(a.0(b.0(b.0(B.1(x0))))))))) → C1.1(c.0(a.0(a.0(a.0(a.1(c.1(c.1(c.0(b.0(b.0(B.1(x0))))))))))))
C1.1(c.0(a.0(b.0(b.0(B.1(x)))))) → C1.0(c.0(b.0(b.0(b.0(B.1(x))))))
C1.1(c.0(b.0(b.1(c.0(a.0(b.0(b.0(B.1(x0))))))))) → C1.1(c.0(a.0(a.0(a.0(a.1(c.1(c.1(c.0(b.0(b.0(b.0(b.0(b.0(B.1(x0)))))))))))))))
C1.1(c.0(a.0(b.0(b.0(B.0(x)))))) → C1.1(c.0(b.0(b.0(b.0(b.0(b.0(B.0(x))))))))
C1.1(c.0(b.0(b.1(c.0(b.0(a.0(b.0(b.0(B.1(x0)))))))))) → C1.0(c.0(a.0(a.0(a.0(a.1(c.1(c.1(c.0(b.0(b.0(b.0(B.1(x0)))))))))))))
C1.1(c.0(a.0(b.0(b.0(B.1(x)))))) → C1.0(c.0(b.0(b.0(b.0(b.0(b.0(B.1(x))))))))
C1.1(c.0(b.0(a.0(b.0(b.0(B.0(x))))))) → C1.1(c.0(b.0(b.0(b.0(b.0(b.0(b.0(B.0(x)))))))))
C1.1(c.0(b.0(b.1(c.0(b.0(a.0(b.0(b.0(B.0(x0)))))))))) → C1.1(c.0(a.0(a.0(a.0(a.1(c.1(c.1(c.0(b.0(b.0(b.0(b.0(b.0(b.0(B.0(x0))))))))))))))))
C1.1(c.0(b.0(b.1(c.0(a.0(b.0(b.0(B.1(x0))))))))) → C1.0(c.0(a.0(a.0(a.0(a.1(c.1(c.1(c.0(b.0(b.0(b.0(b.0(b.0(B.1(x0)))))))))))))))
C1.1(c.0(b.0(b.1(c.0(b.0(a.0(b.0(b.0(B.0(x0)))))))))) → C1.1(c.0(a.0(a.0(a.0(a.1(c.1(c.1(c.0(b.0(b.0(b.0(b.0(b.0(B.0(x0)))))))))))))))
C1.1(c.0(b.0(b.1(c.0(b.0(B.1(x0))))))) → C1.1(c.0(a.0(a.0(a.0(a.1(c.0(C.1(x0))))))))
C1.1(c.0(b.0(a.0(b.0(b.0(B.0(x))))))) → C1.0(c.0(b.0(b.0(b.0(b.0(b.0(B.0(x))))))))
C1.1(c.0(b.0(b.1(c.0(b.0(a.0(b.0(b.0(B.1(x0)))))))))) → C1.0(c.0(a.0(a.0(a.0(a.1(c.1(c.1(c.0(b.0(b.0(B.1(x0))))))))))))
C1.1(c.0(b.0(b.1(c.0(b.0(B.1(x0))))))) → C1.1(c.0(a.0(a.0(a.0(a.1(c.1(c.0(C.1(x0)))))))))
C1.1(c.0(b.0(b.1(c.0(b.0(a.0(b.0(b.0(B.0(x0)))))))))) → C1.1(c.0(a.0(a.0(a.0(a.1(c.1(c.1(c.0(b.0(b.0(B.0(x0))))))))))))
C1.1(c.0(b.0(b.0(x)))) → C1.1(c.0(x))
C1.1(c.0(a.0(b.0(b.0(B.0(x)))))) → C1.0(c.0(b.0(b.0(b.0(b.0(B.0(x)))))))
C1.1(c.0(b.0(a.0(b.0(b.0(B.0(x))))))) → C1.0(c.0(b.0(b.0(B.0(x)))))
C1.1(c.0(b.0(b.1(c.0(x0))))) → C1.1(c.0(b.0(b.0(b.0(a.0(a.0(a.0(x0))))))))
C1.1(c.0(b.0(b.1(c.1(x0))))) → C1.0(c.0(b.0(b.0(b.0(a.0(a.0(a.1(x0))))))))
C1.1(c.0(b.0(b.1(c.0(x0))))) → C1.0(c.0(b.0(b.0(b.0(a.0(a.0(a.0(x0))))))))
C1.1(c.0(b.0(a.0(b.0(b.0(B.1(x))))))) → C1.0(c.0(b.0(b.0(b.0(b.0(B.1(x)))))))
The TRS R consists of the following rules:
c.1(c.0(b.0(a.0(b.0(b.0(B.1(x))))))) → a.0(a.0(a.0(a.1(c.1(c.1(c.0(b.0(b.0(b.0(b.0(b.0(B.1(x)))))))))))))
a.0(b.0(b.0(b.1(x)))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.1(x))))))))
a.0(b.0(b.0(a.0(b.0(b.0(B.0(x))))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.0(x)))))))))))))))
a.0(b.0(b.0(B.0(x)))) → b.0(b.0(B.0(x)))
a.0(b.0(a.0(b.0(b.0(B.1(x)))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.1(x))))))))))))))
a.0(b.0(b.0(a.0(b.0(b.0(B.1(x))))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.1(x)))))))))))))))
c.1(c.0(b.0(a.0(b.0(b.0(B.0(x))))))) → a.0(a.0(a.0(a.1(c.1(c.1(c.0(b.0(b.0(B.0(x))))))))))
a.0(a.0(b.0(b.0(B.1(x))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.1(x)))))))))))))
a.1(x0) → a.0(x0)
c.1(c.0(b.0(a.0(b.0(b.0(B.0(x))))))) → a.0(a.0(a.0(a.1(c.1(c.1(c.0(b.0(b.0(b.0(b.0(b.0(B.0(x)))))))))))))
a.0(a.0(b.0(b.0(B.0(x))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.0(x)))))))))))))
c.1(c.0(b.0(a.0(b.0(b.0(B.1(x))))))) → a.0(a.0(a.0(a.1(c.1(c.1(c.0(b.0(b.0(b.0(b.0(b.0(b.0(B.1(x))))))))))))))
c.0(C.0(x)) → b.0(b.0(B.0(x)))
a.0(b.0(b.0(B.0(x)))) → b.0(B.0(x))
B.1(x0) → B.0(x0)
a.0(b.0(a.0(b.0(b.0(B.0(x)))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.0(x))))))))))))))
a.0(b.0(b.0(B.1(x)))) → b.0(B.1(x))
a.0(b.0(b.0(B.1(x)))) → B.1(x)
c.1(c.0(a.0(b.0(b.0(B.0(x)))))) → a.0(a.0(a.0(a.1(c.1(c.1(c.0(b.0(b.0(b.0(b.0(b.0(B.0(x)))))))))))))
a.0(b.0(b.0(a.0(b.0(b.0(B.1(x))))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.1(x))))))))))))))
a.0(a.0(b.0(b.0(B.0(x))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.0(x))))))))))
c.1(c.0(b.0(b.1(x)))) → a.0(a.0(a.0(a.1(c.1(c.1(c.1(x)))))))
c.1(c.0(b.0(B.1(x)))) → a.0(a.0(a.0(a.1(c.0(C.1(x))))))
a.0(a.0(b.0(b.0(B.0(x))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.0(x))))))))))))
c.1(c.0(a.0(b.0(b.0(B.1(x)))))) → a.0(a.0(a.0(a.1(c.1(c.1(c.0(b.0(b.0(B.1(x))))))))))
c.1(c.0(a.0(b.0(b.0(B.0(x)))))) → a.0(a.0(a.0(a.1(c.1(c.1(c.0(b.0(b.0(B.0(x))))))))))
c.1(c.0(a.0(b.0(b.0(B.1(x)))))) → a.0(a.0(a.0(a.1(c.1(c.1(c.0(b.0(b.0(b.0(b.0(b.0(B.1(x)))))))))))))
a.0(b.0(b.0(a.0(b.0(b.0(B.1(x))))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.1(x)))))))))))))
a.0(b.0(a.0(b.0(b.0(B.0(x)))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.0(x))))))))))))
c.1(c.0(b.0(a.0(b.0(b.0(B.0(x))))))) → a.0(a.0(a.0(a.1(c.1(c.1(c.0(b.0(b.0(b.0(b.0(B.0(x))))))))))))
c.1(c.0(x)) → b.0(b.0(b.0(a.0(a.0(a.0(x))))))
a.0(b.0(a.0(b.0(b.0(B.1(x)))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.1(x)))))))))))))
a.0(a.0(b.0(b.0(B.0(x))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.0(x)))))))))))
c.0(C.1(x)) → b.0(B.1(x))
c.1(c.0(a.0(b.0(b.0(B.0(x)))))) → a.0(a.0(a.0(a.1(c.1(c.1(c.0(b.0(B.0(x)))))))))
c.1(c.0(b.0(b.0(x)))) → a.0(a.0(a.0(a.1(c.1(c.1(c.0(x)))))))
a.0(a.0(b.0(b.0(B.0(x))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.0(x)))))))))
a.0(b.0(a.0(b.0(b.0(B.1(x)))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.1(x))))))))))
a.0(a.0(b.0(b.0(B.1(x))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.1(x)))))))))
b.1(x0) → b.0(x0)
c.1(c.0(b.0(a.0(b.0(b.0(B.1(x))))))) → a.0(a.0(a.0(a.1(c.1(c.1(c.0(b.0(b.0(B.1(x))))))))))
a.0(b.0(b.0(B.1(x)))) → b.0(b.0(B.1(x)))
c.1(c.0(a.0(b.0(b.0(B.1(x)))))) → a.0(a.0(a.0(a.1(c.1(c.1(c.0(b.0(b.0(b.0(b.0(B.1(x))))))))))))
a.0(b.0(a.0(b.0(b.0(B.0(x)))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.0(x)))))))))))
c.1(c.0(b.0(B.0(x)))) → a.0(a.0(a.0(a.1(c.0(C.0(x))))))
c.1(c.0(a.0(b.0(b.0(B.1(x)))))) → a.0(a.0(a.0(a.1(c.1(c.1(c.0(b.0(b.0(b.0(B.1(x)))))))))))
a.0(b.0(b.0(a.0(b.0(b.0(B.1(x))))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.1(x)))))))))))
a.0(a.0(b.0(b.0(B.1(x))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.1(x)))))))))))
a.0(b.0(b.0(a.0(b.0(b.0(B.0(x))))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.0(x))))))))))))
a.0(a.0(b.0(b.0(B.1(x))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.1(x))))))))))))
a.0(b.0(a.0(b.0(b.0(B.1(x)))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.1(x))))))))))))
c.1(c.0(b.0(a.0(b.0(b.0(B.1(x))))))) → a.0(a.0(a.0(a.1(c.1(c.1(c.0(b.0(b.0(b.0(b.0(B.1(x))))))))))))
a.0(b.0(b.0(a.0(b.0(b.0(B.0(x))))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.0(x))))))))))))))
a.0(b.0(b.0(a.0(b.0(b.0(B.0(x))))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.0(x)))))))))))))
a.0(a.0(b.0(b.0(B.1(x))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.1(x))))))))))
c.0(C.0(x)) → b.0(B.0(x))
a.0(b.0(b.0(B.0(x)))) → B.0(x)
c.1(c.0(a.0(b.0(b.0(B.0(x)))))) → a.0(a.0(a.0(a.1(c.1(c.1(c.0(b.0(b.0(b.0(B.0(x)))))))))))
a.0(b.0(b.0(b.0(x)))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(x))))))))
c.1(c.1(x)) → b.0(b.0(b.0(a.0(a.0(a.1(x))))))
a.0(b.0(a.0(b.0(b.0(B.0(x)))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.0(x))))))))))
a.0(b.0(b.0(a.0(b.0(b.0(B.1(x))))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.1(x))))))))))))
c.1(x0) → c.0(x0)
c.1(c.0(b.0(B.1(x)))) → a.0(a.0(a.0(a.1(c.1(c.0(C.1(x)))))))
C.1(x0) → C.0(x0)
c.1(c.0(a.0(b.0(b.0(B.1(x)))))) → a.0(a.0(a.0(a.1(c.1(c.1(c.0(b.0(B.1(x)))))))))
a.0(b.0(b.0(a.0(b.0(b.0(B.0(x))))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.0(x)))))))))))
c.0(C.1(x)) → b.0(b.0(B.1(x)))
a.0(b.0(a.0(b.0(b.0(B.0(x)))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.0(x)))))))))))))
c.1(c.0(b.0(a.0(b.0(b.0(B.0(x))))))) → a.0(a.0(a.0(a.1(c.1(c.1(c.0(b.0(b.0(b.0(B.0(x)))))))))))
c.1(c.0(b.0(B.0(x)))) → a.0(a.0(a.0(a.1(c.1(c.0(C.0(x)))))))
c.1(c.0(a.0(b.0(b.0(B.0(x)))))) → a.0(a.0(a.0(a.1(c.1(c.1(c.0(b.0(b.0(b.0(b.0(B.0(x))))))))))))
c.1(c.0(b.0(a.0(b.0(b.0(B.0(x))))))) → a.0(a.0(a.0(a.1(c.1(c.1(c.0(b.0(b.0(b.0(b.0(b.0(b.0(B.0(x))))))))))))))
a.0(b.0(a.0(b.0(b.0(B.1(x)))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.1(x)))))))))))
c.1(c.0(b.0(a.0(b.0(b.0(B.1(x))))))) → a.0(a.0(a.0(a.1(c.1(c.1(c.0(b.0(b.0(b.0(B.1(x)))))))))))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
↳ QTRS
↳ QTRS Reverse
↳ QTRS Reverse
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ SemLabProof
↳ SemLabProof2
↳ QDP
↳ QDPToSRSProof
↳ QTRS
↳ QTRS Reverse
↳ QTRS
↳ QTRS Reverse
↳ QTRS Reverse
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ SemLabProof
↳ SemLabProof2
↳ QDP
↳ SemLabProof
↳ QDP
↳ DependencyGraphProof
Q DP problem:
The TRS P consists of the following rules:
C1.1(c.0(b.0(a.0(b.0(b.0(B.1(x))))))) → C1.0(c.0(b.0(b.0(b.0(B.1(x))))))
C1.1(c.0(b.0(b.1(c.0(b.0(a.0(b.0(b.0(B.1(x0)))))))))) → C1.0(c.0(a.0(a.0(a.0(a.1(c.1(c.1(c.0(b.0(b.0(b.0(b.0(b.0(b.0(B.1(x0))))))))))))))))
C1.1(c.0(b.0(b.1(c.0(a.0(b.0(b.0(B.0(x0))))))))) → C1.0(c.0(a.0(a.0(a.0(a.1(c.1(c.1(c.0(b.0(b.0(b.0(b.0(b.0(B.0(x0)))))))))))))))
C1.1(c.0(b.0(b.1(c.0(b.0(B.1(x0))))))) → C1.0(c.0(a.0(a.0(a.0(a.1(c.1(c.0(C.1(x0)))))))))
C1.1(c.0(b.0(a.0(b.0(b.0(B.0(x))))))) → C1.1(c.0(b.0(b.0(B.0(x)))))
C1.1(c.0(b.0(b.1(c.0(b.0(B.0(x0))))))) → C1.0(c.0(a.0(a.0(a.0(a.1(c.0(C.0(x0))))))))
C1.1(c.0(b.0(a.0(b.0(b.0(B.0(x))))))) → C1.1(c.0(b.0(b.0(b.0(b.0(B.0(x)))))))
C1.1(c.0(b.0(b.1(c.0(b.0(b.0(x0))))))) → C1.1(c.0(a.0(a.0(a.0(a.1(c.1(c.1(c.0(x0)))))))))
C1.1(c.0(b.0(b.1(c.0(a.0(b.0(b.0(B.0(x0))))))))) → C1.0(c.0(a.0(a.0(a.0(a.1(c.1(c.1(c.0(b.0(b.0(b.0(B.0(x0)))))))))))))
C1.1(c.0(b.0(b.1(c.0(b.0(b.0(x0))))))) → C1.0(c.0(a.0(a.0(a.0(a.1(c.1(c.1(c.0(x0)))))))))
C1.1(c.0(b.0(b.1(c.0(b.0(b.1(c.0(y_4)))))))) → C1.0(c.0(b.0(b.1(c.0(y_4)))))
C1.1(c.0(a.0(b.0(b.0(B.0(x)))))) → C1.0(c.0(b.0(b.0(b.0(b.0(b.0(B.0(x))))))))
C1.1(c.0(b.0(b.1(x)))) → C1.0(c.1(x))
C1.1(c.0(b.0(a.0(b.0(b.0(B.1(x))))))) → C1.1(c.0(b.0(b.0(b.0(b.0(b.0(B.1(x))))))))
C1.1(c.0(b.0(b.1(c.0(b.0(b.1(x0))))))) → C1.1(c.0(a.0(a.0(a.0(a.1(c.1(c.1(c.1(x0)))))))))
C1.1(c.0(b.0(b.1(c.0(b.0(a.0(b.0(b.0(B.0(x0)))))))))) → C1.1(c.0(a.0(a.0(a.0(a.1(c.1(c.1(c.0(b.0(b.0(b.0(b.0(B.0(x0))))))))))))))
C1.1(c.0(b.0(b.1(c.0(a.0(b.0(b.0(B.0(x0))))))))) → C1.0(c.0(a.0(a.0(a.0(a.1(c.1(c.1(c.0(b.0(b.0(B.0(x0))))))))))))
C1.1(c.0(a.0(b.0(b.0(B.1(x)))))) → C1.0(c.0(b.0(b.0(b.0(b.0(B.1(x)))))))
C1.1(c.0(b.0(b.1(c.0(b.0(b.1(x0))))))) → C1.0(c.0(a.0(a.0(a.0(a.1(c.1(c.1(c.1(x0)))))))))
C1.1(c.0(b.0(b.1(c.0(a.0(b.0(b.0(B.1(x0))))))))) → C1.1(c.0(a.0(a.0(a.0(a.1(c.1(c.1(c.0(b.0(B.1(x0)))))))))))
C1.1(c.0(a.0(b.0(b.0(B.1(x)))))) → C1.0(c.0(b.0(b.0(B.1(x)))))
C1.1(c.0(b.0(b.1(c.0(b.0(a.0(b.0(b.0(B.0(x0)))))))))) → C1.0(c.0(a.0(a.0(a.0(a.1(c.1(c.1(c.0(b.0(b.0(B.0(x0))))))))))))
C1.1(c.0(b.0(a.0(b.0(b.0(B.0(x))))))) → C1.0(c.0(b.0(b.0(b.0(B.0(x))))))
C1.1(c.0(b.0(b.1(c.0(b.0(a.0(b.0(b.0(B.1(x0)))))))))) → C1.1(c.0(a.0(a.0(a.0(a.1(c.1(c.1(c.0(b.0(b.0(b.0(b.0(b.0(b.0(B.1(x0))))))))))))))))
C1.1(c.0(b.0(b.1(c.0(b.0(b.1(c.1(y_4)))))))) → C1.1(c.0(b.0(b.1(c.1(y_4)))))
C1.1(c.0(a.0(b.0(b.0(B.0(x)))))) → C1.1(c.0(b.0(b.0(b.0(B.0(x))))))
C1.1(c.0(b.0(b.1(c.1(y_2))))) → C1.0(c.1(y_2))
C1.1(c.0(b.0(b.1(c.0(b.0(a.0(b.0(b.0(B.0(x0)))))))))) → C1.0(c.0(a.0(a.0(a.0(a.1(c.1(c.1(c.0(b.0(b.0(b.0(b.0(b.0(b.0(B.0(x0))))))))))))))))
C1.1(c.0(b.0(b.1(c.0(y_2))))) → C1.1(c.0(y_2))
C1.1(c.0(b.0(b.1(c.0(b.0(a.0(b.0(b.0(B.1(x0)))))))))) → C1.0(c.0(a.0(a.0(a.0(a.1(c.1(c.1(c.0(b.0(b.0(b.0(b.0(B.1(x0))))))))))))))
C1.1(c.0(a.0(b.0(b.0(B.1(x)))))) → C1.1(c.0(b.0(b.0(B.1(x)))))
C1.1(c.0(b.0(b.1(c.0(a.0(b.0(b.0(B.1(x0))))))))) → C1.0(c.0(a.0(a.0(a.0(a.1(c.1(c.1(c.0(b.0(b.0(b.0(B.1(x0)))))))))))))
C1.1(c.0(b.0(a.0(b.0(b.0(B.1(x))))))) → C1.0(c.0(b.0(b.0(B.1(x)))))
C1.1(c.0(b.0(b.1(c.0(b.0(a.0(b.0(b.0(B.0(x0)))))))))) → C1.0(c.0(a.0(a.0(a.0(a.1(c.1(c.1(c.0(b.0(b.0(b.0(B.0(x0)))))))))))))
C1.1(c.0(b.0(b.1(c.0(b.0(a.0(b.0(b.0(B.1(x0)))))))))) → C1.1(c.0(a.0(a.0(a.0(a.1(c.1(c.1(c.0(b.0(b.0(b.0(b.0(b.0(B.1(x0)))))))))))))))
C1.1(c.0(b.0(b.1(c.0(y_2))))) → C1.0(c.0(y_2))
C1.1(c.0(b.0(b.1(c.0(b.0(a.0(b.0(b.0(B.0(x0)))))))))) → C1.1(c.0(a.0(a.0(a.0(a.1(c.1(c.1(c.0(b.0(b.0(b.0(B.0(x0)))))))))))))
C1.1(c.0(b.0(b.1(c.0(a.0(b.0(b.0(B.1(x0))))))))) → C1.0(c.0(a.0(a.0(a.0(a.1(c.1(c.1(c.0(b.0(B.1(x0)))))))))))
C1.1(c.0(b.0(b.1(c.0(b.0(a.0(b.0(b.0(B.0(x0)))))))))) → C1.0(c.0(a.0(a.0(a.0(a.1(c.1(c.1(c.0(b.0(b.0(b.0(b.0(B.0(x0))))))))))))))
C1.1(c.0(b.0(b.1(c.0(b.0(a.0(b.0(b.0(B.1(x0)))))))))) → C1.1(c.0(a.0(a.0(a.0(a.1(c.1(c.1(c.0(b.0(b.0(b.0(B.1(x0)))))))))))))
C1.1(c.0(a.0(b.0(b.0(B.0(x)))))) → C1.1(c.0(b.0(b.0(B.0(x)))))
C1.1(c.0(b.0(b.1(c.0(b.0(a.0(b.0(b.0(B.0(x0)))))))))) → C1.0(c.0(a.0(a.0(a.0(a.1(c.1(c.1(c.0(b.0(b.0(b.0(b.0(b.0(B.0(x0)))))))))))))))
C1.1(c.0(b.0(a.0(b.0(b.0(B.1(x))))))) → C1.1(c.0(b.0(b.0(B.1(x)))))
C1.1(c.0(b.0(b.1(c.0(b.0(b.1(c.0(y_4)))))))) → C1.1(c.0(b.0(b.1(c.0(y_4)))))
C1.1(c.0(b.0(b.1(c.0(b.0(a.0(b.0(b.0(B.1(x0)))))))))) → C1.0(c.0(a.0(a.0(a.0(a.1(c.1(c.1(c.0(b.0(b.0(b.0(b.0(b.0(B.1(x0)))))))))))))))
C1.1(c.0(b.0(b.0(x)))) → C1.0(c.0(x))
C1.1(c.0(b.0(b.1(c.0(a.0(b.0(b.0(B.0(x0))))))))) → C1.1(c.0(a.0(a.0(a.0(a.1(c.1(c.1(c.0(b.0(b.0(b.0(b.0(b.0(B.0(x0)))))))))))))))
C1.1(c.0(b.0(a.0(b.0(b.0(B.0(x))))))) → C1.1(c.0(b.0(b.0(b.0(B.0(x))))))
C1.1(c.0(b.0(b.1(c.0(a.0(b.0(b.0(B.0(x0))))))))) → C1.0(c.0(a.0(a.0(a.0(a.1(c.1(c.1(c.0(b.0(B.0(x0)))))))))))
C1.1(c.0(a.0(b.0(b.0(B.0(x)))))) → C1.1(c.0(b.0(b.0(b.0(b.0(B.0(x)))))))
C1.1(c.0(b.0(b.1(c.0(b.0(B.1(x0))))))) → C1.0(c.0(a.0(a.0(a.0(a.1(c.0(C.1(x0))))))))
C1.1(c.0(b.0(a.0(b.0(b.0(B.1(x))))))) → C1.1(c.0(b.0(b.0(b.0(b.0(b.0(b.0(B.1(x)))))))))
C1.1(c.0(b.0(b.1(c.0(a.0(b.0(b.0(B.0(x0))))))))) → C1.0(c.0(a.0(a.0(a.0(a.1(c.1(c.1(c.0(b.0(b.0(b.0(b.0(B.0(x0))))))))))))))
C1.1(c.0(b.0(b.1(c.0(b.0(a.0(b.0(b.0(B.1(x0)))))))))) → C1.1(c.0(a.0(a.0(a.0(a.1(c.1(c.1(c.0(b.0(b.0(b.0(b.0(B.1(x0))))))))))))))
C1.1(c.0(b.0(b.1(c.0(b.0(B.0(x0))))))) → C1.1(c.0(a.0(a.0(a.0(a.1(c.0(C.0(x0))))))))
C1.1(c.0(a.0(b.0(b.0(B.1(x)))))) → C1.1(c.0(b.0(b.0(b.0(b.0(B.1(x)))))))
C1.1(c.0(b.0(b.1(c.0(a.0(b.0(b.0(B.1(x0))))))))) → C1.1(c.0(a.0(a.0(a.0(a.1(c.1(c.1(c.0(b.0(b.0(b.0(b.0(B.1(x0))))))))))))))
C1.1(c.0(b.0(b.1(c.0(a.0(b.0(b.0(B.1(x0))))))))) → C1.0(c.0(a.0(a.0(a.0(a.1(c.1(c.1(c.0(b.0(b.0(B.1(x0))))))))))))
C1.1(c.0(b.0(b.1(c.0(b.0(b.1(c.1(y_4)))))))) → C1.0(c.0(b.0(b.1(c.1(y_4)))))
C1.1(c.0(b.0(a.0(b.0(b.0(B.0(x))))))) → C1.1(c.0(b.0(b.0(b.0(b.0(b.0(B.0(x))))))))
C1.1(c.0(a.0(b.0(b.0(B.1(x)))))) → C1.1(c.0(b.0(b.0(b.0(b.0(b.0(B.1(x))))))))
C1.1(c.0(b.0(b.1(c.0(a.0(b.0(b.0(B.0(x0))))))))) → C1.1(c.0(a.0(a.0(a.0(a.1(c.1(c.1(c.0(b.0(b.0(B.0(x0))))))))))))
C1.1(c.0(b.0(a.0(b.0(b.0(B.0(x))))))) → C1.0(c.0(b.0(b.0(b.0(b.0(B.0(x)))))))
C1.1(c.0(b.0(b.1(c.0(a.0(b.0(b.0(B.1(x0))))))))) → C1.1(c.0(a.0(a.0(a.0(a.1(c.1(c.1(c.0(b.0(b.0(b.0(B.1(x0)))))))))))))
C1.1(c.0(b.0(a.0(b.0(b.0(B.0(x))))))) → C1.0(c.0(b.0(b.0(b.0(b.0(b.0(b.0(B.0(x)))))))))
C1.1(c.0(b.0(b.1(c.0(a.0(b.0(b.0(B.0(x0))))))))) → C1.1(c.0(a.0(a.0(a.0(a.1(c.1(c.1(c.0(b.0(b.0(b.0(B.0(x0)))))))))))))
C1.1(c.0(a.0(b.0(b.0(B.0(x)))))) → C1.0(c.0(b.0(b.0(B.0(x)))))
C1.1(c.0(b.0(b.1(c.0(a.0(b.0(b.0(B.0(x0))))))))) → C1.1(c.0(a.0(a.0(a.0(a.1(c.1(c.1(c.0(b.0(b.0(b.0(b.0(B.0(x0))))))))))))))
C1.1(c.0(b.0(b.1(c.1(y_2))))) → C1.1(c.1(y_2))
C1.1(c.0(b.0(b.1(c.1(x0))))) → C1.1(c.0(b.0(b.0(b.0(a.0(a.0(a.1(x0))))))))
C1.1(c.0(b.0(b.1(c.0(a.0(b.0(b.0(B.0(x0))))))))) → C1.1(c.0(a.0(a.0(a.0(a.1(c.1(c.1(c.0(b.0(B.0(x0)))))))))))
C1.1(c.0(b.0(a.0(b.0(b.0(B.1(x))))))) → C1.1(c.0(b.0(b.0(b.0(b.0(B.1(x)))))))
C1.1(c.0(a.0(b.0(b.0(B.1(x)))))) → C1.1(c.0(b.0(b.0(b.0(B.1(x))))))
C1.1(c.0(b.0(b.1(c.0(b.0(B.0(x0))))))) → C1.0(c.0(a.0(a.0(a.0(a.1(c.1(c.0(C.0(x0)))))))))
C1.1(c.0(b.0(a.0(b.0(b.0(B.1(x))))))) → C1.0(c.0(b.0(b.0(b.0(b.0(b.0(B.1(x))))))))
C1.1(c.0(b.0(a.0(b.0(b.0(B.1(x))))))) → C1.0(c.0(b.0(b.0(b.0(b.0(b.0(b.0(B.1(x)))))))))
C1.1(c.0(b.0(a.0(b.0(b.0(B.1(x))))))) → C1.1(c.0(b.0(b.0(b.0(B.1(x))))))
C1.1(c.0(b.0(b.1(c.0(a.0(b.0(b.0(B.1(x0))))))))) → C1.0(c.0(a.0(a.0(a.0(a.1(c.1(c.1(c.0(b.0(b.0(b.0(b.0(B.1(x0))))))))))))))
C1.1(c.0(b.0(b.1(c.0(b.0(B.0(x0))))))) → C1.1(c.0(a.0(a.0(a.0(a.1(c.1(c.0(C.0(x0)))))))))
C1.1(c.0(b.0(b.1(c.0(b.0(a.0(b.0(b.0(B.1(x0)))))))))) → C1.1(c.0(a.0(a.0(a.0(a.1(c.1(c.1(c.0(b.0(b.0(B.1(x0))))))))))))
C1.1(c.0(a.0(b.0(b.0(B.0(x)))))) → C1.0(c.0(b.0(b.0(b.0(B.0(x))))))
C1.1(c.0(b.0(b.1(x)))) → C1.1(c.1(x))
C1.1(c.0(b.0(b.1(c.0(a.0(b.0(b.0(B.1(x0))))))))) → C1.1(c.0(a.0(a.0(a.0(a.1(c.1(c.1(c.0(b.0(b.0(B.1(x0))))))))))))
C1.1(c.0(a.0(b.0(b.0(B.1(x)))))) → C1.0(c.0(b.0(b.0(b.0(B.1(x))))))
C1.1(c.0(b.0(b.1(c.0(a.0(b.0(b.0(B.1(x0))))))))) → C1.1(c.0(a.0(a.0(a.0(a.1(c.1(c.1(c.0(b.0(b.0(b.0(b.0(b.0(B.1(x0)))))))))))))))
C1.1(c.0(a.0(b.0(b.0(B.0(x)))))) → C1.1(c.0(b.0(b.0(b.0(b.0(b.0(B.0(x))))))))
C1.1(c.0(b.0(b.1(c.0(b.0(a.0(b.0(b.0(B.1(x0)))))))))) → C1.0(c.0(a.0(a.0(a.0(a.1(c.1(c.1(c.0(b.0(b.0(b.0(B.1(x0)))))))))))))
C1.1(c.0(a.0(b.0(b.0(B.1(x)))))) → C1.0(c.0(b.0(b.0(b.0(b.0(b.0(B.1(x))))))))
C1.1(c.0(b.0(a.0(b.0(b.0(B.0(x))))))) → C1.1(c.0(b.0(b.0(b.0(b.0(b.0(b.0(B.0(x)))))))))
C1.1(c.0(b.0(b.1(c.0(b.0(a.0(b.0(b.0(B.0(x0)))))))))) → C1.1(c.0(a.0(a.0(a.0(a.1(c.1(c.1(c.0(b.0(b.0(b.0(b.0(b.0(b.0(B.0(x0))))))))))))))))
C1.1(c.0(b.0(b.1(c.0(a.0(b.0(b.0(B.1(x0))))))))) → C1.0(c.0(a.0(a.0(a.0(a.1(c.1(c.1(c.0(b.0(b.0(b.0(b.0(b.0(B.1(x0)))))))))))))))
C1.1(c.0(b.0(b.1(c.0(b.0(a.0(b.0(b.0(B.0(x0)))))))))) → C1.1(c.0(a.0(a.0(a.0(a.1(c.1(c.1(c.0(b.0(b.0(b.0(b.0(b.0(B.0(x0)))))))))))))))
C1.1(c.0(b.0(b.1(c.0(b.0(B.1(x0))))))) → C1.1(c.0(a.0(a.0(a.0(a.1(c.0(C.1(x0))))))))
C1.1(c.0(b.0(a.0(b.0(b.0(B.0(x))))))) → C1.0(c.0(b.0(b.0(b.0(b.0(b.0(B.0(x))))))))
C1.1(c.0(b.0(b.1(c.0(b.0(a.0(b.0(b.0(B.1(x0)))))))))) → C1.0(c.0(a.0(a.0(a.0(a.1(c.1(c.1(c.0(b.0(b.0(B.1(x0))))))))))))
C1.1(c.0(b.0(b.1(c.0(b.0(B.1(x0))))))) → C1.1(c.0(a.0(a.0(a.0(a.1(c.1(c.0(C.1(x0)))))))))
C1.1(c.0(b.0(b.1(c.0(b.0(a.0(b.0(b.0(B.0(x0)))))))))) → C1.1(c.0(a.0(a.0(a.0(a.1(c.1(c.1(c.0(b.0(b.0(B.0(x0))))))))))))
C1.1(c.0(b.0(b.0(x)))) → C1.1(c.0(x))
C1.1(c.0(a.0(b.0(b.0(B.0(x)))))) → C1.0(c.0(b.0(b.0(b.0(b.0(B.0(x)))))))
C1.1(c.0(b.0(a.0(b.0(b.0(B.0(x))))))) → C1.0(c.0(b.0(b.0(B.0(x)))))
C1.1(c.0(b.0(b.1(c.0(x0))))) → C1.1(c.0(b.0(b.0(b.0(a.0(a.0(a.0(x0))))))))
C1.1(c.0(b.0(b.1(c.1(x0))))) → C1.0(c.0(b.0(b.0(b.0(a.0(a.0(a.1(x0))))))))
C1.1(c.0(b.0(b.1(c.0(x0))))) → C1.0(c.0(b.0(b.0(b.0(a.0(a.0(a.0(x0))))))))
C1.1(c.0(b.0(a.0(b.0(b.0(B.1(x))))))) → C1.0(c.0(b.0(b.0(b.0(b.0(B.1(x)))))))
The TRS R consists of the following rules:
c.1(c.0(b.0(a.0(b.0(b.0(B.1(x))))))) → a.0(a.0(a.0(a.1(c.1(c.1(c.0(b.0(b.0(b.0(b.0(b.0(B.1(x)))))))))))))
a.0(b.0(b.0(b.1(x)))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.1(x))))))))
a.0(b.0(b.0(a.0(b.0(b.0(B.0(x))))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.0(x)))))))))))))))
a.0(b.0(b.0(B.0(x)))) → b.0(b.0(B.0(x)))
a.0(b.0(a.0(b.0(b.0(B.1(x)))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.1(x))))))))))))))
a.0(b.0(b.0(a.0(b.0(b.0(B.1(x))))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.1(x)))))))))))))))
c.1(c.0(b.0(a.0(b.0(b.0(B.0(x))))))) → a.0(a.0(a.0(a.1(c.1(c.1(c.0(b.0(b.0(B.0(x))))))))))
a.0(a.0(b.0(b.0(B.1(x))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.1(x)))))))))))))
a.1(x0) → a.0(x0)
c.1(c.0(b.0(a.0(b.0(b.0(B.0(x))))))) → a.0(a.0(a.0(a.1(c.1(c.1(c.0(b.0(b.0(b.0(b.0(b.0(B.0(x)))))))))))))
a.0(a.0(b.0(b.0(B.0(x))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.0(x)))))))))))))
c.1(c.0(b.0(a.0(b.0(b.0(B.1(x))))))) → a.0(a.0(a.0(a.1(c.1(c.1(c.0(b.0(b.0(b.0(b.0(b.0(b.0(B.1(x))))))))))))))
c.0(C.0(x)) → b.0(b.0(B.0(x)))
a.0(b.0(b.0(B.0(x)))) → b.0(B.0(x))
B.1(x0) → B.0(x0)
a.0(b.0(a.0(b.0(b.0(B.0(x)))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.0(x))))))))))))))
a.0(b.0(b.0(B.1(x)))) → b.0(B.1(x))
a.0(b.0(b.0(B.1(x)))) → B.1(x)
c.1(c.0(a.0(b.0(b.0(B.0(x)))))) → a.0(a.0(a.0(a.1(c.1(c.1(c.0(b.0(b.0(b.0(b.0(b.0(B.0(x)))))))))))))
a.0(b.0(b.0(a.0(b.0(b.0(B.1(x))))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.1(x))))))))))))))
a.0(a.0(b.0(b.0(B.0(x))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.0(x))))))))))
c.1(c.0(b.0(b.1(x)))) → a.0(a.0(a.0(a.1(c.1(c.1(c.1(x)))))))
c.1(c.0(b.0(B.1(x)))) → a.0(a.0(a.0(a.1(c.0(C.1(x))))))
a.0(a.0(b.0(b.0(B.0(x))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.0(x))))))))))))
c.1(c.0(a.0(b.0(b.0(B.1(x)))))) → a.0(a.0(a.0(a.1(c.1(c.1(c.0(b.0(b.0(B.1(x))))))))))
c.1(c.0(a.0(b.0(b.0(B.0(x)))))) → a.0(a.0(a.0(a.1(c.1(c.1(c.0(b.0(b.0(B.0(x))))))))))
c.1(c.0(a.0(b.0(b.0(B.1(x)))))) → a.0(a.0(a.0(a.1(c.1(c.1(c.0(b.0(b.0(b.0(b.0(b.0(B.1(x)))))))))))))
a.0(b.0(b.0(a.0(b.0(b.0(B.1(x))))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.1(x)))))))))))))
a.0(b.0(a.0(b.0(b.0(B.0(x)))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.0(x))))))))))))
c.1(c.0(b.0(a.0(b.0(b.0(B.0(x))))))) → a.0(a.0(a.0(a.1(c.1(c.1(c.0(b.0(b.0(b.0(b.0(B.0(x))))))))))))
c.1(c.0(x)) → b.0(b.0(b.0(a.0(a.0(a.0(x))))))
a.0(b.0(a.0(b.0(b.0(B.1(x)))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.1(x)))))))))))))
a.0(a.0(b.0(b.0(B.0(x))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.0(x)))))))))))
c.0(C.1(x)) → b.0(B.1(x))
c.1(c.0(a.0(b.0(b.0(B.0(x)))))) → a.0(a.0(a.0(a.1(c.1(c.1(c.0(b.0(B.0(x)))))))))
c.1(c.0(b.0(b.0(x)))) → a.0(a.0(a.0(a.1(c.1(c.1(c.0(x)))))))
a.0(a.0(b.0(b.0(B.0(x))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.0(x)))))))))
a.0(b.0(a.0(b.0(b.0(B.1(x)))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.1(x))))))))))
a.0(a.0(b.0(b.0(B.1(x))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.1(x)))))))))
b.1(x0) → b.0(x0)
c.1(c.0(b.0(a.0(b.0(b.0(B.1(x))))))) → a.0(a.0(a.0(a.1(c.1(c.1(c.0(b.0(b.0(B.1(x))))))))))
a.0(b.0(b.0(B.1(x)))) → b.0(b.0(B.1(x)))
c.1(c.0(a.0(b.0(b.0(B.1(x)))))) → a.0(a.0(a.0(a.1(c.1(c.1(c.0(b.0(b.0(b.0(b.0(B.1(x))))))))))))
a.0(b.0(a.0(b.0(b.0(B.0(x)))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.0(x)))))))))))
c.1(c.0(b.0(B.0(x)))) → a.0(a.0(a.0(a.1(c.0(C.0(x))))))
c.1(c.0(a.0(b.0(b.0(B.1(x)))))) → a.0(a.0(a.0(a.1(c.1(c.1(c.0(b.0(b.0(b.0(B.1(x)))))))))))
a.0(b.0(b.0(a.0(b.0(b.0(B.1(x))))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.1(x)))))))))))
a.0(a.0(b.0(b.0(B.1(x))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.1(x)))))))))))
a.0(b.0(b.0(a.0(b.0(b.0(B.0(x))))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.0(x))))))))))))
a.0(a.0(b.0(b.0(B.1(x))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.1(x))))))))))))
a.0(b.0(a.0(b.0(b.0(B.1(x)))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.1(x))))))))))))
c.1(c.0(b.0(a.0(b.0(b.0(B.1(x))))))) → a.0(a.0(a.0(a.1(c.1(c.1(c.0(b.0(b.0(b.0(b.0(B.1(x))))))))))))
a.0(b.0(b.0(a.0(b.0(b.0(B.0(x))))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.0(x))))))))))))))
a.0(b.0(b.0(a.0(b.0(b.0(B.0(x))))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.0(x)))))))))))))
a.0(a.0(b.0(b.0(B.1(x))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.1(x))))))))))
c.0(C.0(x)) → b.0(B.0(x))
a.0(b.0(b.0(B.0(x)))) → B.0(x)
c.1(c.0(a.0(b.0(b.0(B.0(x)))))) → a.0(a.0(a.0(a.1(c.1(c.1(c.0(b.0(b.0(b.0(B.0(x)))))))))))
a.0(b.0(b.0(b.0(x)))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(x))))))))
c.1(c.1(x)) → b.0(b.0(b.0(a.0(a.0(a.1(x))))))
a.0(b.0(a.0(b.0(b.0(B.0(x)))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.0(x))))))))))
a.0(b.0(b.0(a.0(b.0(b.0(B.1(x))))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.1(x))))))))))))
c.1(x0) → c.0(x0)
c.1(c.0(b.0(B.1(x)))) → a.0(a.0(a.0(a.1(c.1(c.0(C.1(x)))))))
C.1(x0) → C.0(x0)
c.1(c.0(a.0(b.0(b.0(B.1(x)))))) → a.0(a.0(a.0(a.1(c.1(c.1(c.0(b.0(B.1(x)))))))))
a.0(b.0(b.0(a.0(b.0(b.0(B.0(x))))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.0(x)))))))))))
c.0(C.1(x)) → b.0(b.0(B.1(x)))
a.0(b.0(a.0(b.0(b.0(B.0(x)))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.0(x)))))))))))))
c.1(c.0(b.0(a.0(b.0(b.0(B.0(x))))))) → a.0(a.0(a.0(a.1(c.1(c.1(c.0(b.0(b.0(b.0(B.0(x)))))))))))
c.1(c.0(b.0(B.0(x)))) → a.0(a.0(a.0(a.1(c.1(c.0(C.0(x)))))))
c.1(c.0(a.0(b.0(b.0(B.0(x)))))) → a.0(a.0(a.0(a.1(c.1(c.1(c.0(b.0(b.0(b.0(b.0(B.0(x))))))))))))
c.1(c.0(b.0(a.0(b.0(b.0(B.0(x))))))) → a.0(a.0(a.0(a.1(c.1(c.1(c.0(b.0(b.0(b.0(b.0(b.0(b.0(B.0(x))))))))))))))
a.0(b.0(a.0(b.0(b.0(B.1(x)))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.1(x)))))))))))
c.1(c.0(b.0(a.0(b.0(b.0(B.1(x))))))) → a.0(a.0(a.0(a.1(c.1(c.1(c.0(b.0(b.0(b.0(B.1(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 52 less nodes.
↳ QTRS
↳ QTRS Reverse
↳ QTRS Reverse
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ SemLabProof
↳ SemLabProof2
↳ QDP
↳ QDPToSRSProof
↳ QTRS
↳ QTRS Reverse
↳ QTRS
↳ QTRS Reverse
↳ QTRS Reverse
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ SemLabProof
↳ SemLabProof2
↳ QDP
↳ SemLabProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ RuleRemovalProof
Q DP problem:
The TRS P consists of the following rules:
C1.1(c.0(a.0(b.0(b.0(B.0(x)))))) → C1.1(c.0(b.0(b.0(b.0(b.0(B.0(x)))))))
C1.1(c.0(b.0(a.0(b.0(b.0(B.1(x))))))) → C1.1(c.0(b.0(b.0(b.0(b.0(b.0(b.0(B.1(x)))))))))
C1.1(c.0(b.0(a.0(b.0(b.0(B.0(x))))))) → C1.1(c.0(b.0(b.0(B.0(x)))))
C1.1(c.0(b.0(b.1(c.0(b.0(a.0(b.0(b.0(B.1(x0)))))))))) → C1.1(c.0(a.0(a.0(a.0(a.1(c.1(c.1(c.0(b.0(b.0(b.0(b.0(B.1(x0))))))))))))))
C1.1(c.0(b.0(b.1(c.0(b.0(B.0(x0))))))) → C1.1(c.0(a.0(a.0(a.0(a.1(c.0(C.0(x0))))))))
C1.1(c.0(b.0(a.0(b.0(b.0(B.0(x))))))) → C1.1(c.0(b.0(b.0(b.0(b.0(B.0(x)))))))
C1.1(c.0(b.0(b.1(c.0(b.0(b.0(x0))))))) → C1.1(c.0(a.0(a.0(a.0(a.1(c.1(c.1(c.0(x0)))))))))
C1.1(c.0(a.0(b.0(b.0(B.1(x)))))) → C1.1(c.0(b.0(b.0(b.0(b.0(B.1(x)))))))
C1.1(c.0(b.0(b.1(c.0(a.0(b.0(b.0(B.1(x0))))))))) → C1.1(c.0(a.0(a.0(a.0(a.1(c.1(c.1(c.0(b.0(b.0(b.0(b.0(B.1(x0))))))))))))))
C1.1(c.0(b.0(a.0(b.0(b.0(B.0(x))))))) → C1.1(c.0(b.0(b.0(b.0(b.0(b.0(B.0(x))))))))
C1.1(c.0(a.0(b.0(b.0(B.1(x)))))) → C1.1(c.0(b.0(b.0(b.0(b.0(b.0(B.1(x))))))))
C1.1(c.0(b.0(b.1(c.0(a.0(b.0(b.0(B.0(x0))))))))) → C1.1(c.0(a.0(a.0(a.0(a.1(c.1(c.1(c.0(b.0(b.0(B.0(x0))))))))))))
C1.1(c.0(b.0(a.0(b.0(b.0(B.1(x))))))) → C1.1(c.0(b.0(b.0(b.0(b.0(b.0(B.1(x))))))))
C1.1(c.0(b.0(b.1(c.0(b.0(b.1(x0))))))) → C1.1(c.0(a.0(a.0(a.0(a.1(c.1(c.1(c.1(x0)))))))))
C1.1(c.0(b.0(b.1(c.0(a.0(b.0(b.0(B.1(x0))))))))) → C1.1(c.0(a.0(a.0(a.0(a.1(c.1(c.1(c.0(b.0(b.0(b.0(B.1(x0)))))))))))))
C1.1(c.0(b.0(b.1(c.0(b.0(a.0(b.0(b.0(B.0(x0)))))))))) → C1.1(c.0(a.0(a.0(a.0(a.1(c.1(c.1(c.0(b.0(b.0(b.0(b.0(B.0(x0))))))))))))))
C1.1(c.0(b.0(b.1(c.0(a.0(b.0(b.0(B.0(x0))))))))) → C1.1(c.0(a.0(a.0(a.0(a.1(c.1(c.1(c.0(b.0(b.0(b.0(B.0(x0)))))))))))))
C1.1(c.0(b.0(b.1(c.0(a.0(b.0(b.0(B.1(x0))))))))) → C1.1(c.0(a.0(a.0(a.0(a.1(c.1(c.1(c.0(b.0(B.1(x0)))))))))))
C1.1(c.0(b.0(b.1(c.0(a.0(b.0(b.0(B.0(x0))))))))) → C1.1(c.0(a.0(a.0(a.0(a.1(c.1(c.1(c.0(b.0(b.0(b.0(b.0(B.0(x0))))))))))))))
C1.1(c.0(b.0(b.1(c.1(y_2))))) → C1.1(c.1(y_2))
C1.1(c.0(b.0(b.1(c.1(x0))))) → C1.1(c.0(b.0(b.0(b.0(a.0(a.0(a.1(x0))))))))
C1.1(c.0(b.0(b.1(c.0(a.0(b.0(b.0(B.0(x0))))))))) → C1.1(c.0(a.0(a.0(a.0(a.1(c.1(c.1(c.0(b.0(B.0(x0)))))))))))
C1.1(c.0(b.0(a.0(b.0(b.0(B.1(x))))))) → C1.1(c.0(b.0(b.0(b.0(b.0(B.1(x)))))))
C1.1(c.0(a.0(b.0(b.0(B.1(x)))))) → C1.1(c.0(b.0(b.0(b.0(B.1(x))))))
C1.1(c.0(b.0(b.1(c.0(b.0(a.0(b.0(b.0(B.1(x0)))))))))) → C1.1(c.0(a.0(a.0(a.0(a.1(c.1(c.1(c.0(b.0(b.0(b.0(b.0(b.0(b.0(B.1(x0))))))))))))))))
C1.1(c.0(b.0(b.1(c.0(b.0(b.1(c.1(y_4)))))))) → C1.1(c.0(b.0(b.1(c.1(y_4)))))
C1.1(c.0(b.0(a.0(b.0(b.0(B.1(x))))))) → C1.1(c.0(b.0(b.0(b.0(B.1(x))))))
C1.1(c.0(a.0(b.0(b.0(B.0(x)))))) → C1.1(c.0(b.0(b.0(b.0(B.0(x))))))
C1.1(c.0(b.0(b.1(c.0(b.0(B.0(x0))))))) → C1.1(c.0(a.0(a.0(a.0(a.1(c.1(c.0(C.0(x0)))))))))
C1.1(c.0(b.0(b.1(c.0(b.0(a.0(b.0(b.0(B.1(x0)))))))))) → C1.1(c.0(a.0(a.0(a.0(a.1(c.1(c.1(c.0(b.0(b.0(B.1(x0))))))))))))
C1.1(c.0(b.0(b.1(x)))) → C1.1(c.1(x))
C1.1(c.0(b.0(b.1(c.0(a.0(b.0(b.0(B.1(x0))))))))) → C1.1(c.0(a.0(a.0(a.0(a.1(c.1(c.1(c.0(b.0(b.0(B.1(x0))))))))))))
C1.1(c.0(b.0(b.1(c.0(y_2))))) → C1.1(c.0(y_2))
C1.1(c.0(b.0(b.1(c.0(a.0(b.0(b.0(B.1(x0))))))))) → C1.1(c.0(a.0(a.0(a.0(a.1(c.1(c.1(c.0(b.0(b.0(b.0(b.0(b.0(B.1(x0)))))))))))))))
C1.1(c.0(a.0(b.0(b.0(B.0(x)))))) → C1.1(c.0(b.0(b.0(b.0(b.0(b.0(B.0(x))))))))
C1.1(c.0(a.0(b.0(b.0(B.1(x)))))) → C1.1(c.0(b.0(b.0(B.1(x)))))
C1.1(c.0(b.0(a.0(b.0(b.0(B.0(x))))))) → C1.1(c.0(b.0(b.0(b.0(b.0(b.0(b.0(B.0(x)))))))))
C1.1(c.0(b.0(b.1(c.0(b.0(a.0(b.0(b.0(B.0(x0)))))))))) → C1.1(c.0(a.0(a.0(a.0(a.1(c.1(c.1(c.0(b.0(b.0(b.0(b.0(b.0(b.0(B.0(x0))))))))))))))))
C1.1(c.0(b.0(b.1(c.0(b.0(a.0(b.0(b.0(B.1(x0)))))))))) → C1.1(c.0(a.0(a.0(a.0(a.1(c.1(c.1(c.0(b.0(b.0(b.0(b.0(b.0(B.1(x0)))))))))))))))
C1.1(c.0(b.0(b.1(c.0(b.0(a.0(b.0(b.0(B.0(x0)))))))))) → C1.1(c.0(a.0(a.0(a.0(a.1(c.1(c.1(c.0(b.0(b.0(b.0(b.0(b.0(B.0(x0)))))))))))))))
C1.1(c.0(b.0(b.1(c.0(b.0(B.1(x0))))))) → C1.1(c.0(a.0(a.0(a.0(a.1(c.0(C.1(x0))))))))
C1.1(c.0(b.0(b.1(c.0(b.0(a.0(b.0(b.0(B.0(x0)))))))))) → C1.1(c.0(a.0(a.0(a.0(a.1(c.1(c.1(c.0(b.0(b.0(b.0(B.0(x0)))))))))))))
C1.1(c.0(b.0(b.1(c.0(b.0(a.0(b.0(b.0(B.1(x0)))))))))) → C1.1(c.0(a.0(a.0(a.0(a.1(c.1(c.1(c.0(b.0(b.0(b.0(B.1(x0)))))))))))))
C1.1(c.0(a.0(b.0(b.0(B.0(x)))))) → C1.1(c.0(b.0(b.0(B.0(x)))))
C1.1(c.0(b.0(a.0(b.0(b.0(B.1(x))))))) → C1.1(c.0(b.0(b.0(B.1(x)))))
C1.1(c.0(b.0(b.1(c.0(b.0(b.1(c.0(y_4)))))))) → C1.1(c.0(b.0(b.1(c.0(y_4)))))
C1.1(c.0(b.0(b.1(c.0(b.0(B.1(x0))))))) → C1.1(c.0(a.0(a.0(a.0(a.1(c.1(c.0(C.1(x0)))))))))
C1.1(c.0(b.0(b.1(c.0(b.0(a.0(b.0(b.0(B.0(x0)))))))))) → C1.1(c.0(a.0(a.0(a.0(a.1(c.1(c.1(c.0(b.0(b.0(B.0(x0))))))))))))
C1.1(c.0(b.0(b.0(x)))) → C1.1(c.0(x))
C1.1(c.0(b.0(b.1(c.0(a.0(b.0(b.0(B.0(x0))))))))) → C1.1(c.0(a.0(a.0(a.0(a.1(c.1(c.1(c.0(b.0(b.0(b.0(b.0(b.0(B.0(x0)))))))))))))))
C1.1(c.0(b.0(b.1(c.0(x0))))) → C1.1(c.0(b.0(b.0(b.0(a.0(a.0(a.0(x0))))))))
C1.1(c.0(b.0(a.0(b.0(b.0(B.0(x))))))) → C1.1(c.0(b.0(b.0(b.0(B.0(x))))))
The TRS R consists of the following rules:
c.1(c.0(b.0(a.0(b.0(b.0(B.1(x))))))) → a.0(a.0(a.0(a.1(c.1(c.1(c.0(b.0(b.0(b.0(b.0(b.0(B.1(x)))))))))))))
a.0(b.0(b.0(b.1(x)))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.1(x))))))))
a.0(b.0(b.0(a.0(b.0(b.0(B.0(x))))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.0(x)))))))))))))))
a.0(b.0(b.0(B.0(x)))) → b.0(b.0(B.0(x)))
a.0(b.0(a.0(b.0(b.0(B.1(x)))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.1(x))))))))))))))
a.0(b.0(b.0(a.0(b.0(b.0(B.1(x))))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.1(x)))))))))))))))
c.1(c.0(b.0(a.0(b.0(b.0(B.0(x))))))) → a.0(a.0(a.0(a.1(c.1(c.1(c.0(b.0(b.0(B.0(x))))))))))
a.0(a.0(b.0(b.0(B.1(x))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.1(x)))))))))))))
a.1(x0) → a.0(x0)
c.1(c.0(b.0(a.0(b.0(b.0(B.0(x))))))) → a.0(a.0(a.0(a.1(c.1(c.1(c.0(b.0(b.0(b.0(b.0(b.0(B.0(x)))))))))))))
a.0(a.0(b.0(b.0(B.0(x))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.0(x)))))))))))))
c.1(c.0(b.0(a.0(b.0(b.0(B.1(x))))))) → a.0(a.0(a.0(a.1(c.1(c.1(c.0(b.0(b.0(b.0(b.0(b.0(b.0(B.1(x))))))))))))))
c.0(C.0(x)) → b.0(b.0(B.0(x)))
a.0(b.0(b.0(B.0(x)))) → b.0(B.0(x))
B.1(x0) → B.0(x0)
a.0(b.0(a.0(b.0(b.0(B.0(x)))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.0(x))))))))))))))
a.0(b.0(b.0(B.1(x)))) → b.0(B.1(x))
a.0(b.0(b.0(B.1(x)))) → B.1(x)
c.1(c.0(a.0(b.0(b.0(B.0(x)))))) → a.0(a.0(a.0(a.1(c.1(c.1(c.0(b.0(b.0(b.0(b.0(b.0(B.0(x)))))))))))))
a.0(b.0(b.0(a.0(b.0(b.0(B.1(x))))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.1(x))))))))))))))
a.0(a.0(b.0(b.0(B.0(x))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.0(x))))))))))
c.1(c.0(b.0(b.1(x)))) → a.0(a.0(a.0(a.1(c.1(c.1(c.1(x)))))))
c.1(c.0(b.0(B.1(x)))) → a.0(a.0(a.0(a.1(c.0(C.1(x))))))
a.0(a.0(b.0(b.0(B.0(x))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.0(x))))))))))))
c.1(c.0(a.0(b.0(b.0(B.1(x)))))) → a.0(a.0(a.0(a.1(c.1(c.1(c.0(b.0(b.0(B.1(x))))))))))
c.1(c.0(a.0(b.0(b.0(B.0(x)))))) → a.0(a.0(a.0(a.1(c.1(c.1(c.0(b.0(b.0(B.0(x))))))))))
c.1(c.0(a.0(b.0(b.0(B.1(x)))))) → a.0(a.0(a.0(a.1(c.1(c.1(c.0(b.0(b.0(b.0(b.0(b.0(B.1(x)))))))))))))
a.0(b.0(b.0(a.0(b.0(b.0(B.1(x))))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.1(x)))))))))))))
a.0(b.0(a.0(b.0(b.0(B.0(x)))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.0(x))))))))))))
c.1(c.0(b.0(a.0(b.0(b.0(B.0(x))))))) → a.0(a.0(a.0(a.1(c.1(c.1(c.0(b.0(b.0(b.0(b.0(B.0(x))))))))))))
c.1(c.0(x)) → b.0(b.0(b.0(a.0(a.0(a.0(x))))))
a.0(b.0(a.0(b.0(b.0(B.1(x)))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.1(x)))))))))))))
a.0(a.0(b.0(b.0(B.0(x))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.0(x)))))))))))
c.0(C.1(x)) → b.0(B.1(x))
c.1(c.0(a.0(b.0(b.0(B.0(x)))))) → a.0(a.0(a.0(a.1(c.1(c.1(c.0(b.0(B.0(x)))))))))
c.1(c.0(b.0(b.0(x)))) → a.0(a.0(a.0(a.1(c.1(c.1(c.0(x)))))))
a.0(a.0(b.0(b.0(B.0(x))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.0(x)))))))))
a.0(b.0(a.0(b.0(b.0(B.1(x)))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.1(x))))))))))
a.0(a.0(b.0(b.0(B.1(x))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.1(x)))))))))
b.1(x0) → b.0(x0)
c.1(c.0(b.0(a.0(b.0(b.0(B.1(x))))))) → a.0(a.0(a.0(a.1(c.1(c.1(c.0(b.0(b.0(B.1(x))))))))))
a.0(b.0(b.0(B.1(x)))) → b.0(b.0(B.1(x)))
c.1(c.0(a.0(b.0(b.0(B.1(x)))))) → a.0(a.0(a.0(a.1(c.1(c.1(c.0(b.0(b.0(b.0(b.0(B.1(x))))))))))))
a.0(b.0(a.0(b.0(b.0(B.0(x)))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.0(x)))))))))))
c.1(c.0(b.0(B.0(x)))) → a.0(a.0(a.0(a.1(c.0(C.0(x))))))
c.1(c.0(a.0(b.0(b.0(B.1(x)))))) → a.0(a.0(a.0(a.1(c.1(c.1(c.0(b.0(b.0(b.0(B.1(x)))))))))))
a.0(b.0(b.0(a.0(b.0(b.0(B.1(x))))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.1(x)))))))))))
a.0(a.0(b.0(b.0(B.1(x))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.1(x)))))))))))
a.0(b.0(b.0(a.0(b.0(b.0(B.0(x))))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.0(x))))))))))))
a.0(a.0(b.0(b.0(B.1(x))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.1(x))))))))))))
a.0(b.0(a.0(b.0(b.0(B.1(x)))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.1(x))))))))))))
c.1(c.0(b.0(a.0(b.0(b.0(B.1(x))))))) → a.0(a.0(a.0(a.1(c.1(c.1(c.0(b.0(b.0(b.0(b.0(B.1(x))))))))))))
a.0(b.0(b.0(a.0(b.0(b.0(B.0(x))))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.0(x))))))))))))))
a.0(b.0(b.0(a.0(b.0(b.0(B.0(x))))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.0(x)))))))))))))
a.0(a.0(b.0(b.0(B.1(x))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.1(x))))))))))
c.0(C.0(x)) → b.0(B.0(x))
a.0(b.0(b.0(B.0(x)))) → B.0(x)
c.1(c.0(a.0(b.0(b.0(B.0(x)))))) → a.0(a.0(a.0(a.1(c.1(c.1(c.0(b.0(b.0(b.0(B.0(x)))))))))))
a.0(b.0(b.0(b.0(x)))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(x))))))))
c.1(c.1(x)) → b.0(b.0(b.0(a.0(a.0(a.1(x))))))
a.0(b.0(a.0(b.0(b.0(B.0(x)))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.0(x))))))))))
a.0(b.0(b.0(a.0(b.0(b.0(B.1(x))))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.1(x))))))))))))
c.1(x0) → c.0(x0)
c.1(c.0(b.0(B.1(x)))) → a.0(a.0(a.0(a.1(c.1(c.0(C.1(x)))))))
C.1(x0) → C.0(x0)
c.1(c.0(a.0(b.0(b.0(B.1(x)))))) → a.0(a.0(a.0(a.1(c.1(c.1(c.0(b.0(B.1(x)))))))))
a.0(b.0(b.0(a.0(b.0(b.0(B.0(x))))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.0(x)))))))))))
c.0(C.1(x)) → b.0(b.0(B.1(x)))
a.0(b.0(a.0(b.0(b.0(B.0(x)))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.0(x)))))))))))))
c.1(c.0(b.0(a.0(b.0(b.0(B.0(x))))))) → a.0(a.0(a.0(a.1(c.1(c.1(c.0(b.0(b.0(b.0(B.0(x)))))))))))
c.1(c.0(b.0(B.0(x)))) → a.0(a.0(a.0(a.1(c.1(c.0(C.0(x)))))))
c.1(c.0(a.0(b.0(b.0(B.0(x)))))) → a.0(a.0(a.0(a.1(c.1(c.1(c.0(b.0(b.0(b.0(b.0(B.0(x))))))))))))
c.1(c.0(b.0(a.0(b.0(b.0(B.0(x))))))) → a.0(a.0(a.0(a.1(c.1(c.1(c.0(b.0(b.0(b.0(b.0(b.0(b.0(B.0(x))))))))))))))
a.0(b.0(a.0(b.0(b.0(B.1(x)))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.1(x)))))))))))
c.1(c.0(b.0(a.0(b.0(b.0(B.1(x))))))) → a.0(a.0(a.0(a.1(c.1(c.1(c.0(b.0(b.0(b.0(B.1(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:
C1.1(c.0(b.0(b.1(c.0(b.0(a.0(b.0(b.0(B.1(x0)))))))))) → C1.1(c.0(a.0(a.0(a.0(a.1(c.1(c.1(c.0(b.0(b.0(b.0(b.0(B.1(x0))))))))))))))
C1.1(c.0(b.0(b.1(c.0(b.0(B.0(x0))))))) → C1.1(c.0(a.0(a.0(a.0(a.1(c.0(C.0(x0))))))))
C1.1(c.0(b.0(b.1(c.0(b.0(b.0(x0))))))) → C1.1(c.0(a.0(a.0(a.0(a.1(c.1(c.1(c.0(x0)))))))))
C1.1(c.0(b.0(b.1(c.0(a.0(b.0(b.0(B.1(x0))))))))) → C1.1(c.0(a.0(a.0(a.0(a.1(c.1(c.1(c.0(b.0(b.0(b.0(b.0(B.1(x0))))))))))))))
C1.1(c.0(b.0(b.1(c.0(a.0(b.0(b.0(B.0(x0))))))))) → C1.1(c.0(a.0(a.0(a.0(a.1(c.1(c.1(c.0(b.0(b.0(B.0(x0))))))))))))
C1.1(c.0(b.0(b.1(c.0(b.0(b.1(x0))))))) → C1.1(c.0(a.0(a.0(a.0(a.1(c.1(c.1(c.1(x0)))))))))
C1.1(c.0(b.0(b.1(c.0(a.0(b.0(b.0(B.1(x0))))))))) → C1.1(c.0(a.0(a.0(a.0(a.1(c.1(c.1(c.0(b.0(b.0(b.0(B.1(x0)))))))))))))
C1.1(c.0(b.0(b.1(c.0(b.0(a.0(b.0(b.0(B.0(x0)))))))))) → C1.1(c.0(a.0(a.0(a.0(a.1(c.1(c.1(c.0(b.0(b.0(b.0(b.0(B.0(x0))))))))))))))
C1.1(c.0(b.0(b.1(c.0(a.0(b.0(b.0(B.0(x0))))))))) → C1.1(c.0(a.0(a.0(a.0(a.1(c.1(c.1(c.0(b.0(b.0(b.0(B.0(x0)))))))))))))
C1.1(c.0(b.0(b.1(c.0(a.0(b.0(b.0(B.1(x0))))))))) → C1.1(c.0(a.0(a.0(a.0(a.1(c.1(c.1(c.0(b.0(B.1(x0)))))))))))
C1.1(c.0(b.0(b.1(c.0(a.0(b.0(b.0(B.0(x0))))))))) → C1.1(c.0(a.0(a.0(a.0(a.1(c.1(c.1(c.0(b.0(b.0(b.0(b.0(B.0(x0))))))))))))))
C1.1(c.0(b.0(b.1(c.1(y_2))))) → C1.1(c.1(y_2))
C1.1(c.0(b.0(b.1(c.1(x0))))) → C1.1(c.0(b.0(b.0(b.0(a.0(a.0(a.1(x0))))))))
C1.1(c.0(b.0(b.1(c.0(a.0(b.0(b.0(B.0(x0))))))))) → C1.1(c.0(a.0(a.0(a.0(a.1(c.1(c.1(c.0(b.0(B.0(x0)))))))))))
C1.1(c.0(b.0(b.1(c.0(b.0(a.0(b.0(b.0(B.1(x0)))))))))) → C1.1(c.0(a.0(a.0(a.0(a.1(c.1(c.1(c.0(b.0(b.0(b.0(b.0(b.0(b.0(B.1(x0))))))))))))))))
C1.1(c.0(b.0(b.1(c.0(b.0(b.1(c.1(y_4)))))))) → C1.1(c.0(b.0(b.1(c.1(y_4)))))
C1.1(c.0(b.0(b.1(c.0(b.0(B.0(x0))))))) → C1.1(c.0(a.0(a.0(a.0(a.1(c.1(c.0(C.0(x0)))))))))
C1.1(c.0(b.0(b.1(c.0(b.0(a.0(b.0(b.0(B.1(x0)))))))))) → C1.1(c.0(a.0(a.0(a.0(a.1(c.1(c.1(c.0(b.0(b.0(B.1(x0))))))))))))
C1.1(c.0(b.0(b.1(x)))) → C1.1(c.1(x))
C1.1(c.0(b.0(b.1(c.0(a.0(b.0(b.0(B.1(x0))))))))) → C1.1(c.0(a.0(a.0(a.0(a.1(c.1(c.1(c.0(b.0(b.0(B.1(x0))))))))))))
C1.1(c.0(b.0(b.1(c.0(y_2))))) → C1.1(c.0(y_2))
C1.1(c.0(b.0(b.1(c.0(a.0(b.0(b.0(B.1(x0))))))))) → C1.1(c.0(a.0(a.0(a.0(a.1(c.1(c.1(c.0(b.0(b.0(b.0(b.0(b.0(B.1(x0)))))))))))))))
C1.1(c.0(b.0(b.1(c.0(b.0(a.0(b.0(b.0(B.0(x0)))))))))) → C1.1(c.0(a.0(a.0(a.0(a.1(c.1(c.1(c.0(b.0(b.0(b.0(b.0(b.0(b.0(B.0(x0))))))))))))))))
C1.1(c.0(b.0(b.1(c.0(b.0(a.0(b.0(b.0(B.1(x0)))))))))) → C1.1(c.0(a.0(a.0(a.0(a.1(c.1(c.1(c.0(b.0(b.0(b.0(b.0(b.0(B.1(x0)))))))))))))))
C1.1(c.0(b.0(b.1(c.0(b.0(a.0(b.0(b.0(B.0(x0)))))))))) → C1.1(c.0(a.0(a.0(a.0(a.1(c.1(c.1(c.0(b.0(b.0(b.0(b.0(b.0(B.0(x0)))))))))))))))
C1.1(c.0(b.0(b.1(c.0(b.0(B.1(x0))))))) → C1.1(c.0(a.0(a.0(a.0(a.1(c.0(C.1(x0))))))))
C1.1(c.0(b.0(b.1(c.0(b.0(a.0(b.0(b.0(B.0(x0)))))))))) → C1.1(c.0(a.0(a.0(a.0(a.1(c.1(c.1(c.0(b.0(b.0(b.0(B.0(x0)))))))))))))
C1.1(c.0(b.0(b.1(c.0(b.0(a.0(b.0(b.0(B.1(x0)))))))))) → C1.1(c.0(a.0(a.0(a.0(a.1(c.1(c.1(c.0(b.0(b.0(b.0(B.1(x0)))))))))))))
C1.1(c.0(b.0(b.1(c.0(b.0(b.1(c.0(y_4)))))))) → C1.1(c.0(b.0(b.1(c.0(y_4)))))
C1.1(c.0(b.0(b.1(c.0(b.0(B.1(x0))))))) → C1.1(c.0(a.0(a.0(a.0(a.1(c.1(c.0(C.1(x0)))))))))
C1.1(c.0(b.0(b.1(c.0(b.0(a.0(b.0(b.0(B.0(x0)))))))))) → C1.1(c.0(a.0(a.0(a.0(a.1(c.1(c.1(c.0(b.0(b.0(B.0(x0))))))))))))
C1.1(c.0(b.0(b.1(c.0(a.0(b.0(b.0(B.0(x0))))))))) → C1.1(c.0(a.0(a.0(a.0(a.1(c.1(c.1(c.0(b.0(b.0(b.0(b.0(b.0(B.0(x0)))))))))))))))
C1.1(c.0(b.0(b.1(c.0(x0))))) → C1.1(c.0(b.0(b.0(b.0(a.0(a.0(a.0(x0))))))))
Strictly oriented rules of the TRS R:
c.1(c.0(b.0(b.1(x)))) → a.0(a.0(a.0(a.1(c.1(c.1(c.1(x)))))))
b.1(x0) → b.0(x0)
Used ordering: POLO with Polynomial interpretation [25]:
POL(B.0(x1)) = x1
POL(B.1(x1)) = x1
POL(C.0(x1)) = x1
POL(C.1(x1)) = x1
POL(C1.1(x1)) = x1
POL(a.0(x1)) = x1
POL(a.1(x1)) = x1
POL(b.0(x1)) = x1
POL(b.1(x1)) = 1 + x1
POL(c.0(x1)) = x1
POL(c.1(x1)) = x1
↳ QTRS
↳ QTRS Reverse
↳ QTRS Reverse
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ SemLabProof
↳ SemLabProof2
↳ QDP
↳ QDPToSRSProof
↳ QTRS
↳ QTRS Reverse
↳ QTRS
↳ QTRS Reverse
↳ QTRS Reverse
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ SemLabProof
↳ SemLabProof2
↳ QDP
↳ SemLabProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ RuleRemovalProof
↳ QDP
↳ UsableRulesReductionPairsProof
Q DP problem:
The TRS P consists of the following rules:
C1.1(c.0(b.0(a.0(b.0(b.0(B.1(x))))))) → C1.1(c.0(b.0(b.0(b.0(B.1(x))))))
C1.1(c.0(a.0(b.0(b.0(B.0(x)))))) → C1.1(c.0(b.0(b.0(b.0(B.0(x))))))
C1.1(c.0(a.0(b.0(b.0(B.0(x)))))) → C1.1(c.0(b.0(b.0(b.0(b.0(B.0(x)))))))
C1.1(c.0(b.0(a.0(b.0(b.0(B.1(x))))))) → C1.1(c.0(b.0(b.0(b.0(b.0(b.0(B.1(x))))))))
C1.1(c.0(a.0(b.0(b.0(B.0(x)))))) → C1.1(c.0(b.0(b.0(B.0(x)))))
C1.1(c.0(b.0(a.0(b.0(b.0(B.1(x))))))) → C1.1(c.0(b.0(b.0(b.0(b.0(b.0(b.0(B.1(x)))))))))
C1.1(c.0(b.0(a.0(b.0(b.0(B.0(x))))))) → C1.1(c.0(b.0(b.0(B.0(x)))))
C1.1(c.0(a.0(b.0(b.0(B.0(x)))))) → C1.1(c.0(b.0(b.0(b.0(b.0(b.0(B.0(x))))))))
C1.1(c.0(b.0(a.0(b.0(b.0(B.1(x))))))) → C1.1(c.0(b.0(b.0(B.1(x)))))
C1.1(c.0(a.0(b.0(b.0(B.1(x)))))) → C1.1(c.0(b.0(b.0(B.1(x)))))
C1.1(c.0(b.0(a.0(b.0(b.0(B.0(x))))))) → C1.1(c.0(b.0(b.0(b.0(b.0(b.0(b.0(B.0(x)))))))))
C1.1(c.0(b.0(a.0(b.0(b.0(B.0(x))))))) → C1.1(c.0(b.0(b.0(b.0(b.0(B.0(x)))))))
C1.1(c.0(a.0(b.0(b.0(B.1(x)))))) → C1.1(c.0(b.0(b.0(b.0(b.0(B.1(x)))))))
C1.1(c.0(b.0(b.0(x)))) → C1.1(c.0(x))
C1.1(c.0(b.0(a.0(b.0(b.0(B.0(x))))))) → C1.1(c.0(b.0(b.0(b.0(b.0(b.0(B.0(x))))))))
C1.1(c.0(b.0(a.0(b.0(b.0(B.1(x))))))) → C1.1(c.0(b.0(b.0(b.0(b.0(B.1(x)))))))
C1.1(c.0(b.0(a.0(b.0(b.0(B.0(x))))))) → C1.1(c.0(b.0(b.0(b.0(B.0(x))))))
C1.1(c.0(a.0(b.0(b.0(B.1(x)))))) → C1.1(c.0(b.0(b.0(b.0(B.1(x))))))
C1.1(c.0(a.0(b.0(b.0(B.1(x)))))) → C1.1(c.0(b.0(b.0(b.0(b.0(b.0(B.1(x))))))))
The TRS R consists of the following rules:
c.1(c.0(b.0(a.0(b.0(b.0(B.1(x))))))) → a.0(a.0(a.0(a.1(c.1(c.1(c.0(b.0(b.0(b.0(b.0(b.0(B.1(x)))))))))))))
a.0(b.0(b.0(b.1(x)))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.1(x))))))))
a.0(b.0(b.0(a.0(b.0(b.0(B.0(x))))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.0(x)))))))))))))))
a.0(b.0(b.0(B.0(x)))) → b.0(b.0(B.0(x)))
a.0(b.0(a.0(b.0(b.0(B.1(x)))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.1(x))))))))))))))
a.0(b.0(b.0(a.0(b.0(b.0(B.1(x))))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.1(x)))))))))))))))
c.1(c.0(b.0(a.0(b.0(b.0(B.0(x))))))) → a.0(a.0(a.0(a.1(c.1(c.1(c.0(b.0(b.0(B.0(x))))))))))
a.0(a.0(b.0(b.0(B.1(x))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.1(x)))))))))))))
a.1(x0) → a.0(x0)
c.1(c.0(b.0(a.0(b.0(b.0(B.0(x))))))) → a.0(a.0(a.0(a.1(c.1(c.1(c.0(b.0(b.0(b.0(b.0(b.0(B.0(x)))))))))))))
a.0(a.0(b.0(b.0(B.0(x))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.0(x)))))))))))))
c.1(c.0(b.0(a.0(b.0(b.0(B.1(x))))))) → a.0(a.0(a.0(a.1(c.1(c.1(c.0(b.0(b.0(b.0(b.0(b.0(b.0(B.1(x))))))))))))))
c.0(C.0(x)) → b.0(b.0(B.0(x)))
a.0(b.0(b.0(B.0(x)))) → b.0(B.0(x))
B.1(x0) → B.0(x0)
a.0(b.0(a.0(b.0(b.0(B.0(x)))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.0(x))))))))))))))
a.0(b.0(b.0(B.1(x)))) → b.0(B.1(x))
a.0(b.0(b.0(B.1(x)))) → B.1(x)
c.1(c.0(a.0(b.0(b.0(B.0(x)))))) → a.0(a.0(a.0(a.1(c.1(c.1(c.0(b.0(b.0(b.0(b.0(b.0(B.0(x)))))))))))))
a.0(b.0(b.0(a.0(b.0(b.0(B.1(x))))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.1(x))))))))))))))
a.0(a.0(b.0(b.0(B.0(x))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.0(x))))))))))
c.1(c.0(b.0(B.1(x)))) → a.0(a.0(a.0(a.1(c.0(C.1(x))))))
a.0(a.0(b.0(b.0(B.0(x))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.0(x))))))))))))
c.1(c.0(a.0(b.0(b.0(B.1(x)))))) → a.0(a.0(a.0(a.1(c.1(c.1(c.0(b.0(b.0(B.1(x))))))))))
c.1(c.0(a.0(b.0(b.0(B.0(x)))))) → a.0(a.0(a.0(a.1(c.1(c.1(c.0(b.0(b.0(B.0(x))))))))))
c.1(c.0(a.0(b.0(b.0(B.1(x)))))) → a.0(a.0(a.0(a.1(c.1(c.1(c.0(b.0(b.0(b.0(b.0(b.0(B.1(x)))))))))))))
a.0(b.0(b.0(a.0(b.0(b.0(B.1(x))))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.1(x)))))))))))))
a.0(b.0(a.0(b.0(b.0(B.0(x)))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.0(x))))))))))))
c.1(c.0(b.0(a.0(b.0(b.0(B.0(x))))))) → a.0(a.0(a.0(a.1(c.1(c.1(c.0(b.0(b.0(b.0(b.0(B.0(x))))))))))))
c.1(c.0(x)) → b.0(b.0(b.0(a.0(a.0(a.0(x))))))
a.0(b.0(a.0(b.0(b.0(B.1(x)))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.1(x)))))))))))))
a.0(a.0(b.0(b.0(B.0(x))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.0(x)))))))))))
c.0(C.1(x)) → b.0(B.1(x))
c.1(c.0(a.0(b.0(b.0(B.0(x)))))) → a.0(a.0(a.0(a.1(c.1(c.1(c.0(b.0(B.0(x)))))))))
c.1(c.0(b.0(b.0(x)))) → a.0(a.0(a.0(a.1(c.1(c.1(c.0(x)))))))
a.0(a.0(b.0(b.0(B.0(x))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.0(x)))))))))
a.0(b.0(a.0(b.0(b.0(B.1(x)))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.1(x))))))))))
a.0(a.0(b.0(b.0(B.1(x))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.1(x)))))))))
c.1(c.0(b.0(a.0(b.0(b.0(B.1(x))))))) → a.0(a.0(a.0(a.1(c.1(c.1(c.0(b.0(b.0(B.1(x))))))))))
a.0(b.0(b.0(B.1(x)))) → b.0(b.0(B.1(x)))
c.1(c.0(a.0(b.0(b.0(B.1(x)))))) → a.0(a.0(a.0(a.1(c.1(c.1(c.0(b.0(b.0(b.0(b.0(B.1(x))))))))))))
a.0(b.0(a.0(b.0(b.0(B.0(x)))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.0(x)))))))))))
c.1(c.0(b.0(B.0(x)))) → a.0(a.0(a.0(a.1(c.0(C.0(x))))))
c.1(c.0(a.0(b.0(b.0(B.1(x)))))) → a.0(a.0(a.0(a.1(c.1(c.1(c.0(b.0(b.0(b.0(B.1(x)))))))))))
a.0(b.0(b.0(a.0(b.0(b.0(B.1(x))))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.1(x)))))))))))
a.0(a.0(b.0(b.0(B.1(x))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.1(x)))))))))))
a.0(b.0(b.0(a.0(b.0(b.0(B.0(x))))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.0(x))))))))))))
a.0(a.0(b.0(b.0(B.1(x))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.1(x))))))))))))
a.0(b.0(a.0(b.0(b.0(B.1(x)))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.1(x))))))))))))
c.1(c.0(b.0(a.0(b.0(b.0(B.1(x))))))) → a.0(a.0(a.0(a.1(c.1(c.1(c.0(b.0(b.0(b.0(b.0(B.1(x))))))))))))
a.0(b.0(b.0(a.0(b.0(b.0(B.0(x))))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.0(x))))))))))))))
a.0(b.0(b.0(a.0(b.0(b.0(B.0(x))))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.0(x)))))))))))))
a.0(a.0(b.0(b.0(B.1(x))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.1(x))))))))))
c.0(C.0(x)) → b.0(B.0(x))
a.0(b.0(b.0(B.0(x)))) → B.0(x)
c.1(c.0(a.0(b.0(b.0(B.0(x)))))) → a.0(a.0(a.0(a.1(c.1(c.1(c.0(b.0(b.0(b.0(B.0(x)))))))))))
a.0(b.0(b.0(b.0(x)))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(x))))))))
c.1(c.1(x)) → b.0(b.0(b.0(a.0(a.0(a.1(x))))))
a.0(b.0(a.0(b.0(b.0(B.0(x)))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.0(x))))))))))
a.0(b.0(b.0(a.0(b.0(b.0(B.1(x))))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.1(x))))))))))))
c.1(x0) → c.0(x0)
c.1(c.0(b.0(B.1(x)))) → a.0(a.0(a.0(a.1(c.1(c.0(C.1(x)))))))
C.1(x0) → C.0(x0)
c.1(c.0(a.0(b.0(b.0(B.1(x)))))) → a.0(a.0(a.0(a.1(c.1(c.1(c.0(b.0(B.1(x)))))))))
a.0(b.0(b.0(a.0(b.0(b.0(B.0(x))))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.0(x)))))))))))
c.0(C.1(x)) → b.0(b.0(B.1(x)))
a.0(b.0(a.0(b.0(b.0(B.0(x)))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.0(x)))))))))))))
c.1(c.0(b.0(a.0(b.0(b.0(B.0(x))))))) → a.0(a.0(a.0(a.1(c.1(c.1(c.0(b.0(b.0(b.0(B.0(x)))))))))))
c.1(c.0(b.0(B.0(x)))) → a.0(a.0(a.0(a.1(c.1(c.0(C.0(x)))))))
c.1(c.0(a.0(b.0(b.0(B.0(x)))))) → a.0(a.0(a.0(a.1(c.1(c.1(c.0(b.0(b.0(b.0(b.0(B.0(x))))))))))))
c.1(c.0(b.0(a.0(b.0(b.0(B.0(x))))))) → a.0(a.0(a.0(a.1(c.1(c.1(c.0(b.0(b.0(b.0(b.0(b.0(b.0(B.0(x))))))))))))))
a.0(b.0(a.0(b.0(b.0(B.1(x)))))) → b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(b.0(B.1(x)))))))))))
c.1(c.0(b.0(a.0(b.0(b.0(B.1(x))))))) → a.0(a.0(a.0(a.1(c.1(c.1(c.0(b.0(b.0(b.0(B.1(x)))))))))))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By using the usable rules with reduction pair processor [15] with a polynomial ordering [25], all dependency pairs and the corresponding usable rules [17] can be oriented non-strictly. All non-usable rules are removed, and those dependency pairs and usable rules that have been oriented strictly or contain non-usable symbols in their left-hand side are removed as well.
The following dependency pairs can be deleted:
C1.1(c.0(b.0(a.0(b.0(b.0(B.1(x))))))) → C1.1(c.0(b.0(b.0(b.0(B.1(x))))))
C1.1(c.0(a.0(b.0(b.0(B.0(x)))))) → C1.1(c.0(b.0(b.0(b.0(B.0(x))))))
C1.1(c.0(a.0(b.0(b.0(B.0(x)))))) → C1.1(c.0(b.0(b.0(b.0(b.0(B.0(x)))))))
C1.1(c.0(b.0(a.0(b.0(b.0(B.1(x))))))) → C1.1(c.0(b.0(b.0(b.0(b.0(b.0(B.1(x))))))))
C1.1(c.0(a.0(b.0(b.0(B.0(x)))))) → C1.1(c.0(b.0(b.0(B.0(x)))))
C1.1(c.0(b.0(a.0(b.0(b.0(B.1(x))))))) → C1.1(c.0(b.0(b.0(b.0(b.0(b.0(b.0(B.1(x)))))))))
C1.1(c.0(b.0(a.0(b.0(b.0(B.0(x))))))) → C1.1(c.0(b.0(b.0(B.0(x)))))
C1.1(c.0(a.0(b.0(b.0(B.0(x)))))) → C1.1(c.0(b.0(b.0(b.0(b.0(b.0(B.0(x))))))))
C1.1(c.0(b.0(a.0(b.0(b.0(B.1(x))))))) → C1.1(c.0(b.0(b.0(B.1(x)))))
C1.1(c.0(a.0(b.0(b.0(B.1(x)))))) → C1.1(c.0(b.0(b.0(B.1(x)))))
C1.1(c.0(b.0(a.0(b.0(b.0(B.0(x))))))) → C1.1(c.0(b.0(b.0(b.0(b.0(b.0(b.0(B.0(x)))))))))
C1.1(c.0(b.0(a.0(b.0(b.0(B.0(x))))))) → C1.1(c.0(b.0(b.0(b.0(b.0(B.0(x)))))))
C1.1(c.0(a.0(b.0(b.0(B.1(x)))))) → C1.1(c.0(b.0(b.0(b.0(b.0(B.1(x)))))))
C1.1(c.0(b.0(a.0(b.0(b.0(B.0(x))))))) → C1.1(c.0(b.0(b.0(b.0(b.0(b.0(B.0(x))))))))
C1.1(c.0(b.0(a.0(b.0(b.0(B.1(x))))))) → C1.1(c.0(b.0(b.0(b.0(b.0(B.1(x)))))))
C1.1(c.0(b.0(a.0(b.0(b.0(B.0(x))))))) → C1.1(c.0(b.0(b.0(b.0(B.0(x))))))
C1.1(c.0(a.0(b.0(b.0(B.1(x)))))) → C1.1(c.0(b.0(b.0(b.0(B.1(x))))))
C1.1(c.0(a.0(b.0(b.0(B.1(x)))))) → C1.1(c.0(b.0(b.0(b.0(b.0(b.0(B.1(x))))))))
The following rules are removed from R:
c.0(C.0(x)) → b.0(b.0(B.0(x)))
c.0(C.1(x)) → b.0(B.1(x))
c.0(C.0(x)) → b.0(B.0(x))
c.0(C.1(x)) → b.0(b.0(B.1(x)))
Used ordering: POLO with Polynomial interpretation [25]:
POL(B.0(x1)) = x1
POL(B.1(x1)) = x1
POL(C.0(x1)) = 1 + x1
POL(C.1(x1)) = 1 + x1
POL(C1.1(x1)) = x1
POL(a.0(x1)) = 1 + x1
POL(b.0(x1)) = x1
POL(c.0(x1)) = x1
↳ QTRS
↳ QTRS Reverse
↳ QTRS Reverse
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ SemLabProof
↳ SemLabProof2
↳ QDP
↳ QDPToSRSProof
↳ QTRS
↳ QTRS Reverse
↳ QTRS
↳ QTRS Reverse
↳ QTRS Reverse
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ SemLabProof
↳ SemLabProof2
↳ QDP
↳ SemLabProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ RuleRemovalProof
↳ QDP
↳ UsableRulesReductionPairsProof
↳ QDP
↳ UsableRulesReductionPairsProof
Q DP problem:
The TRS P consists of the following rules:
C1.1(c.0(b.0(b.0(x)))) → C1.1(c.0(x))
The TRS R consists of the following rules:
B.1(x0) → B.0(x0)
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By using the usable rules with reduction pair processor [15] with a polynomial ordering [25], all dependency pairs and the corresponding usable rules [17] can be oriented non-strictly. All non-usable rules are removed, and those dependency pairs and usable rules that have been oriented strictly or contain non-usable symbols in their left-hand side are removed as well.
The following dependency pairs can be deleted:
C1.1(c.0(b.0(b.0(x)))) → C1.1(c.0(x))
No rules are removed from R.
Used ordering: POLO with Polynomial interpretation [25]:
POL(C1.1(x1)) = x1
POL(b.0(x1)) = x1
POL(c.0(x1)) = x1
↳ QTRS
↳ QTRS Reverse
↳ QTRS Reverse
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ SemLabProof
↳ SemLabProof2
↳ QDP
↳ QDPToSRSProof
↳ QTRS
↳ QTRS Reverse
↳ QTRS
↳ QTRS Reverse
↳ QTRS Reverse
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ SemLabProof
↳ SemLabProof2
↳ QDP
↳ SemLabProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ RuleRemovalProof
↳ QDP
↳ UsableRulesReductionPairsProof
↳ QDP
↳ UsableRulesReductionPairsProof
↳ QDP
↳ PisEmptyProof
Q DP problem:
P is empty.
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The TRS P is empty. Hence, there is no (P,Q,R) chain.