Termination w.r.t. Q of the following Term Rewriting System could not be shown:
Q restricted rewrite system:
The TRS R consists of the following rules:
b(b(b(x1))) → b(b(a(b(a(b(a(b(x1))))))))
b(a(b(b(x1)))) → b(a(b(a(b(a(a(b(x1))))))))
b(b(a(b(a(b(a(a(b(x1))))))))) → b(b(b(b(a(b(x1))))))
Q is empty.
↳ QTRS
↳ QTRS Reverse
↳ QTRS Reverse
↳ DependencyPairsProof
Q restricted rewrite system:
The TRS R consists of the following rules:
b(b(b(x1))) → b(b(a(b(a(b(a(b(x1))))))))
b(a(b(b(x1)))) → b(a(b(a(b(a(a(b(x1))))))))
b(b(a(b(a(b(a(a(b(x1))))))))) → b(b(b(b(a(b(x1))))))
Q is empty.
We have reversed the following QTRS:
The set of rules R is
b(b(b(x1))) → b(b(a(b(a(b(a(b(x1))))))))
b(a(b(b(x1)))) → b(a(b(a(b(a(a(b(x1))))))))
b(b(a(b(a(b(a(a(b(x1))))))))) → b(b(b(b(a(b(x1))))))
The set Q is empty.
We have obtained the following QTRS:
b(b(b(x))) → b(a(b(a(b(a(b(b(x))))))))
b(b(a(b(x)))) → b(a(a(b(a(b(a(b(x))))))))
b(a(a(b(a(b(a(b(b(x))))))))) → b(a(b(b(b(b(x))))))
The set Q is empty.
↳ QTRS
↳ QTRS Reverse
↳ QTRS
↳ QTRS Reverse
↳ DependencyPairsProof
Q restricted rewrite system:
The TRS R consists of the following rules:
b(b(b(x))) → b(a(b(a(b(a(b(b(x))))))))
b(b(a(b(x)))) → b(a(a(b(a(b(a(b(x))))))))
b(a(a(b(a(b(a(b(b(x))))))))) → b(a(b(b(b(b(x))))))
Q is empty.
We have reversed the following QTRS:
The set of rules R is
b(b(b(x1))) → b(b(a(b(a(b(a(b(x1))))))))
b(a(b(b(x1)))) → b(a(b(a(b(a(a(b(x1))))))))
b(b(a(b(a(b(a(a(b(x1))))))))) → b(b(b(b(a(b(x1))))))
The set Q is empty.
We have obtained the following QTRS:
b(b(b(x))) → b(a(b(a(b(a(b(b(x))))))))
b(b(a(b(x)))) → b(a(a(b(a(b(a(b(x))))))))
b(a(a(b(a(b(a(b(b(x))))))))) → b(a(b(b(b(b(x))))))
The set Q is empty.
↳ QTRS
↳ QTRS Reverse
↳ QTRS Reverse
↳ QTRS
↳ DependencyPairsProof
Q restricted rewrite system:
The TRS R consists of the following rules:
b(b(b(x))) → b(a(b(a(b(a(b(b(x))))))))
b(b(a(b(x)))) → b(a(a(b(a(b(a(b(x))))))))
b(a(a(b(a(b(a(b(b(x))))))))) → b(a(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:
B(a(b(b(x1)))) → B(a(a(b(x1))))
B(a(b(b(x1)))) → B(a(b(a(b(a(a(b(x1))))))))
B(b(a(b(a(b(a(a(b(x1))))))))) → B(b(b(b(a(b(x1))))))
B(b(b(x1))) → B(a(b(a(b(x1)))))
B(b(a(b(a(b(a(a(b(x1))))))))) → B(b(b(a(b(x1)))))
B(a(b(b(x1)))) → B(a(b(a(a(b(x1))))))
B(b(b(x1))) → B(a(b(a(b(a(b(x1)))))))
B(b(b(x1))) → B(b(a(b(a(b(a(b(x1))))))))
B(b(a(b(a(b(a(a(b(x1))))))))) → B(b(a(b(x1))))
B(b(b(x1))) → B(a(b(x1)))
B(b(a(b(a(b(a(a(b(x1))))))))) → B(a(b(x1)))
The TRS R consists of the following rules:
b(b(b(x1))) → b(b(a(b(a(b(a(b(x1))))))))
b(a(b(b(x1)))) → b(a(b(a(b(a(a(b(x1))))))))
b(b(a(b(a(b(a(a(b(x1))))))))) → b(b(b(b(a(b(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(a(b(b(x1)))) → B(a(a(b(x1))))
B(a(b(b(x1)))) → B(a(b(a(b(a(a(b(x1))))))))
B(b(a(b(a(b(a(a(b(x1))))))))) → B(b(b(b(a(b(x1))))))
B(b(b(x1))) → B(a(b(a(b(x1)))))
B(b(a(b(a(b(a(a(b(x1))))))))) → B(b(b(a(b(x1)))))
B(a(b(b(x1)))) → B(a(b(a(a(b(x1))))))
B(b(b(x1))) → B(a(b(a(b(a(b(x1)))))))
B(b(b(x1))) → B(b(a(b(a(b(a(b(x1))))))))
B(b(a(b(a(b(a(a(b(x1))))))))) → B(b(a(b(x1))))
B(b(b(x1))) → B(a(b(x1)))
B(b(a(b(a(b(a(a(b(x1))))))))) → B(a(b(x1)))
The TRS R consists of the following rules:
b(b(b(x1))) → b(b(a(b(a(b(a(b(x1))))))))
b(a(b(b(x1)))) → b(a(b(a(b(a(a(b(x1))))))))
b(b(a(b(a(b(a(a(b(x1))))))))) → b(b(b(b(a(b(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 7 less nodes.
↳ QTRS
↳ QTRS Reverse
↳ QTRS Reverse
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
Q DP problem:
The TRS P consists of the following rules:
B(b(a(b(a(b(a(a(b(x1))))))))) → B(b(b(b(a(b(x1))))))
B(b(a(b(a(b(a(a(b(x1))))))))) → B(b(b(a(b(x1)))))
B(b(b(x1))) → B(b(a(b(a(b(a(b(x1))))))))
B(b(a(b(a(b(a(a(b(x1))))))))) → B(b(a(b(x1))))
The TRS R consists of the following rules:
b(b(b(x1))) → b(b(a(b(a(b(a(b(x1))))))))
b(a(b(b(x1)))) → b(a(b(a(b(a(a(b(x1))))))))
b(b(a(b(a(b(a(a(b(x1))))))))) → b(b(b(b(a(b(x1))))))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule B(b(b(x1))) → B(b(a(b(a(b(a(b(x1)))))))) at position [0] we obtained the following new rules:
B(b(b(a(b(b(x0)))))) → B(b(a(b(a(b(a(b(a(b(a(b(a(a(b(x0)))))))))))))))
B(b(b(b(a(b(a(b(a(a(b(x0))))))))))) → B(b(a(b(a(b(a(b(b(b(b(a(b(x0)))))))))))))
B(b(b(b(x0)))) → B(b(a(b(a(b(a(b(a(b(a(a(b(x0)))))))))))))
B(b(b(b(b(x0))))) → B(b(a(b(a(b(a(b(b(a(b(a(b(a(b(x0)))))))))))))))
↳ QTRS
↳ QTRS Reverse
↳ QTRS Reverse
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
Q DP problem:
The TRS P consists of the following rules:
B(b(b(b(a(b(a(b(a(a(b(x0))))))))))) → B(b(a(b(a(b(a(b(b(b(b(a(b(x0)))))))))))))
B(b(b(b(x0)))) → B(b(a(b(a(b(a(b(a(b(a(a(b(x0)))))))))))))
B(b(a(b(a(b(a(a(b(x1))))))))) → B(b(b(b(a(b(x1))))))
B(b(b(b(b(x0))))) → B(b(a(b(a(b(a(b(b(a(b(a(b(a(b(x0)))))))))))))))
B(b(a(b(a(b(a(a(b(x1))))))))) → B(b(b(a(b(x1)))))
B(b(b(a(b(b(x0)))))) → B(b(a(b(a(b(a(b(a(b(a(b(a(a(b(x0)))))))))))))))
B(b(a(b(a(b(a(a(b(x1))))))))) → B(b(a(b(x1))))
The TRS R consists of the following rules:
b(b(b(x1))) → b(b(a(b(a(b(a(b(x1))))))))
b(a(b(b(x1)))) → b(a(b(a(b(a(a(b(x1))))))))
b(b(a(b(a(b(a(a(b(x1))))))))) → b(b(b(b(a(b(x1))))))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 1 SCC with 2 less nodes.
↳ QTRS
↳ QTRS Reverse
↳ QTRS Reverse
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
Q DP problem:
The TRS P consists of the following rules:
B(b(b(b(a(b(a(b(a(a(b(x0))))))))))) → B(b(a(b(a(b(a(b(b(b(b(a(b(x0)))))))))))))
B(b(a(b(a(b(a(a(b(x1))))))))) → B(b(b(b(a(b(x1))))))
B(b(b(b(b(x0))))) → B(b(a(b(a(b(a(b(b(a(b(a(b(a(b(x0)))))))))))))))
B(b(a(b(a(b(a(a(b(x1))))))))) → B(b(b(a(b(x1)))))
B(b(a(b(a(b(a(a(b(x1))))))))) → B(b(a(b(x1))))
The TRS R consists of the following rules:
b(b(b(x1))) → b(b(a(b(a(b(a(b(x1))))))))
b(a(b(b(x1)))) → b(a(b(a(b(a(a(b(x1))))))))
b(b(a(b(a(b(a(a(b(x1))))))))) → b(b(b(b(a(b(x1))))))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule B(b(a(b(a(b(a(a(b(x1))))))))) → B(b(b(a(b(x1))))) at position [0] we obtained the following new rules:
B(b(a(b(a(b(a(a(b(b(x0)))))))))) → B(b(b(a(b(a(b(a(a(b(x0))))))))))
B(b(a(b(a(b(a(a(b(a(b(a(a(b(x0)))))))))))))) → B(b(b(b(b(a(b(x0)))))))
B(b(a(b(a(b(a(a(b(b(b(x0))))))))))) → B(b(b(a(b(b(a(b(a(b(a(b(x0))))))))))))
B(b(a(b(a(b(a(a(b(a(b(b(x0)))))))))))) → B(b(b(a(b(a(b(a(b(a(a(b(x0))))))))))))
B(b(a(b(a(b(a(a(b(b(a(b(a(b(a(a(b(x0))))))))))))))))) → B(b(b(a(b(b(b(b(a(b(x0))))))))))
↳ 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:
B(b(a(b(a(b(a(a(b(b(x0)))))))))) → B(b(b(a(b(a(b(a(a(b(x0))))))))))
B(b(b(b(a(b(a(b(a(a(b(x0))))))))))) → B(b(a(b(a(b(a(b(b(b(b(a(b(x0)))))))))))))
B(b(a(b(a(b(a(a(b(x1))))))))) → B(b(b(b(a(b(x1))))))
B(b(b(b(b(x0))))) → B(b(a(b(a(b(a(b(b(a(b(a(b(a(b(x0)))))))))))))))
B(b(a(b(a(b(a(a(b(a(b(a(a(b(x0)))))))))))))) → B(b(b(b(b(a(b(x0)))))))
B(b(a(b(a(b(a(a(b(b(b(x0))))))))))) → B(b(b(a(b(b(a(b(a(b(a(b(x0))))))))))))
B(b(a(b(a(b(a(a(b(a(b(b(x0)))))))))))) → B(b(b(a(b(a(b(a(b(a(a(b(x0))))))))))))
B(b(a(b(a(b(a(a(b(x1))))))))) → B(b(a(b(x1))))
B(b(a(b(a(b(a(a(b(b(a(b(a(b(a(a(b(x0))))))))))))))))) → B(b(b(a(b(b(b(b(a(b(x0))))))))))
The TRS R consists of the following rules:
b(b(b(x1))) → b(b(a(b(a(b(a(b(x1))))))))
b(a(b(b(x1)))) → b(a(b(a(b(a(a(b(x1))))))))
b(b(a(b(a(b(a(a(b(x1))))))))) → b(b(b(b(a(b(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
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPToSRSProof
Q DP problem:
The TRS P consists of the following rules:
B(b(a(b(a(b(a(a(b(b(x0)))))))))) → B(b(b(a(b(a(b(a(a(b(x0))))))))))
B(b(b(b(a(b(a(b(a(a(b(x0))))))))))) → B(b(a(b(a(b(a(b(b(b(b(a(b(x0)))))))))))))
B(b(a(b(a(b(a(a(b(x1))))))))) → B(b(b(b(a(b(x1))))))
B(b(b(b(b(x0))))) → B(b(a(b(a(b(a(b(b(a(b(a(b(a(b(x0)))))))))))))))
B(b(a(b(a(b(a(a(b(a(b(a(a(b(x0)))))))))))))) → B(b(b(b(b(a(b(x0)))))))
B(b(a(b(a(b(a(a(b(b(b(x0))))))))))) → B(b(b(a(b(b(a(b(a(b(a(b(x0))))))))))))
B(b(a(b(a(b(a(a(b(x1))))))))) → B(b(a(b(x1))))
B(b(a(b(a(b(a(a(b(b(a(b(a(b(a(a(b(x0))))))))))))))))) → B(b(b(a(b(b(b(b(a(b(x0))))))))))
The TRS R consists of the following rules:
b(b(b(x1))) → b(b(a(b(a(b(a(b(x1))))))))
b(a(b(b(x1)))) → b(a(b(a(b(a(a(b(x1))))))))
b(b(a(b(a(b(a(a(b(x1))))))))) → b(b(b(b(a(b(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
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPToSRSProof
↳ QTRS
↳ QTRS Reverse
Q restricted rewrite system:
The TRS R consists of the following rules:
b(b(b(x1))) → b(b(a(b(a(b(a(b(x1))))))))
b(a(b(b(x1)))) → b(a(b(a(b(a(a(b(x1))))))))
b(b(a(b(a(b(a(a(b(x1))))))))) → b(b(b(b(a(b(x1))))))
B(b(a(b(a(b(a(a(b(b(x0)))))))))) → B(b(b(a(b(a(b(a(a(b(x0))))))))))
B(b(b(b(a(b(a(b(a(a(b(x0))))))))))) → B(b(a(b(a(b(a(b(b(b(b(a(b(x0)))))))))))))
B(b(a(b(a(b(a(a(b(x1))))))))) → B(b(b(b(a(b(x1))))))
B(b(b(b(b(x0))))) → B(b(a(b(a(b(a(b(b(a(b(a(b(a(b(x0)))))))))))))))
B(b(a(b(a(b(a(a(b(a(b(a(a(b(x0)))))))))))))) → B(b(b(b(b(a(b(x0)))))))
B(b(a(b(a(b(a(a(b(b(b(x0))))))))))) → B(b(b(a(b(b(a(b(a(b(a(b(x0))))))))))))
B(b(a(b(a(b(a(a(b(x1))))))))) → B(b(a(b(x1))))
B(b(a(b(a(b(a(a(b(b(a(b(a(b(a(a(b(x0))))))))))))))))) → B(b(b(a(b(b(b(b(a(b(x0))))))))))
Q is empty.
We have reversed the following QTRS:
The set of rules R is
b(b(b(x1))) → b(b(a(b(a(b(a(b(x1))))))))
b(a(b(b(x1)))) → b(a(b(a(b(a(a(b(x1))))))))
b(b(a(b(a(b(a(a(b(x1))))))))) → b(b(b(b(a(b(x1))))))
B(b(a(b(a(b(a(a(b(b(x0)))))))))) → B(b(b(a(b(a(b(a(a(b(x0))))))))))
B(b(b(b(a(b(a(b(a(a(b(x0))))))))))) → B(b(a(b(a(b(a(b(b(b(b(a(b(x0)))))))))))))
B(b(a(b(a(b(a(a(b(x1))))))))) → B(b(b(b(a(b(x1))))))
B(b(b(b(b(x0))))) → B(b(a(b(a(b(a(b(b(a(b(a(b(a(b(x0)))))))))))))))
B(b(a(b(a(b(a(a(b(a(b(a(a(b(x0)))))))))))))) → B(b(b(b(b(a(b(x0)))))))
B(b(a(b(a(b(a(a(b(b(b(x0))))))))))) → B(b(b(a(b(b(a(b(a(b(a(b(x0))))))))))))
B(b(a(b(a(b(a(a(b(x1))))))))) → B(b(a(b(x1))))
B(b(a(b(a(b(a(a(b(b(a(b(a(b(a(a(b(x0))))))))))))))))) → B(b(b(a(b(b(b(b(a(b(x0))))))))))
The set Q is empty.
We have obtained the following QTRS:
b(b(b(x))) → b(a(b(a(b(a(b(b(x))))))))
b(b(a(b(x)))) → b(a(a(b(a(b(a(b(x))))))))
b(a(a(b(a(b(a(b(b(x))))))))) → b(a(b(b(b(b(x))))))
b(b(a(a(b(a(b(a(b(B(x)))))))))) → b(a(a(b(a(b(a(b(b(B(x))))))))))
b(a(a(b(a(b(a(b(b(b(B(x))))))))))) → b(a(b(b(b(b(a(b(a(b(a(b(B(x)))))))))))))
b(a(a(b(a(b(a(b(B(x))))))))) → b(a(b(b(b(B(x))))))
b(b(b(b(B(x))))) → b(a(b(a(b(a(b(b(a(b(a(b(a(b(B(x)))))))))))))))
b(a(a(b(a(b(a(a(b(a(b(a(b(B(x)))))))))))))) → b(a(b(b(b(b(B(x)))))))
b(b(b(a(a(b(a(b(a(b(B(x))))))))))) → b(a(b(a(b(a(b(b(a(b(b(B(x))))))))))))
b(a(a(b(a(b(a(b(B(x))))))))) → b(a(b(B(x))))
b(a(a(b(a(b(a(b(b(a(a(b(a(b(a(b(B(x))))))))))))))))) → b(a(b(b(b(b(a(b(b(B(x))))))))))
The set Q is empty.
↳ QTRS
↳ QTRS Reverse
↳ QTRS Reverse
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPToSRSProof
↳ QTRS
↳ QTRS Reverse
↳ QTRS
↳ QTRS Reverse
↳ QTRS Reverse
↳ DependencyPairsProof
Q restricted rewrite system:
The TRS R consists of the following rules:
b(b(b(x))) → b(a(b(a(b(a(b(b(x))))))))
b(b(a(b(x)))) → b(a(a(b(a(b(a(b(x))))))))
b(a(a(b(a(b(a(b(b(x))))))))) → b(a(b(b(b(b(x))))))
b(b(a(a(b(a(b(a(b(B(x)))))))))) → b(a(a(b(a(b(a(b(b(B(x))))))))))
b(a(a(b(a(b(a(b(b(b(B(x))))))))))) → b(a(b(b(b(b(a(b(a(b(a(b(B(x)))))))))))))
b(a(a(b(a(b(a(b(B(x))))))))) → b(a(b(b(b(B(x))))))
b(b(b(b(B(x))))) → b(a(b(a(b(a(b(b(a(b(a(b(a(b(B(x)))))))))))))))
b(a(a(b(a(b(a(a(b(a(b(a(b(B(x)))))))))))))) → b(a(b(b(b(b(B(x)))))))
b(b(b(a(a(b(a(b(a(b(B(x))))))))))) → b(a(b(a(b(a(b(b(a(b(b(B(x))))))))))))
b(a(a(b(a(b(a(b(B(x))))))))) → b(a(b(B(x))))
b(a(a(b(a(b(a(b(b(a(a(b(a(b(a(b(B(x))))))))))))))))) → b(a(b(b(b(b(a(b(b(B(x))))))))))
Q is empty.
We have reversed the following QTRS:
The set of rules R is
b(b(b(x))) → b(a(b(a(b(a(b(b(x))))))))
b(b(a(b(x)))) → b(a(a(b(a(b(a(b(x))))))))
b(a(a(b(a(b(a(b(b(x))))))))) → b(a(b(b(b(b(x))))))
b(b(a(a(b(a(b(a(b(B(x)))))))))) → b(a(a(b(a(b(a(b(b(B(x))))))))))
b(a(a(b(a(b(a(b(b(b(B(x))))))))))) → b(a(b(b(b(b(a(b(a(b(a(b(B(x)))))))))))))
b(a(a(b(a(b(a(b(B(x))))))))) → b(a(b(b(b(B(x))))))
b(b(b(b(B(x))))) → b(a(b(a(b(a(b(b(a(b(a(b(a(b(B(x)))))))))))))))
b(a(a(b(a(b(a(a(b(a(b(a(b(B(x)))))))))))))) → b(a(b(b(b(b(B(x)))))))
b(b(b(a(a(b(a(b(a(b(B(x))))))))))) → b(a(b(a(b(a(b(b(a(b(b(B(x))))))))))))
b(a(a(b(a(b(a(b(B(x))))))))) → b(a(b(B(x))))
b(a(a(b(a(b(a(b(b(a(a(b(a(b(a(b(B(x))))))))))))))))) → b(a(b(b(b(b(a(b(b(B(x))))))))))
The set Q is empty.
We have obtained the following QTRS:
b(b(b(x))) → b(b(a(b(a(b(a(b(x))))))))
b(a(b(b(x)))) → b(a(b(a(b(a(a(b(x))))))))
b(b(a(b(a(b(a(a(b(x))))))))) → b(b(b(b(a(b(x))))))
B(b(a(b(a(b(a(a(b(b(x)))))))))) → B(b(b(a(b(a(b(a(a(b(x))))))))))
B(b(b(b(a(b(a(b(a(a(b(x))))))))))) → B(b(a(b(a(b(a(b(b(b(b(a(b(x)))))))))))))
B(b(a(b(a(b(a(a(b(x))))))))) → B(b(b(b(a(b(x))))))
B(b(b(b(b(x))))) → B(b(a(b(a(b(a(b(b(a(b(a(b(a(b(x)))))))))))))))
B(b(a(b(a(b(a(a(b(a(b(a(a(b(x)))))))))))))) → B(b(b(b(b(a(b(x)))))))
B(b(a(b(a(b(a(a(b(b(b(x))))))))))) → B(b(b(a(b(b(a(b(a(b(a(b(x))))))))))))
B(b(a(b(a(b(a(a(b(x))))))))) → B(b(a(b(x))))
B(b(a(b(a(b(a(a(b(b(a(b(a(b(a(a(b(x))))))))))))))))) → B(b(b(a(b(b(b(b(a(b(x))))))))))
The set Q is empty.
↳ QTRS
↳ QTRS Reverse
↳ QTRS Reverse
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPToSRSProof
↳ QTRS
↳ QTRS Reverse
↳ QTRS
↳ QTRS Reverse
↳ QTRS
↳ QTRS Reverse
↳ DependencyPairsProof
Q restricted rewrite system:
The TRS R consists of the following rules:
b(b(b(x))) → b(b(a(b(a(b(a(b(x))))))))
b(a(b(b(x)))) → b(a(b(a(b(a(a(b(x))))))))
b(b(a(b(a(b(a(a(b(x))))))))) → b(b(b(b(a(b(x))))))
B(b(a(b(a(b(a(a(b(b(x)))))))))) → B(b(b(a(b(a(b(a(a(b(x))))))))))
B(b(b(b(a(b(a(b(a(a(b(x))))))))))) → B(b(a(b(a(b(a(b(b(b(b(a(b(x)))))))))))))
B(b(a(b(a(b(a(a(b(x))))))))) → B(b(b(b(a(b(x))))))
B(b(b(b(b(x))))) → B(b(a(b(a(b(a(b(b(a(b(a(b(a(b(x)))))))))))))))
B(b(a(b(a(b(a(a(b(a(b(a(a(b(x)))))))))))))) → B(b(b(b(b(a(b(x)))))))
B(b(a(b(a(b(a(a(b(b(b(x))))))))))) → B(b(b(a(b(b(a(b(a(b(a(b(x))))))))))))
B(b(a(b(a(b(a(a(b(x))))))))) → B(b(a(b(x))))
B(b(a(b(a(b(a(a(b(b(a(b(a(b(a(a(b(x))))))))))))))))) → B(b(b(a(b(b(b(b(a(b(x))))))))))
Q is empty.
We have reversed the following QTRS:
The set of rules R is
b(b(b(x))) → b(a(b(a(b(a(b(b(x))))))))
b(b(a(b(x)))) → b(a(a(b(a(b(a(b(x))))))))
b(a(a(b(a(b(a(b(b(x))))))))) → b(a(b(b(b(b(x))))))
b(b(a(a(b(a(b(a(b(B(x)))))))))) → b(a(a(b(a(b(a(b(b(B(x))))))))))
b(a(a(b(a(b(a(b(b(b(B(x))))))))))) → b(a(b(b(b(b(a(b(a(b(a(b(B(x)))))))))))))
b(a(a(b(a(b(a(b(B(x))))))))) → b(a(b(b(b(B(x))))))
b(b(b(b(B(x))))) → b(a(b(a(b(a(b(b(a(b(a(b(a(b(B(x)))))))))))))))
b(a(a(b(a(b(a(a(b(a(b(a(b(B(x)))))))))))))) → b(a(b(b(b(b(B(x)))))))
b(b(b(a(a(b(a(b(a(b(B(x))))))))))) → b(a(b(a(b(a(b(b(a(b(b(B(x))))))))))))
b(a(a(b(a(b(a(b(B(x))))))))) → b(a(b(B(x))))
b(a(a(b(a(b(a(b(b(a(a(b(a(b(a(b(B(x))))))))))))))))) → b(a(b(b(b(b(a(b(b(B(x))))))))))
The set Q is empty.
We have obtained the following QTRS:
b(b(b(x))) → b(b(a(b(a(b(a(b(x))))))))
b(a(b(b(x)))) → b(a(b(a(b(a(a(b(x))))))))
b(b(a(b(a(b(a(a(b(x))))))))) → b(b(b(b(a(b(x))))))
B(b(a(b(a(b(a(a(b(b(x)))))))))) → B(b(b(a(b(a(b(a(a(b(x))))))))))
B(b(b(b(a(b(a(b(a(a(b(x))))))))))) → B(b(a(b(a(b(a(b(b(b(b(a(b(x)))))))))))))
B(b(a(b(a(b(a(a(b(x))))))))) → B(b(b(b(a(b(x))))))
B(b(b(b(b(x))))) → B(b(a(b(a(b(a(b(b(a(b(a(b(a(b(x)))))))))))))))
B(b(a(b(a(b(a(a(b(a(b(a(a(b(x)))))))))))))) → B(b(b(b(b(a(b(x)))))))
B(b(a(b(a(b(a(a(b(b(b(x))))))))))) → B(b(b(a(b(b(a(b(a(b(a(b(x))))))))))))
B(b(a(b(a(b(a(a(b(x))))))))) → B(b(a(b(x))))
B(b(a(b(a(b(a(a(b(b(a(b(a(b(a(a(b(x))))))))))))))))) → B(b(b(a(b(b(b(b(a(b(x))))))))))
The set Q is empty.
↳ QTRS
↳ QTRS Reverse
↳ QTRS Reverse
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPToSRSProof
↳ QTRS
↳ QTRS Reverse
↳ QTRS
↳ QTRS Reverse
↳ QTRS Reverse
↳ QTRS
↳ DependencyPairsProof
Q restricted rewrite system:
The TRS R consists of the following rules:
b(b(b(x))) → b(b(a(b(a(b(a(b(x))))))))
b(a(b(b(x)))) → b(a(b(a(b(a(a(b(x))))))))
b(b(a(b(a(b(a(a(b(x))))))))) → b(b(b(b(a(b(x))))))
B(b(a(b(a(b(a(a(b(b(x)))))))))) → B(b(b(a(b(a(b(a(a(b(x))))))))))
B(b(b(b(a(b(a(b(a(a(b(x))))))))))) → B(b(a(b(a(b(a(b(b(b(b(a(b(x)))))))))))))
B(b(a(b(a(b(a(a(b(x))))))))) → B(b(b(b(a(b(x))))))
B(b(b(b(b(x))))) → B(b(a(b(a(b(a(b(b(a(b(a(b(a(b(x)))))))))))))))
B(b(a(b(a(b(a(a(b(a(b(a(a(b(x)))))))))))))) → B(b(b(b(b(a(b(x)))))))
B(b(a(b(a(b(a(a(b(b(b(x))))))))))) → B(b(b(a(b(b(a(b(a(b(a(b(x))))))))))))
B(b(a(b(a(b(a(a(b(x))))))))) → B(b(a(b(x))))
B(b(a(b(a(b(a(a(b(b(a(b(a(b(a(a(b(x))))))))))))))))) → B(b(b(a(b(b(b(b(a(b(x))))))))))
Q is empty.
Using Dependency Pairs [1,15] we result in the following initial DP problem:
Q DP problem:
The TRS P consists of the following rules:
B1(a(a(b(a(b(a(b(b(x))))))))) → B1(b(b(b(x))))
B1(b(b(a(a(b(a(b(a(b(B(x))))))))))) → B1(a(b(b(B(x)))))
B1(a(a(b(a(b(a(b(b(a(a(b(a(b(a(b(B(x))))))))))))))))) → B1(b(a(b(b(B(x))))))
B1(b(b(x))) → B1(a(b(b(x))))
B1(a(a(b(a(b(a(b(B(x))))))))) → B1(b(B(x)))
B1(b(b(a(a(b(a(b(a(b(B(x))))))))))) → B1(b(a(b(b(B(x))))))
B1(b(b(b(B(x))))) → B1(a(b(a(b(B(x))))))
B1(b(a(b(x)))) → B1(a(a(b(a(b(a(b(x))))))))
B1(b(b(a(a(b(a(b(a(b(B(x))))))))))) → B1(a(b(a(b(a(b(b(a(b(b(B(x))))))))))))
B1(b(b(b(B(x))))) → B1(a(b(B(x))))
B1(b(a(a(b(a(b(a(b(B(x)))))))))) → B1(a(b(b(B(x)))))
B1(a(a(b(a(b(a(b(b(b(B(x))))))))))) → B1(a(b(b(b(b(a(b(a(b(a(b(B(x)))))))))))))
B1(b(a(b(x)))) → B1(a(b(a(b(x)))))
B1(b(a(a(b(a(b(a(b(B(x)))))))))) → B1(a(b(a(b(b(B(x)))))))
B1(a(a(b(a(b(a(b(b(a(a(b(a(b(a(b(B(x))))))))))))))))) → B1(a(b(b(b(b(a(b(b(B(x))))))))))
B1(a(a(b(a(b(a(b(b(b(B(x))))))))))) → B1(b(b(a(b(a(b(a(b(B(x))))))))))
B1(b(a(a(b(a(b(a(b(B(x)))))))))) → B1(b(B(x)))
B1(a(a(b(a(b(a(b(b(b(B(x))))))))))) → B1(b(b(b(a(b(a(b(a(b(B(x)))))))))))
B1(a(a(b(a(b(a(b(b(b(B(x))))))))))) → B1(a(b(B(x))))
B1(a(a(b(a(b(a(a(b(a(b(a(b(B(x)))))))))))))) → B1(b(b(b(B(x)))))
B1(a(a(b(a(b(a(b(b(b(B(x))))))))))) → B1(a(b(a(b(B(x))))))
B1(b(b(b(B(x))))) → B1(a(b(a(b(a(b(B(x))))))))
B1(b(b(x))) → B1(a(b(a(b(b(x))))))
B1(b(b(a(a(b(a(b(a(b(B(x))))))))))) → B1(a(b(a(b(b(a(b(b(B(x))))))))))
B1(a(a(b(a(b(a(b(b(a(a(b(a(b(a(b(B(x))))))))))))))))) → B1(a(b(b(B(x)))))
B1(a(a(b(a(b(a(b(b(a(a(b(a(b(a(b(B(x))))))))))))))))) → B1(b(b(a(b(b(B(x)))))))
B1(b(b(b(B(x))))) → B1(b(a(b(a(b(a(b(B(x)))))))))
B1(a(a(b(a(b(a(b(B(x))))))))) → B1(b(b(B(x))))
B1(a(a(b(a(b(a(a(b(a(b(a(b(B(x)))))))))))))) → B1(a(b(b(b(b(B(x)))))))
B1(b(b(x))) → B1(a(b(a(b(a(b(b(x))))))))
B1(a(a(b(a(b(a(a(b(a(b(a(b(B(x)))))))))))))) → B1(b(B(x)))
B1(a(a(b(a(b(a(a(b(a(b(a(b(B(x)))))))))))))) → B1(b(b(B(x))))
B1(b(b(b(B(x))))) → B1(a(b(a(b(b(a(b(a(b(a(b(B(x)))))))))))))
B1(a(a(b(a(b(a(b(b(x))))))))) → B1(b(b(x)))
B1(b(b(b(B(x))))) → B1(a(b(b(a(b(a(b(a(b(B(x)))))))))))
B1(a(a(b(a(b(a(b(b(a(a(b(a(b(a(b(B(x))))))))))))))))) → B1(b(b(b(a(b(b(B(x))))))))
B1(b(b(a(a(b(a(b(a(b(B(x))))))))))) → B1(a(b(b(a(b(b(B(x))))))))
B1(b(b(b(B(x))))) → B1(a(b(a(b(a(b(b(a(b(a(b(a(b(B(x)))))))))))))))
B1(a(a(b(a(b(a(b(b(a(a(b(a(b(a(b(B(x))))))))))))))))) → B1(b(B(x)))
B1(a(a(b(a(b(a(b(B(x))))))))) → B1(a(b(b(b(B(x))))))
B1(b(b(a(a(b(a(b(a(b(B(x))))))))))) → B1(b(B(x)))
B1(a(a(b(a(b(a(b(b(b(B(x))))))))))) → B1(b(a(b(a(b(a(b(B(x)))))))))
B1(a(a(b(a(b(a(b(b(b(B(x))))))))))) → B1(a(b(a(b(a(b(B(x))))))))
B1(a(a(b(a(b(a(b(b(x))))))))) → B1(a(b(b(b(b(x))))))
B1(b(a(a(b(a(b(a(b(B(x)))))))))) → B1(a(a(b(a(b(a(b(b(B(x))))))))))
The TRS R consists of the following rules:
b(b(b(x))) → b(a(b(a(b(a(b(b(x))))))))
b(b(a(b(x)))) → b(a(a(b(a(b(a(b(x))))))))
b(a(a(b(a(b(a(b(b(x))))))))) → b(a(b(b(b(b(x))))))
b(b(a(a(b(a(b(a(b(B(x)))))))))) → b(a(a(b(a(b(a(b(b(B(x))))))))))
b(a(a(b(a(b(a(b(b(b(B(x))))))))))) → b(a(b(b(b(b(a(b(a(b(a(b(B(x)))))))))))))
b(a(a(b(a(b(a(b(B(x))))))))) → b(a(b(b(b(B(x))))))
b(b(b(b(B(x))))) → b(a(b(a(b(a(b(b(a(b(a(b(a(b(B(x)))))))))))))))
b(a(a(b(a(b(a(a(b(a(b(a(b(B(x)))))))))))))) → b(a(b(b(b(b(B(x)))))))
b(b(b(a(a(b(a(b(a(b(B(x))))))))))) → b(a(b(a(b(a(b(b(a(b(b(B(x))))))))))))
b(a(a(b(a(b(a(b(B(x))))))))) → b(a(b(B(x))))
b(a(a(b(a(b(a(b(b(a(a(b(a(b(a(b(B(x))))))))))))))))) → b(a(b(b(b(b(a(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
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPToSRSProof
↳ QTRS
↳ QTRS Reverse
↳ QTRS
↳ QTRS Reverse
↳ QTRS Reverse
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
Q DP problem:
The TRS P consists of the following rules:
B1(a(a(b(a(b(a(b(b(x))))))))) → B1(b(b(b(x))))
B1(b(b(a(a(b(a(b(a(b(B(x))))))))))) → B1(a(b(b(B(x)))))
B1(a(a(b(a(b(a(b(b(a(a(b(a(b(a(b(B(x))))))))))))))))) → B1(b(a(b(b(B(x))))))
B1(b(b(x))) → B1(a(b(b(x))))
B1(a(a(b(a(b(a(b(B(x))))))))) → B1(b(B(x)))
B1(b(b(a(a(b(a(b(a(b(B(x))))))))))) → B1(b(a(b(b(B(x))))))
B1(b(b(b(B(x))))) → B1(a(b(a(b(B(x))))))
B1(b(a(b(x)))) → B1(a(a(b(a(b(a(b(x))))))))
B1(b(b(a(a(b(a(b(a(b(B(x))))))))))) → B1(a(b(a(b(a(b(b(a(b(b(B(x))))))))))))
B1(b(b(b(B(x))))) → B1(a(b(B(x))))
B1(b(a(a(b(a(b(a(b(B(x)))))))))) → B1(a(b(b(B(x)))))
B1(a(a(b(a(b(a(b(b(b(B(x))))))))))) → B1(a(b(b(b(b(a(b(a(b(a(b(B(x)))))))))))))
B1(b(a(b(x)))) → B1(a(b(a(b(x)))))
B1(b(a(a(b(a(b(a(b(B(x)))))))))) → B1(a(b(a(b(b(B(x)))))))
B1(a(a(b(a(b(a(b(b(a(a(b(a(b(a(b(B(x))))))))))))))))) → B1(a(b(b(b(b(a(b(b(B(x))))))))))
B1(a(a(b(a(b(a(b(b(b(B(x))))))))))) → B1(b(b(a(b(a(b(a(b(B(x))))))))))
B1(b(a(a(b(a(b(a(b(B(x)))))))))) → B1(b(B(x)))
B1(a(a(b(a(b(a(b(b(b(B(x))))))))))) → B1(b(b(b(a(b(a(b(a(b(B(x)))))))))))
B1(a(a(b(a(b(a(b(b(b(B(x))))))))))) → B1(a(b(B(x))))
B1(a(a(b(a(b(a(a(b(a(b(a(b(B(x)))))))))))))) → B1(b(b(b(B(x)))))
B1(a(a(b(a(b(a(b(b(b(B(x))))))))))) → B1(a(b(a(b(B(x))))))
B1(b(b(b(B(x))))) → B1(a(b(a(b(a(b(B(x))))))))
B1(b(b(x))) → B1(a(b(a(b(b(x))))))
B1(b(b(a(a(b(a(b(a(b(B(x))))))))))) → B1(a(b(a(b(b(a(b(b(B(x))))))))))
B1(a(a(b(a(b(a(b(b(a(a(b(a(b(a(b(B(x))))))))))))))))) → B1(a(b(b(B(x)))))
B1(a(a(b(a(b(a(b(b(a(a(b(a(b(a(b(B(x))))))))))))))))) → B1(b(b(a(b(b(B(x)))))))
B1(b(b(b(B(x))))) → B1(b(a(b(a(b(a(b(B(x)))))))))
B1(a(a(b(a(b(a(b(B(x))))))))) → B1(b(b(B(x))))
B1(a(a(b(a(b(a(a(b(a(b(a(b(B(x)))))))))))))) → B1(a(b(b(b(b(B(x)))))))
B1(b(b(x))) → B1(a(b(a(b(a(b(b(x))))))))
B1(a(a(b(a(b(a(a(b(a(b(a(b(B(x)))))))))))))) → B1(b(B(x)))
B1(a(a(b(a(b(a(a(b(a(b(a(b(B(x)))))))))))))) → B1(b(b(B(x))))
B1(b(b(b(B(x))))) → B1(a(b(a(b(b(a(b(a(b(a(b(B(x)))))))))))))
B1(a(a(b(a(b(a(b(b(x))))))))) → B1(b(b(x)))
B1(b(b(b(B(x))))) → B1(a(b(b(a(b(a(b(a(b(B(x)))))))))))
B1(a(a(b(a(b(a(b(b(a(a(b(a(b(a(b(B(x))))))))))))))))) → B1(b(b(b(a(b(b(B(x))))))))
B1(b(b(a(a(b(a(b(a(b(B(x))))))))))) → B1(a(b(b(a(b(b(B(x))))))))
B1(b(b(b(B(x))))) → B1(a(b(a(b(a(b(b(a(b(a(b(a(b(B(x)))))))))))))))
B1(a(a(b(a(b(a(b(b(a(a(b(a(b(a(b(B(x))))))))))))))))) → B1(b(B(x)))
B1(a(a(b(a(b(a(b(B(x))))))))) → B1(a(b(b(b(B(x))))))
B1(b(b(a(a(b(a(b(a(b(B(x))))))))))) → B1(b(B(x)))
B1(a(a(b(a(b(a(b(b(b(B(x))))))))))) → B1(b(a(b(a(b(a(b(B(x)))))))))
B1(a(a(b(a(b(a(b(b(b(B(x))))))))))) → B1(a(b(a(b(a(b(B(x))))))))
B1(a(a(b(a(b(a(b(b(x))))))))) → B1(a(b(b(b(b(x))))))
B1(b(a(a(b(a(b(a(b(B(x)))))))))) → B1(a(a(b(a(b(a(b(b(B(x))))))))))
The TRS R consists of the following rules:
b(b(b(x))) → b(a(b(a(b(a(b(b(x))))))))
b(b(a(b(x)))) → b(a(a(b(a(b(a(b(x))))))))
b(a(a(b(a(b(a(b(b(x))))))))) → b(a(b(b(b(b(x))))))
b(b(a(a(b(a(b(a(b(B(x)))))))))) → b(a(a(b(a(b(a(b(b(B(x))))))))))
b(a(a(b(a(b(a(b(b(b(B(x))))))))))) → b(a(b(b(b(b(a(b(a(b(a(b(B(x)))))))))))))
b(a(a(b(a(b(a(b(B(x))))))))) → b(a(b(b(b(B(x))))))
b(b(b(b(B(x))))) → b(a(b(a(b(a(b(b(a(b(a(b(a(b(B(x)))))))))))))))
b(a(a(b(a(b(a(a(b(a(b(a(b(B(x)))))))))))))) → b(a(b(b(b(b(B(x)))))))
b(b(b(a(a(b(a(b(a(b(B(x))))))))))) → b(a(b(a(b(a(b(b(a(b(b(B(x))))))))))))
b(a(a(b(a(b(a(b(B(x))))))))) → b(a(b(B(x))))
b(a(a(b(a(b(a(b(b(a(a(b(a(b(a(b(B(x))))))))))))))))) → b(a(b(b(b(b(a(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 32 less nodes.
↳ QTRS
↳ QTRS Reverse
↳ QTRS Reverse
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ 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:
B1(a(a(b(a(b(a(b(b(x))))))))) → B1(b(b(b(x))))
B1(a(a(b(a(b(a(b(b(a(a(b(a(b(a(b(B(x))))))))))))))))) → B1(b(a(b(b(B(x))))))
B1(b(b(a(a(b(a(b(a(b(B(x))))))))))) → B1(b(a(b(b(B(x))))))
B1(a(a(b(a(b(a(b(b(a(a(b(a(b(a(b(B(x))))))))))))))))) → B1(b(b(a(b(b(B(x)))))))
B1(b(b(b(B(x))))) → B1(b(a(b(a(b(a(b(B(x)))))))))
B1(b(a(b(x)))) → B1(a(a(b(a(b(a(b(x))))))))
B1(a(a(b(a(b(a(b(b(x))))))))) → B1(b(b(x)))
B1(a(a(b(a(b(a(b(b(a(a(b(a(b(a(b(B(x))))))))))))))))) → B1(b(b(b(a(b(b(B(x))))))))
B1(a(a(b(a(b(a(b(b(b(B(x))))))))))) → B1(b(a(b(a(b(a(b(B(x)))))))))
B1(a(a(b(a(b(a(b(b(b(B(x))))))))))) → B1(b(b(a(b(a(b(a(b(B(x))))))))))
B1(a(a(b(a(b(a(b(b(b(B(x))))))))))) → B1(b(b(b(a(b(a(b(a(b(B(x)))))))))))
B1(b(a(a(b(a(b(a(b(B(x)))))))))) → B1(a(a(b(a(b(a(b(b(B(x))))))))))
B1(a(a(b(a(b(a(a(b(a(b(a(b(B(x)))))))))))))) → B1(b(b(b(B(x)))))
The TRS R consists of the following rules:
b(b(b(x))) → b(a(b(a(b(a(b(b(x))))))))
b(b(a(b(x)))) → b(a(a(b(a(b(a(b(x))))))))
b(a(a(b(a(b(a(b(b(x))))))))) → b(a(b(b(b(b(x))))))
b(b(a(a(b(a(b(a(b(B(x)))))))))) → b(a(a(b(a(b(a(b(b(B(x))))))))))
b(a(a(b(a(b(a(b(b(b(B(x))))))))))) → b(a(b(b(b(b(a(b(a(b(a(b(B(x)))))))))))))
b(a(a(b(a(b(a(b(B(x))))))))) → b(a(b(b(b(B(x))))))
b(b(b(b(B(x))))) → b(a(b(a(b(a(b(b(a(b(a(b(a(b(B(x)))))))))))))))
b(a(a(b(a(b(a(a(b(a(b(a(b(B(x)))))))))))))) → b(a(b(b(b(b(B(x)))))))
b(b(b(a(a(b(a(b(a(b(B(x))))))))))) → b(a(b(a(b(a(b(b(a(b(b(B(x))))))))))))
b(a(a(b(a(b(a(b(B(x))))))))) → b(a(b(B(x))))
b(a(a(b(a(b(a(b(b(a(a(b(a(b(a(b(B(x))))))))))))))))) → b(a(b(b(b(b(a(b(b(B(x))))))))))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule B1(a(a(b(a(b(a(b(b(b(B(x))))))))))) → B1(b(b(a(b(a(b(a(b(B(x)))))))))) at position [0] we obtained the following new rules:
B1(a(a(b(a(b(a(b(b(b(B(y0))))))))))) → B1(b(a(a(b(a(b(a(b(a(b(a(b(B(y0))))))))))))))
↳ QTRS
↳ QTRS Reverse
↳ QTRS Reverse
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ 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:
B1(a(a(b(a(b(a(b(b(x))))))))) → B1(b(b(b(x))))
B1(a(a(b(a(b(a(b(b(b(B(y0))))))))))) → B1(b(a(a(b(a(b(a(b(a(b(a(b(B(y0))))))))))))))
B1(a(a(b(a(b(a(b(b(a(a(b(a(b(a(b(B(x))))))))))))))))) → B1(b(a(b(b(B(x))))))
B1(b(b(a(a(b(a(b(a(b(B(x))))))))))) → B1(b(a(b(b(B(x))))))
B1(a(a(b(a(b(a(b(b(a(a(b(a(b(a(b(B(x))))))))))))))))) → B1(b(b(a(b(b(B(x)))))))
B1(b(b(b(B(x))))) → B1(b(a(b(a(b(a(b(B(x)))))))))
B1(b(a(b(x)))) → B1(a(a(b(a(b(a(b(x))))))))
B1(a(a(b(a(b(a(b(b(x))))))))) → B1(b(b(x)))
B1(a(a(b(a(b(a(b(b(a(a(b(a(b(a(b(B(x))))))))))))))))) → B1(b(b(b(a(b(b(B(x))))))))
B1(a(a(b(a(b(a(b(b(b(B(x))))))))))) → B1(b(a(b(a(b(a(b(B(x)))))))))
B1(a(a(b(a(b(a(b(b(b(B(x))))))))))) → B1(b(b(b(a(b(a(b(a(b(B(x)))))))))))
B1(b(a(a(b(a(b(a(b(B(x)))))))))) → B1(a(a(b(a(b(a(b(b(B(x))))))))))
B1(a(a(b(a(b(a(a(b(a(b(a(b(B(x)))))))))))))) → B1(b(b(b(B(x)))))
The TRS R consists of the following rules:
b(b(b(x))) → b(a(b(a(b(a(b(b(x))))))))
b(b(a(b(x)))) → b(a(a(b(a(b(a(b(x))))))))
b(a(a(b(a(b(a(b(b(x))))))))) → b(a(b(b(b(b(x))))))
b(b(a(a(b(a(b(a(b(B(x)))))))))) → b(a(a(b(a(b(a(b(b(B(x))))))))))
b(a(a(b(a(b(a(b(b(b(B(x))))))))))) → b(a(b(b(b(b(a(b(a(b(a(b(B(x)))))))))))))
b(a(a(b(a(b(a(b(B(x))))))))) → b(a(b(b(b(B(x))))))
b(b(b(b(B(x))))) → b(a(b(a(b(a(b(b(a(b(a(b(a(b(B(x)))))))))))))))
b(a(a(b(a(b(a(a(b(a(b(a(b(B(x)))))))))))))) → b(a(b(b(b(b(B(x)))))))
b(b(b(a(a(b(a(b(a(b(B(x))))))))))) → b(a(b(a(b(a(b(b(a(b(b(B(x))))))))))))
b(a(a(b(a(b(a(b(B(x))))))))) → b(a(b(B(x))))
b(a(a(b(a(b(a(b(b(a(a(b(a(b(a(b(B(x))))))))))))))))) → b(a(b(b(b(b(a(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 1 less node.
↳ QTRS
↳ QTRS Reverse
↳ QTRS Reverse
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ 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:
B1(a(a(b(a(b(a(b(b(x))))))))) → B1(b(b(b(x))))
B1(a(a(b(a(b(a(b(b(x))))))))) → B1(b(b(x)))
B1(a(a(b(a(b(a(b(b(a(a(b(a(b(a(b(B(x))))))))))))))))) → B1(b(b(b(a(b(b(B(x))))))))
B1(a(a(b(a(b(a(b(b(a(a(b(a(b(a(b(B(x))))))))))))))))) → B1(b(a(b(b(B(x))))))
B1(b(b(a(a(b(a(b(a(b(B(x))))))))))) → B1(b(a(b(b(B(x))))))
B1(a(a(b(a(b(a(b(b(a(a(b(a(b(a(b(B(x))))))))))))))))) → B1(b(b(a(b(b(B(x)))))))
B1(a(a(b(a(b(a(b(b(b(B(x))))))))))) → B1(b(a(b(a(b(a(b(B(x)))))))))
B1(b(b(b(B(x))))) → B1(b(a(b(a(b(a(b(B(x)))))))))
B1(a(a(b(a(b(a(b(b(b(B(x))))))))))) → B1(b(b(b(a(b(a(b(a(b(B(x)))))))))))
B1(b(a(a(b(a(b(a(b(B(x)))))))))) → B1(a(a(b(a(b(a(b(b(B(x))))))))))
B1(b(a(b(x)))) → B1(a(a(b(a(b(a(b(x))))))))
B1(a(a(b(a(b(a(a(b(a(b(a(b(B(x)))))))))))))) → B1(b(b(b(B(x)))))
The TRS R consists of the following rules:
b(b(b(x))) → b(a(b(a(b(a(b(b(x))))))))
b(b(a(b(x)))) → b(a(a(b(a(b(a(b(x))))))))
b(a(a(b(a(b(a(b(b(x))))))))) → b(a(b(b(b(b(x))))))
b(b(a(a(b(a(b(a(b(B(x)))))))))) → b(a(a(b(a(b(a(b(b(B(x))))))))))
b(a(a(b(a(b(a(b(b(b(B(x))))))))))) → b(a(b(b(b(b(a(b(a(b(a(b(B(x)))))))))))))
b(a(a(b(a(b(a(b(B(x))))))))) → b(a(b(b(b(B(x))))))
b(b(b(b(B(x))))) → b(a(b(a(b(a(b(b(a(b(a(b(a(b(B(x)))))))))))))))
b(a(a(b(a(b(a(a(b(a(b(a(b(B(x)))))))))))))) → b(a(b(b(b(b(B(x)))))))
b(b(b(a(a(b(a(b(a(b(B(x))))))))))) → b(a(b(a(b(a(b(b(a(b(b(B(x))))))))))))
b(a(a(b(a(b(a(b(B(x))))))))) → b(a(b(B(x))))
b(a(a(b(a(b(a(b(b(a(a(b(a(b(a(b(B(x))))))))))))))))) → b(a(b(b(b(b(a(b(b(B(x))))))))))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule B1(a(a(b(a(b(a(b(b(b(B(x))))))))))) → B1(b(b(b(a(b(a(b(a(b(B(x))))))))))) at position [0] we obtained the following new rules:
B1(a(a(b(a(b(a(b(b(b(B(y0))))))))))) → B1(b(b(a(a(b(a(b(a(b(a(b(a(b(B(y0)))))))))))))))
B1(a(a(b(a(b(a(b(b(b(B(y0))))))))))) → B1(b(a(b(a(b(a(b(b(a(b(a(b(a(b(B(y0))))))))))))))))
↳ QTRS
↳ QTRS Reverse
↳ QTRS Reverse
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ 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:
B1(a(a(b(a(b(a(b(b(b(B(y0))))))))))) → B1(b(b(a(a(b(a(b(a(b(a(b(a(b(B(y0)))))))))))))))
B1(a(a(b(a(b(a(b(b(x))))))))) → B1(b(b(b(x))))
B1(a(a(b(a(b(a(b(b(a(a(b(a(b(a(b(B(x))))))))))))))))) → B1(b(a(b(b(B(x))))))
B1(b(b(a(a(b(a(b(a(b(B(x))))))))))) → B1(b(a(b(b(B(x))))))
B1(a(a(b(a(b(a(b(b(a(a(b(a(b(a(b(B(x))))))))))))))))) → B1(b(b(a(b(b(B(x)))))))
B1(b(b(b(B(x))))) → B1(b(a(b(a(b(a(b(B(x)))))))))
B1(b(a(b(x)))) → B1(a(a(b(a(b(a(b(x))))))))
B1(a(a(b(a(b(a(b(b(b(B(y0))))))))))) → B1(b(a(b(a(b(a(b(b(a(b(a(b(a(b(B(y0))))))))))))))))
B1(a(a(b(a(b(a(b(b(x))))))))) → B1(b(b(x)))
B1(a(a(b(a(b(a(b(b(a(a(b(a(b(a(b(B(x))))))))))))))))) → B1(b(b(b(a(b(b(B(x))))))))
B1(a(a(b(a(b(a(b(b(b(B(x))))))))))) → B1(b(a(b(a(b(a(b(B(x)))))))))
B1(b(a(a(b(a(b(a(b(B(x)))))))))) → B1(a(a(b(a(b(a(b(b(B(x))))))))))
B1(a(a(b(a(b(a(a(b(a(b(a(b(B(x)))))))))))))) → B1(b(b(b(B(x)))))
The TRS R consists of the following rules:
b(b(b(x))) → b(a(b(a(b(a(b(b(x))))))))
b(b(a(b(x)))) → b(a(a(b(a(b(a(b(x))))))))
b(a(a(b(a(b(a(b(b(x))))))))) → b(a(b(b(b(b(x))))))
b(b(a(a(b(a(b(a(b(B(x)))))))))) → b(a(a(b(a(b(a(b(b(B(x))))))))))
b(a(a(b(a(b(a(b(b(b(B(x))))))))))) → b(a(b(b(b(b(a(b(a(b(a(b(B(x)))))))))))))
b(a(a(b(a(b(a(b(B(x))))))))) → b(a(b(b(b(B(x))))))
b(b(b(b(B(x))))) → b(a(b(a(b(a(b(b(a(b(a(b(a(b(B(x)))))))))))))))
b(a(a(b(a(b(a(a(b(a(b(a(b(B(x)))))))))))))) → b(a(b(b(b(b(B(x)))))))
b(b(b(a(a(b(a(b(a(b(B(x))))))))))) → b(a(b(a(b(a(b(b(a(b(b(B(x))))))))))))
b(a(a(b(a(b(a(b(B(x))))))))) → b(a(b(B(x))))
b(a(a(b(a(b(a(b(b(a(a(b(a(b(a(b(B(x))))))))))))))))) → b(a(b(b(b(b(a(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 1 less node.
↳ QTRS
↳ QTRS Reverse
↳ QTRS Reverse
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ 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:
B1(a(a(b(a(b(a(b(b(x))))))))) → B1(b(b(b(x))))
B1(a(a(b(a(b(a(b(b(x))))))))) → B1(b(b(x)))
B1(a(a(b(a(b(a(b(b(a(a(b(a(b(a(b(B(x))))))))))))))))) → B1(b(b(b(a(b(b(B(x))))))))
B1(a(a(b(a(b(a(b(b(a(a(b(a(b(a(b(B(x))))))))))))))))) → B1(b(a(b(b(B(x))))))
B1(b(b(a(a(b(a(b(a(b(B(x))))))))))) → B1(b(a(b(b(B(x))))))
B1(a(a(b(a(b(a(b(b(a(a(b(a(b(a(b(B(x))))))))))))))))) → B1(b(b(a(b(b(B(x)))))))
B1(a(a(b(a(b(a(b(b(b(B(x))))))))))) → B1(b(a(b(a(b(a(b(B(x)))))))))
B1(b(b(b(B(x))))) → B1(b(a(b(a(b(a(b(B(x)))))))))
B1(b(a(a(b(a(b(a(b(B(x)))))))))) → B1(a(a(b(a(b(a(b(b(B(x))))))))))
B1(b(a(b(x)))) → B1(a(a(b(a(b(a(b(x))))))))
B1(a(a(b(a(b(a(b(b(b(B(y0))))))))))) → B1(b(a(b(a(b(a(b(b(a(b(a(b(a(b(B(y0))))))))))))))))
B1(a(a(b(a(b(a(a(b(a(b(a(b(B(x)))))))))))))) → B1(b(b(b(B(x)))))
The TRS R consists of the following rules:
b(b(b(x))) → b(a(b(a(b(a(b(b(x))))))))
b(b(a(b(x)))) → b(a(a(b(a(b(a(b(x))))))))
b(a(a(b(a(b(a(b(b(x))))))))) → b(a(b(b(b(b(x))))))
b(b(a(a(b(a(b(a(b(B(x)))))))))) → b(a(a(b(a(b(a(b(b(B(x))))))))))
b(a(a(b(a(b(a(b(b(b(B(x))))))))))) → b(a(b(b(b(b(a(b(a(b(a(b(B(x)))))))))))))
b(a(a(b(a(b(a(b(B(x))))))))) → b(a(b(b(b(B(x))))))
b(b(b(b(B(x))))) → b(a(b(a(b(a(b(b(a(b(a(b(a(b(B(x)))))))))))))))
b(a(a(b(a(b(a(a(b(a(b(a(b(B(x)))))))))))))) → b(a(b(b(b(b(B(x)))))))
b(b(b(a(a(b(a(b(a(b(B(x))))))))))) → b(a(b(a(b(a(b(b(a(b(b(B(x))))))))))))
b(a(a(b(a(b(a(b(B(x))))))))) → b(a(b(B(x))))
b(a(a(b(a(b(a(b(b(a(a(b(a(b(a(b(B(x))))))))))))))))) → b(a(b(b(b(b(a(b(b(B(x))))))))))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule B1(a(a(b(a(b(a(b(b(a(a(b(a(b(a(b(B(x))))))))))))))))) → B1(b(b(b(a(b(b(B(x)))))))) at position [0] we obtained the following new rules:
B1(a(a(b(a(b(a(b(b(a(a(b(a(b(a(b(B(y0))))))))))))))))) → B1(b(a(b(a(b(a(b(b(a(b(b(B(y0)))))))))))))
B1(a(a(b(a(b(a(b(b(a(a(b(a(b(a(b(B(y0))))))))))))))))) → B1(b(b(a(a(b(a(b(a(b(b(B(y0))))))))))))
↳ QTRS
↳ QTRS Reverse
↳ QTRS Reverse
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPToSRSProof
↳ QTRS
↳ QTRS Reverse
↳ QTRS
↳ QTRS Reverse
↳ QTRS Reverse
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
Q DP problem:
The TRS P consists of the following rules:
B1(a(a(b(a(b(a(b(b(x))))))))) → B1(b(b(b(x))))
B1(a(a(b(a(b(a(b(b(a(a(b(a(b(a(b(B(x))))))))))))))))) → B1(b(a(b(b(B(x))))))
B1(b(b(a(a(b(a(b(a(b(B(x))))))))))) → B1(b(a(b(b(B(x))))))
B1(a(a(b(a(b(a(b(b(a(a(b(a(b(a(b(B(y0))))))))))))))))) → B1(b(b(a(a(b(a(b(a(b(b(B(y0))))))))))))
B1(a(a(b(a(b(a(b(b(a(a(b(a(b(a(b(B(x))))))))))))))))) → B1(b(b(a(b(b(B(x)))))))
B1(b(b(b(B(x))))) → B1(b(a(b(a(b(a(b(B(x)))))))))
B1(b(a(b(x)))) → B1(a(a(b(a(b(a(b(x))))))))
B1(a(a(b(a(b(a(b(b(b(B(y0))))))))))) → B1(b(a(b(a(b(a(b(b(a(b(a(b(a(b(B(y0))))))))))))))))
B1(a(a(b(a(b(a(b(b(a(a(b(a(b(a(b(B(y0))))))))))))))))) → B1(b(a(b(a(b(a(b(b(a(b(b(B(y0)))))))))))))
B1(a(a(b(a(b(a(b(b(x))))))))) → B1(b(b(x)))
B1(a(a(b(a(b(a(b(b(b(B(x))))))))))) → B1(b(a(b(a(b(a(b(B(x)))))))))
B1(b(a(a(b(a(b(a(b(B(x)))))))))) → B1(a(a(b(a(b(a(b(b(B(x))))))))))
B1(a(a(b(a(b(a(a(b(a(b(a(b(B(x)))))))))))))) → B1(b(b(b(B(x)))))
The TRS R consists of the following rules:
b(b(b(x))) → b(a(b(a(b(a(b(b(x))))))))
b(b(a(b(x)))) → b(a(a(b(a(b(a(b(x))))))))
b(a(a(b(a(b(a(b(b(x))))))))) → b(a(b(b(b(b(x))))))
b(b(a(a(b(a(b(a(b(B(x)))))))))) → b(a(a(b(a(b(a(b(b(B(x))))))))))
b(a(a(b(a(b(a(b(b(b(B(x))))))))))) → b(a(b(b(b(b(a(b(a(b(a(b(B(x)))))))))))))
b(a(a(b(a(b(a(b(B(x))))))))) → b(a(b(b(b(B(x))))))
b(b(b(b(B(x))))) → b(a(b(a(b(a(b(b(a(b(a(b(a(b(B(x)))))))))))))))
b(a(a(b(a(b(a(a(b(a(b(a(b(B(x)))))))))))))) → b(a(b(b(b(b(B(x)))))))
b(b(b(a(a(b(a(b(a(b(B(x))))))))))) → b(a(b(a(b(a(b(b(a(b(b(B(x))))))))))))
b(a(a(b(a(b(a(b(B(x))))))))) → b(a(b(B(x))))
b(a(a(b(a(b(a(b(b(a(a(b(a(b(a(b(B(x))))))))))))))))) → b(a(b(b(b(b(a(b(b(B(x))))))))))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule B1(a(a(b(a(b(a(b(b(a(a(b(a(b(a(b(B(x))))))))))))))))) → B1(b(b(a(b(b(B(x))))))) at position [0] we obtained the following new rules:
B1(a(a(b(a(b(a(b(b(a(a(b(a(b(a(b(B(y0))))))))))))))))) → B1(b(a(a(b(a(b(a(b(b(B(y0)))))))))))
↳ QTRS
↳ QTRS Reverse
↳ QTRS Reverse
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPToSRSProof
↳ QTRS
↳ QTRS Reverse
↳ QTRS
↳ QTRS Reverse
↳ QTRS Reverse
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ SemLabProof
↳ SemLabProof2
Q DP problem:
The TRS P consists of the following rules:
B1(a(a(b(a(b(a(b(b(x))))))))) → B1(b(b(b(x))))
B1(a(a(b(a(b(a(b(b(a(a(b(a(b(a(b(B(x))))))))))))))))) → B1(b(a(b(b(B(x))))))
B1(b(b(a(a(b(a(b(a(b(B(x))))))))))) → B1(b(a(b(b(B(x))))))
B1(a(a(b(a(b(a(b(b(a(a(b(a(b(a(b(B(y0))))))))))))))))) → B1(b(b(a(a(b(a(b(a(b(b(B(y0))))))))))))
B1(b(b(b(B(x))))) → B1(b(a(b(a(b(a(b(B(x)))))))))
B1(b(a(b(x)))) → B1(a(a(b(a(b(a(b(x))))))))
B1(a(a(b(a(b(a(b(b(b(B(y0))))))))))) → B1(b(a(b(a(b(a(b(b(a(b(a(b(a(b(B(y0))))))))))))))))
B1(a(a(b(a(b(a(b(b(x))))))))) → B1(b(b(x)))
B1(a(a(b(a(b(a(b(b(a(a(b(a(b(a(b(B(y0))))))))))))))))) → B1(b(a(b(a(b(a(b(b(a(b(b(B(y0)))))))))))))
B1(a(a(b(a(b(a(b(b(a(a(b(a(b(a(b(B(y0))))))))))))))))) → B1(b(a(a(b(a(b(a(b(b(B(y0)))))))))))
B1(a(a(b(a(b(a(b(b(b(B(x))))))))))) → B1(b(a(b(a(b(a(b(B(x)))))))))
B1(b(a(a(b(a(b(a(b(B(x)))))))))) → B1(a(a(b(a(b(a(b(b(B(x))))))))))
B1(a(a(b(a(b(a(a(b(a(b(a(b(B(x)))))))))))))) → B1(b(b(b(B(x)))))
The TRS R consists of the following rules:
b(b(b(x))) → b(a(b(a(b(a(b(b(x))))))))
b(b(a(b(x)))) → b(a(a(b(a(b(a(b(x))))))))
b(a(a(b(a(b(a(b(b(x))))))))) → b(a(b(b(b(b(x))))))
b(b(a(a(b(a(b(a(b(B(x)))))))))) → b(a(a(b(a(b(a(b(b(B(x))))))))))
b(a(a(b(a(b(a(b(b(b(B(x))))))))))) → b(a(b(b(b(b(a(b(a(b(a(b(B(x)))))))))))))
b(a(a(b(a(b(a(b(B(x))))))))) → b(a(b(b(b(B(x))))))
b(b(b(b(B(x))))) → b(a(b(a(b(a(b(b(a(b(a(b(a(b(B(x)))))))))))))))
b(a(a(b(a(b(a(a(b(a(b(a(b(B(x)))))))))))))) → b(a(b(b(b(b(B(x)))))))
b(b(b(a(a(b(a(b(a(b(B(x))))))))))) → b(a(b(a(b(a(b(b(a(b(b(B(x))))))))))))
b(a(a(b(a(b(a(b(B(x))))))))) → b(a(b(B(x))))
b(a(a(b(a(b(a(b(b(a(a(b(a(b(a(b(B(x))))))))))))))))) → b(a(b(b(b(b(a(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.B: 0
a: 1
B1: 0
b: 0
By semantic labelling [33] we obtain the following labelled TRS:Q DP problem:
The TRS P consists of the following rules:
B1.0(b.1(a.0(b.1(x)))) → B1.1(a.1(a.0(b.1(a.0(b.1(a.0(b.1(x))))))))
B1.1(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.1(a.1(a.0(b.1(a.0(b.1(a.0(b.0(B.0(y0))))))))))))))))) → B1.0(b.0(b.1(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(B.0(y0))))))))))))
B1.1(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(x))))))))) → B1.0(b.0(b.0(b.0(x))))
B1.1(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.1(a.1(a.0(b.1(a.0(b.1(a.0(b.0(B.1(y0))))))))))))))))) → B1.0(b.0(b.1(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(B.1(y0))))))))))))
B1.0(b.1(a.1(a.0(b.1(a.0(b.1(a.0(b.0(B.1(x)))))))))) → B1.1(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(B.1(x))))))))))
B1.1(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(b.0(B.1(y0))))))))))) → B1.0(b.1(a.0(b.1(a.0(b.1(a.0(b.0(b.1(a.0(b.1(a.0(b.1(a.0(b.0(B.1(y0))))))))))))))))
B1.1(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.1(a.1(a.0(b.1(a.0(b.1(a.0(b.0(B.0(x))))))))))))))))) → B1.0(b.1(a.0(b.0(b.0(B.0(x))))))
B1.1(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(b.0(B.1(x))))))))))) → B1.0(b.1(a.0(b.1(a.0(b.1(a.0(b.0(B.1(x)))))))))
B1.1(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.1(a.1(a.0(b.1(a.0(b.1(a.0(b.0(B.0(y0))))))))))))))))) → B1.0(b.1(a.0(b.1(a.0(b.1(a.0(b.0(b.1(a.0(b.0(b.0(B.0(y0)))))))))))))
B1.1(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.1(a.1(a.0(b.1(a.0(b.1(a.0(b.0(B.1(x))))))))))))))))) → B1.0(b.1(a.0(b.0(b.0(B.1(x))))))
B1.1(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.1(a.1(a.0(b.1(a.0(b.1(a.0(b.0(B.1(y0))))))))))))))))) → B1.0(b.1(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(B.1(y0)))))))))))
B1.0(b.1(a.0(b.0(x)))) → B1.1(a.1(a.0(b.1(a.0(b.1(a.0(b.0(x))))))))
B1.1(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.1(a.1(a.0(b.1(a.0(b.1(a.0(b.0(B.1(y0))))))))))))))))) → B1.0(b.1(a.0(b.1(a.0(b.1(a.0(b.0(b.1(a.0(b.0(b.0(B.1(y0)))))))))))))
B1.1(a.1(a.0(b.1(a.0(b.1(a.1(a.0(b.1(a.0(b.1(a.0(b.0(B.1(x)))))))))))))) → B1.0(b.0(b.0(b.0(B.1(x)))))
B1.1(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(x))))))))) → B1.0(b.0(b.0(x)))
B1.0(b.0(b.0(b.0(B.1(x))))) → B1.0(b.1(a.0(b.1(a.0(b.1(a.0(b.0(B.1(x)))))))))
B1.0(b.0(b.1(a.1(a.0(b.1(a.0(b.1(a.0(b.0(B.0(x))))))))))) → B1.0(b.1(a.0(b.0(b.0(B.0(x))))))
B1.0(b.0(b.0(b.0(B.0(x))))) → B1.0(b.1(a.0(b.1(a.0(b.1(a.0(b.0(B.0(x)))))))))
B1.1(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(b.0(B.0(x))))))))))) → B1.0(b.1(a.0(b.1(a.0(b.1(a.0(b.0(B.0(x)))))))))
B1.0(b.1(a.1(a.0(b.1(a.0(b.1(a.0(b.0(B.0(x)))))))))) → B1.1(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(B.0(x))))))))))
B1.1(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.1(x))))))))) → B1.0(b.0(b.1(x)))
B1.1(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(b.0(B.0(y0))))))))))) → B1.0(b.1(a.0(b.1(a.0(b.1(a.0(b.0(b.1(a.0(b.1(a.0(b.1(a.0(b.0(B.0(y0))))))))))))))))
B1.1(a.1(a.0(b.1(a.0(b.1(a.1(a.0(b.1(a.0(b.1(a.0(b.0(B.0(x)))))))))))))) → B1.0(b.0(b.0(b.0(B.0(x)))))
B1.0(b.0(b.1(a.1(a.0(b.1(a.0(b.1(a.0(b.0(B.1(x))))))))))) → B1.0(b.1(a.0(b.0(b.0(B.1(x))))))
B1.1(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.1(x))))))))) → B1.0(b.0(b.0(b.1(x))))
B1.1(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.1(a.1(a.0(b.1(a.0(b.1(a.0(b.0(B.0(y0))))))))))))))))) → B1.0(b.1(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(B.0(y0)))))))))))
The TRS R consists of the following rules:
b.1(a.1(a.0(b.1(a.0(b.1(a.1(a.0(b.1(a.0(b.1(a.0(b.0(B.0(x)))))))))))))) → b.1(a.0(b.0(b.0(b.0(b.0(B.0(x)))))))
b.1(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.1(a.1(a.0(b.1(a.0(b.1(a.0(b.0(B.0(x))))))))))))))))) → b.1(a.0(b.0(b.0(b.0(b.1(a.0(b.0(b.0(B.0(x))))))))))
b.1(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(b.0(B.1(x))))))))))) → b.1(a.0(b.0(b.0(b.0(b.1(a.0(b.1(a.0(b.1(a.0(b.0(B.1(x)))))))))))))
b.0(b.1(a.0(b.1(x)))) → b.1(a.1(a.0(b.1(a.0(b.1(a.0(b.1(x))))))))
b.1(a.1(a.0(b.1(a.0(b.1(a.0(b.0(B.1(x))))))))) → b.1(a.0(b.0(b.0(b.0(B.1(x))))))
b.1(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.1(x))))))))) → b.1(a.0(b.0(b.0(b.0(b.1(x))))))
b.0(b.0(b.1(a.1(a.0(b.1(a.0(b.1(a.0(b.0(B.1(x))))))))))) → b.1(a.0(b.1(a.0(b.1(a.0(b.0(b.1(a.0(b.0(b.0(B.1(x))))))))))))
b.1(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(b.0(B.0(x))))))))))) → b.1(a.0(b.0(b.0(b.0(b.1(a.0(b.1(a.0(b.1(a.0(b.0(B.0(x)))))))))))))
b.0(b.0(b.1(x))) → b.1(a.0(b.1(a.0(b.1(a.0(b.0(b.1(x))))))))
b.1(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.1(a.1(a.0(b.1(a.0(b.1(a.0(b.0(B.1(x))))))))))))))))) → b.1(a.0(b.0(b.0(b.0(b.1(a.0(b.0(b.0(B.1(x))))))))))
b.0(b.0(b.0(x))) → b.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(x))))))))
b.0(b.0(b.1(a.1(a.0(b.1(a.0(b.1(a.0(b.0(B.0(x))))))))))) → b.1(a.0(b.1(a.0(b.1(a.0(b.0(b.1(a.0(b.0(b.0(B.0(x))))))))))))
b.0(b.0(b.0(b.0(B.0(x))))) → b.1(a.0(b.1(a.0(b.1(a.0(b.0(b.1(a.0(b.1(a.0(b.1(a.0(b.0(B.0(x)))))))))))))))
b.0(b.1(a.1(a.0(b.1(a.0(b.1(a.0(b.0(B.0(x)))))))))) → b.1(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(B.0(x))))))))))
b.0(b.1(a.1(a.0(b.1(a.0(b.1(a.0(b.0(B.1(x)))))))))) → b.1(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(B.1(x))))))))))
b.1(a.1(a.0(b.1(a.0(b.1(a.0(b.0(B.1(x))))))))) → b.1(a.0(b.0(B.1(x))))
b.1(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(x))))))))) → b.1(a.0(b.0(b.0(b.0(b.0(x))))))
b.1(a.1(a.0(b.1(a.0(b.1(a.1(a.0(b.1(a.0(b.1(a.0(b.0(B.1(x)))))))))))))) → b.1(a.0(b.0(b.0(b.0(b.0(B.1(x)))))))
b.1(a.1(a.0(b.1(a.0(b.1(a.0(b.0(B.0(x))))))))) → b.1(a.0(b.0(b.0(b.0(B.0(x))))))
b.0(b.0(b.0(b.0(B.1(x))))) → b.1(a.0(b.1(a.0(b.1(a.0(b.0(b.1(a.0(b.1(a.0(b.1(a.0(b.0(B.1(x)))))))))))))))
b.1(a.1(a.0(b.1(a.0(b.1(a.0(b.0(B.0(x))))))))) → b.1(a.0(b.0(B.0(x))))
b.0(b.1(a.0(b.0(x)))) → b.1(a.1(a.0(b.1(a.0(b.1(a.0(b.0(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
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPToSRSProof
↳ QTRS
↳ QTRS Reverse
↳ QTRS
↳ QTRS Reverse
↳ QTRS Reverse
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ SemLabProof
↳ QDP
↳ DependencyGraphProof
↳ SemLabProof2
Q DP problem:
The TRS P consists of the following rules:
B1.0(b.1(a.0(b.1(x)))) → B1.1(a.1(a.0(b.1(a.0(b.1(a.0(b.1(x))))))))
B1.1(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.1(a.1(a.0(b.1(a.0(b.1(a.0(b.0(B.0(y0))))))))))))))))) → B1.0(b.0(b.1(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(B.0(y0))))))))))))
B1.1(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(x))))))))) → B1.0(b.0(b.0(b.0(x))))
B1.1(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.1(a.1(a.0(b.1(a.0(b.1(a.0(b.0(B.1(y0))))))))))))))))) → B1.0(b.0(b.1(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(B.1(y0))))))))))))
B1.0(b.1(a.1(a.0(b.1(a.0(b.1(a.0(b.0(B.1(x)))))))))) → B1.1(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(B.1(x))))))))))
B1.1(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(b.0(B.1(y0))))))))))) → B1.0(b.1(a.0(b.1(a.0(b.1(a.0(b.0(b.1(a.0(b.1(a.0(b.1(a.0(b.0(B.1(y0))))))))))))))))
B1.1(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.1(a.1(a.0(b.1(a.0(b.1(a.0(b.0(B.0(x))))))))))))))))) → B1.0(b.1(a.0(b.0(b.0(B.0(x))))))
B1.1(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(b.0(B.1(x))))))))))) → B1.0(b.1(a.0(b.1(a.0(b.1(a.0(b.0(B.1(x)))))))))
B1.1(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.1(a.1(a.0(b.1(a.0(b.1(a.0(b.0(B.0(y0))))))))))))))))) → B1.0(b.1(a.0(b.1(a.0(b.1(a.0(b.0(b.1(a.0(b.0(b.0(B.0(y0)))))))))))))
B1.1(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.1(a.1(a.0(b.1(a.0(b.1(a.0(b.0(B.1(x))))))))))))))))) → B1.0(b.1(a.0(b.0(b.0(B.1(x))))))
B1.1(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.1(a.1(a.0(b.1(a.0(b.1(a.0(b.0(B.1(y0))))))))))))))))) → B1.0(b.1(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(B.1(y0)))))))))))
B1.0(b.1(a.0(b.0(x)))) → B1.1(a.1(a.0(b.1(a.0(b.1(a.0(b.0(x))))))))
B1.1(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.1(a.1(a.0(b.1(a.0(b.1(a.0(b.0(B.1(y0))))))))))))))))) → B1.0(b.1(a.0(b.1(a.0(b.1(a.0(b.0(b.1(a.0(b.0(b.0(B.1(y0)))))))))))))
B1.1(a.1(a.0(b.1(a.0(b.1(a.1(a.0(b.1(a.0(b.1(a.0(b.0(B.1(x)))))))))))))) → B1.0(b.0(b.0(b.0(B.1(x)))))
B1.1(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(x))))))))) → B1.0(b.0(b.0(x)))
B1.0(b.0(b.0(b.0(B.1(x))))) → B1.0(b.1(a.0(b.1(a.0(b.1(a.0(b.0(B.1(x)))))))))
B1.0(b.0(b.1(a.1(a.0(b.1(a.0(b.1(a.0(b.0(B.0(x))))))))))) → B1.0(b.1(a.0(b.0(b.0(B.0(x))))))
B1.0(b.0(b.0(b.0(B.0(x))))) → B1.0(b.1(a.0(b.1(a.0(b.1(a.0(b.0(B.0(x)))))))))
B1.1(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(b.0(B.0(x))))))))))) → B1.0(b.1(a.0(b.1(a.0(b.1(a.0(b.0(B.0(x)))))))))
B1.0(b.1(a.1(a.0(b.1(a.0(b.1(a.0(b.0(B.0(x)))))))))) → B1.1(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(B.0(x))))))))))
B1.1(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.1(x))))))))) → B1.0(b.0(b.1(x)))
B1.1(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(b.0(B.0(y0))))))))))) → B1.0(b.1(a.0(b.1(a.0(b.1(a.0(b.0(b.1(a.0(b.1(a.0(b.1(a.0(b.0(B.0(y0))))))))))))))))
B1.1(a.1(a.0(b.1(a.0(b.1(a.1(a.0(b.1(a.0(b.1(a.0(b.0(B.0(x)))))))))))))) → B1.0(b.0(b.0(b.0(B.0(x)))))
B1.0(b.0(b.1(a.1(a.0(b.1(a.0(b.1(a.0(b.0(B.1(x))))))))))) → B1.0(b.1(a.0(b.0(b.0(B.1(x))))))
B1.1(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.1(x))))))))) → B1.0(b.0(b.0(b.1(x))))
B1.1(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.1(a.1(a.0(b.1(a.0(b.1(a.0(b.0(B.0(y0))))))))))))))))) → B1.0(b.1(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(B.0(y0)))))))))))
The TRS R consists of the following rules:
b.1(a.1(a.0(b.1(a.0(b.1(a.1(a.0(b.1(a.0(b.1(a.0(b.0(B.0(x)))))))))))))) → b.1(a.0(b.0(b.0(b.0(b.0(B.0(x)))))))
b.1(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.1(a.1(a.0(b.1(a.0(b.1(a.0(b.0(B.0(x))))))))))))))))) → b.1(a.0(b.0(b.0(b.0(b.1(a.0(b.0(b.0(B.0(x))))))))))
b.1(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(b.0(B.1(x))))))))))) → b.1(a.0(b.0(b.0(b.0(b.1(a.0(b.1(a.0(b.1(a.0(b.0(B.1(x)))))))))))))
b.0(b.1(a.0(b.1(x)))) → b.1(a.1(a.0(b.1(a.0(b.1(a.0(b.1(x))))))))
b.1(a.1(a.0(b.1(a.0(b.1(a.0(b.0(B.1(x))))))))) → b.1(a.0(b.0(b.0(b.0(B.1(x))))))
b.1(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.1(x))))))))) → b.1(a.0(b.0(b.0(b.0(b.1(x))))))
b.0(b.0(b.1(a.1(a.0(b.1(a.0(b.1(a.0(b.0(B.1(x))))))))))) → b.1(a.0(b.1(a.0(b.1(a.0(b.0(b.1(a.0(b.0(b.0(B.1(x))))))))))))
b.1(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(b.0(B.0(x))))))))))) → b.1(a.0(b.0(b.0(b.0(b.1(a.0(b.1(a.0(b.1(a.0(b.0(B.0(x)))))))))))))
b.0(b.0(b.1(x))) → b.1(a.0(b.1(a.0(b.1(a.0(b.0(b.1(x))))))))
b.1(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.1(a.1(a.0(b.1(a.0(b.1(a.0(b.0(B.1(x))))))))))))))))) → b.1(a.0(b.0(b.0(b.0(b.1(a.0(b.0(b.0(B.1(x))))))))))
b.0(b.0(b.0(x))) → b.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(x))))))))
b.0(b.0(b.1(a.1(a.0(b.1(a.0(b.1(a.0(b.0(B.0(x))))))))))) → b.1(a.0(b.1(a.0(b.1(a.0(b.0(b.1(a.0(b.0(b.0(B.0(x))))))))))))
b.0(b.0(b.0(b.0(B.0(x))))) → b.1(a.0(b.1(a.0(b.1(a.0(b.0(b.1(a.0(b.1(a.0(b.1(a.0(b.0(B.0(x)))))))))))))))
b.0(b.1(a.1(a.0(b.1(a.0(b.1(a.0(b.0(B.0(x)))))))))) → b.1(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(B.0(x))))))))))
b.0(b.1(a.1(a.0(b.1(a.0(b.1(a.0(b.0(B.1(x)))))))))) → b.1(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(B.1(x))))))))))
b.1(a.1(a.0(b.1(a.0(b.1(a.0(b.0(B.1(x))))))))) → b.1(a.0(b.0(B.1(x))))
b.1(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(x))))))))) → b.1(a.0(b.0(b.0(b.0(b.0(x))))))
b.1(a.1(a.0(b.1(a.0(b.1(a.1(a.0(b.1(a.0(b.1(a.0(b.0(B.1(x)))))))))))))) → b.1(a.0(b.0(b.0(b.0(b.0(B.1(x)))))))
b.1(a.1(a.0(b.1(a.0(b.1(a.0(b.0(B.0(x))))))))) → b.1(a.0(b.0(b.0(b.0(B.0(x))))))
b.0(b.0(b.0(b.0(B.1(x))))) → b.1(a.0(b.1(a.0(b.1(a.0(b.0(b.1(a.0(b.1(a.0(b.1(a.0(b.0(B.1(x)))))))))))))))
b.1(a.1(a.0(b.1(a.0(b.1(a.0(b.0(B.0(x))))))))) → b.1(a.0(b.0(B.0(x))))
b.0(b.1(a.0(b.0(x)))) → b.1(a.1(a.0(b.1(a.0(b.1(a.0(b.0(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
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPToSRSProof
↳ QTRS
↳ QTRS Reverse
↳ QTRS
↳ QTRS Reverse
↳ QTRS Reverse
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ SemLabProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ SemLabProof2
Q DP problem:
The TRS P consists of the following rules:
B1.0(b.1(a.0(b.1(x)))) → B1.1(a.1(a.0(b.1(a.0(b.1(a.0(b.1(x))))))))
B1.0(b.0(b.0(b.0(B.1(x))))) → B1.0(b.1(a.0(b.1(a.0(b.1(a.0(b.0(B.1(x)))))))))
B1.1(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.1(a.1(a.0(b.1(a.0(b.1(a.0(b.0(B.0(y0))))))))))))))))) → B1.0(b.0(b.1(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(B.0(y0))))))))))))
B1.1(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(x))))))))) → B1.0(b.0(b.0(b.0(x))))
B1.1(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.1(a.1(a.0(b.1(a.0(b.1(a.0(b.0(B.1(y0))))))))))))))))) → B1.0(b.0(b.1(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(B.1(y0))))))))))))
B1.0(b.0(b.1(a.1(a.0(b.1(a.0(b.1(a.0(b.0(B.0(x))))))))))) → B1.0(b.1(a.0(b.0(b.0(B.0(x))))))
B1.0(b.1(a.1(a.0(b.1(a.0(b.1(a.0(b.0(B.1(x)))))))))) → B1.1(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(B.1(x))))))))))
B1.1(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(b.0(B.1(y0))))))))))) → B1.0(b.1(a.0(b.1(a.0(b.1(a.0(b.0(b.1(a.0(b.1(a.0(b.1(a.0(b.0(B.1(y0))))))))))))))))
B1.1(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.1(a.1(a.0(b.1(a.0(b.1(a.0(b.0(B.0(x))))))))))))))))) → B1.0(b.1(a.0(b.0(b.0(B.0(x))))))
B1.1(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(b.0(B.1(x))))))))))) → B1.0(b.1(a.0(b.1(a.0(b.1(a.0(b.0(B.1(x)))))))))
B1.1(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.1(a.1(a.0(b.1(a.0(b.1(a.0(b.0(B.0(y0))))))))))))))))) → B1.0(b.1(a.0(b.1(a.0(b.1(a.0(b.0(b.1(a.0(b.0(b.0(B.0(y0)))))))))))))
B1.1(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.1(a.1(a.0(b.1(a.0(b.1(a.0(b.0(B.1(x))))))))))))))))) → B1.0(b.1(a.0(b.0(b.0(B.1(x))))))
B1.0(b.0(b.0(b.0(B.0(x))))) → B1.0(b.1(a.0(b.1(a.0(b.1(a.0(b.0(B.0(x)))))))))
B1.1(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.1(a.1(a.0(b.1(a.0(b.1(a.0(b.0(B.1(y0))))))))))))))))) → B1.0(b.1(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(B.1(y0)))))))))))
B1.1(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(b.0(B.0(x))))))))))) → B1.0(b.1(a.0(b.1(a.0(b.1(a.0(b.0(B.0(x)))))))))
B1.0(b.1(a.0(b.0(x)))) → B1.1(a.1(a.0(b.1(a.0(b.1(a.0(b.0(x))))))))
B1.0(b.1(a.1(a.0(b.1(a.0(b.1(a.0(b.0(B.0(x)))))))))) → B1.1(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(B.0(x))))))))))
B1.1(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.1(x))))))))) → B1.0(b.0(b.1(x)))
B1.1(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.1(a.1(a.0(b.1(a.0(b.1(a.0(b.0(B.1(y0))))))))))))))))) → B1.0(b.1(a.0(b.1(a.0(b.1(a.0(b.0(b.1(a.0(b.0(b.0(B.1(y0)))))))))))))
B1.1(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(b.0(B.0(y0))))))))))) → B1.0(b.1(a.0(b.1(a.0(b.1(a.0(b.0(b.1(a.0(b.1(a.0(b.1(a.0(b.0(B.0(y0))))))))))))))))
B1.0(b.0(b.1(a.1(a.0(b.1(a.0(b.1(a.0(b.0(B.1(x))))))))))) → B1.0(b.1(a.0(b.0(b.0(B.1(x))))))
B1.1(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(x))))))))) → B1.0(b.0(b.0(x)))
B1.1(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.1(x))))))))) → B1.0(b.0(b.0(b.1(x))))
B1.1(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.1(a.1(a.0(b.1(a.0(b.1(a.0(b.0(B.0(y0))))))))))))))))) → B1.0(b.1(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(B.0(y0)))))))))))
The TRS R consists of the following rules:
b.1(a.1(a.0(b.1(a.0(b.1(a.1(a.0(b.1(a.0(b.1(a.0(b.0(B.0(x)))))))))))))) → b.1(a.0(b.0(b.0(b.0(b.0(B.0(x)))))))
b.1(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.1(a.1(a.0(b.1(a.0(b.1(a.0(b.0(B.0(x))))))))))))))))) → b.1(a.0(b.0(b.0(b.0(b.1(a.0(b.0(b.0(B.0(x))))))))))
b.1(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(b.0(B.1(x))))))))))) → b.1(a.0(b.0(b.0(b.0(b.1(a.0(b.1(a.0(b.1(a.0(b.0(B.1(x)))))))))))))
b.0(b.1(a.0(b.1(x)))) → b.1(a.1(a.0(b.1(a.0(b.1(a.0(b.1(x))))))))
b.1(a.1(a.0(b.1(a.0(b.1(a.0(b.0(B.1(x))))))))) → b.1(a.0(b.0(b.0(b.0(B.1(x))))))
b.1(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.1(x))))))))) → b.1(a.0(b.0(b.0(b.0(b.1(x))))))
b.0(b.0(b.1(a.1(a.0(b.1(a.0(b.1(a.0(b.0(B.1(x))))))))))) → b.1(a.0(b.1(a.0(b.1(a.0(b.0(b.1(a.0(b.0(b.0(B.1(x))))))))))))
b.1(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(b.0(B.0(x))))))))))) → b.1(a.0(b.0(b.0(b.0(b.1(a.0(b.1(a.0(b.1(a.0(b.0(B.0(x)))))))))))))
b.0(b.0(b.1(x))) → b.1(a.0(b.1(a.0(b.1(a.0(b.0(b.1(x))))))))
b.1(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.1(a.1(a.0(b.1(a.0(b.1(a.0(b.0(B.1(x))))))))))))))))) → b.1(a.0(b.0(b.0(b.0(b.1(a.0(b.0(b.0(B.1(x))))))))))
b.0(b.0(b.0(x))) → b.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(x))))))))
b.0(b.0(b.1(a.1(a.0(b.1(a.0(b.1(a.0(b.0(B.0(x))))))))))) → b.1(a.0(b.1(a.0(b.1(a.0(b.0(b.1(a.0(b.0(b.0(B.0(x))))))))))))
b.0(b.0(b.0(b.0(B.0(x))))) → b.1(a.0(b.1(a.0(b.1(a.0(b.0(b.1(a.0(b.1(a.0(b.1(a.0(b.0(B.0(x)))))))))))))))
b.0(b.1(a.1(a.0(b.1(a.0(b.1(a.0(b.0(B.0(x)))))))))) → b.1(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(B.0(x))))))))))
b.0(b.1(a.1(a.0(b.1(a.0(b.1(a.0(b.0(B.1(x)))))))))) → b.1(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(B.1(x))))))))))
b.1(a.1(a.0(b.1(a.0(b.1(a.0(b.0(B.1(x))))))))) → b.1(a.0(b.0(B.1(x))))
b.1(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(x))))))))) → b.1(a.0(b.0(b.0(b.0(b.0(x))))))
b.1(a.1(a.0(b.1(a.0(b.1(a.1(a.0(b.1(a.0(b.1(a.0(b.0(B.1(x)))))))))))))) → b.1(a.0(b.0(b.0(b.0(b.0(B.1(x)))))))
b.1(a.1(a.0(b.1(a.0(b.1(a.0(b.0(B.0(x))))))))) → b.1(a.0(b.0(b.0(b.0(B.0(x))))))
b.0(b.0(b.0(b.0(B.1(x))))) → b.1(a.0(b.1(a.0(b.1(a.0(b.0(b.1(a.0(b.1(a.0(b.1(a.0(b.0(B.1(x)))))))))))))))
b.1(a.1(a.0(b.1(a.0(b.1(a.0(b.0(B.0(x))))))))) → b.1(a.0(b.0(B.0(x))))
b.0(b.1(a.0(b.0(x)))) → b.1(a.1(a.0(b.1(a.0(b.1(a.0(b.0(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
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPToSRSProof
↳ QTRS
↳ QTRS Reverse
↳ QTRS
↳ QTRS Reverse
↳ QTRS Reverse
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ SemLabProof
↳ SemLabProof2
↳ QDP
↳ SemLabProof
↳ SemLabProof2
Q DP problem:
The TRS P consists of the following rules:
B1(a(a(b(a(b(a(b(b(x))))))))) → B1(b(b(b(x))))
B1(a(a(b(a(b(a(b(b(x))))))))) → B1(b(b(x)))
B1(a(a(b(a(b(a(b(b(a(a(b(a(b(a(b(B(y0))))))))))))))))) → B1(b(a(b(a(b(a(b(b(a(b(b(B(y0)))))))))))))
B1(a(a(b(a(b(a(b(b(a(a(b(a(b(a(b(B(y0))))))))))))))))) → B1(b(a(a(b(a(b(a(b(b(B(y0)))))))))))
B1(a(a(b(a(b(a(b(b(a(a(b(a(b(a(b(B(x))))))))))))))))) → B1(b(a(b(b(B(x))))))
B1(b(b(a(a(b(a(b(a(b(B(x))))))))))) → B1(b(a(b(b(B(x))))))
B1(a(a(b(a(b(a(b(b(a(a(b(a(b(a(b(B(y0))))))))))))))))) → B1(b(b(a(a(b(a(b(a(b(b(B(y0))))))))))))
B1(a(a(b(a(b(a(b(b(b(B(x))))))))))) → B1(b(a(b(a(b(a(b(B(x)))))))))
B1(b(b(b(B(x))))) → B1(b(a(b(a(b(a(b(B(x)))))))))
B1(b(a(a(b(a(b(a(b(B(x)))))))))) → B1(a(a(b(a(b(a(b(b(B(x))))))))))
B1(b(a(b(x)))) → B1(a(a(b(a(b(a(b(x))))))))
B1(a(a(b(a(b(a(b(b(b(B(y0))))))))))) → B1(b(a(b(a(b(a(b(b(a(b(a(b(a(b(B(y0))))))))))))))))
The TRS R consists of the following rules:
b(b(b(x))) → b(a(b(a(b(a(b(b(x))))))))
b(b(a(b(x)))) → b(a(a(b(a(b(a(b(x))))))))
b(a(a(b(a(b(a(b(b(x))))))))) → b(a(b(b(b(b(x))))))
b(b(a(a(b(a(b(a(b(B(x)))))))))) → b(a(a(b(a(b(a(b(b(B(x))))))))))
b(a(a(b(a(b(a(b(b(b(B(x))))))))))) → b(a(b(b(b(b(a(b(a(b(a(b(B(x)))))))))))))
b(a(a(b(a(b(a(b(B(x))))))))) → b(a(b(b(b(B(x))))))
b(b(b(b(B(x))))) → b(a(b(a(b(a(b(b(a(b(a(b(a(b(B(x)))))))))))))))
b(a(a(b(a(b(a(a(b(a(b(a(b(B(x)))))))))))))) → b(a(b(b(b(b(B(x)))))))
b(b(b(a(a(b(a(b(a(b(B(x))))))))))) → b(a(b(a(b(a(b(b(a(b(b(B(x))))))))))))
b(a(a(b(a(b(a(b(B(x))))))))) → b(a(b(B(x))))
b(a(a(b(a(b(a(b(b(a(a(b(a(b(a(b(B(x))))))))))))))))) → b(a(b(b(b(b(a(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.B: 0
a: 1 + x0
B1: 0
b: 0
By semantic labelling [33] we obtain the following labelled TRS:Q DP problem:
The TRS P consists of the following rules:
B1.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(B.0(y0))))))))))))))))) → B1.0(b.1(a.0(b.1(a.0(b.1(a.0(b.0(b.1(a.0(b.0(b.0(B.0(y0)))))))))))))
B1.0(b.0(b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(B.1(x))))))))))) → B1.0(b.1(a.0(b.0(b.0(B.1(x))))))
B1.0(b.0(b.0(b.0(B.1(x))))) → B1.0(b.1(a.0(b.1(a.0(b.1(a.0(b.0(B.1(x)))))))))
B1.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(B.0(y0))))))))))))))))) → B1.0(b.0(b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(B.0(y0))))))))))))
B1.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(B.1(x))))))))))))))))) → B1.0(b.1(a.0(b.0(b.0(B.1(x))))))
B1.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(B.1(y0))))))))))))))))) → B1.0(b.1(a.0(b.1(a.0(b.1(a.0(b.0(b.1(a.0(b.0(b.0(B.1(y0)))))))))))))
B1.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(b.0(B.0(x))))))))))) → B1.0(b.1(a.0(b.1(a.0(b.1(a.0(b.0(B.0(x)))))))))
B1.0(b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(B.0(x)))))))))) → B1.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(B.0(x))))))))))
B1.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.1(x))))))))) → B1.0(b.0(b.0(b.1(x))))
B1.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(B.1(y0))))))))))))))))) → B1.0(b.0(b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(B.1(y0))))))))))))
B1.0(b.0(b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(B.0(x))))))))))) → B1.0(b.1(a.0(b.0(b.0(B.0(x))))))
B1.0(b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(B.1(x)))))))))) → B1.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(B.1(x))))))))))
B1.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.1(x))))))))) → B1.0(b.0(b.1(x)))
B1.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(b.0(B.1(x))))))))))) → B1.0(b.1(a.0(b.1(a.0(b.1(a.0(b.0(B.1(x)))))))))
B1.0(b.0(b.0(b.0(B.0(x))))) → B1.0(b.1(a.0(b.1(a.0(b.1(a.0(b.0(B.0(x)))))))))
B1.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(B.0(x))))))))))))))))) → B1.0(b.1(a.0(b.0(b.0(B.0(x))))))
B1.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(x))))))))) → B1.0(b.0(b.0(x)))
B1.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(b.0(B.0(y0))))))))))) → B1.0(b.1(a.0(b.1(a.0(b.1(a.0(b.0(b.1(a.0(b.1(a.0(b.1(a.0(b.0(B.0(y0))))))))))))))))
B1.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(x))))))))) → B1.0(b.0(b.0(b.0(x))))
B1.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(b.0(B.1(y0))))))))))) → B1.0(b.1(a.0(b.1(a.0(b.1(a.0(b.0(b.1(a.0(b.1(a.0(b.1(a.0(b.0(B.1(y0))))))))))))))))
B1.0(b.1(a.0(b.0(x)))) → B1.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(x))))))))
B1.0(b.1(a.0(b.1(x)))) → B1.0(a.1(a.0(b.1(a.0(b.1(a.0(b.1(x))))))))
B1.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(B.1(y0))))))))))))))))) → B1.0(b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(B.1(y0)))))))))))
B1.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(B.0(y0))))))))))))))))) → B1.0(b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(B.0(y0)))))))))))
The TRS R consists of the following rules:
b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(B.0(x))))))))) → b.1(a.0(b.0(b.0(b.0(B.0(x))))))
b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(B.0(x))))))))) → b.1(a.0(b.0(B.0(x))))
b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(B.1(x))))))))))))))))) → b.1(a.0(b.0(b.0(b.0(b.1(a.0(b.0(b.0(B.1(x))))))))))
b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(B.0(x))))))))))))))))) → b.1(a.0(b.0(b.0(b.0(b.1(a.0(b.0(b.0(B.0(x))))))))))
b.0(b.0(b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(B.1(x))))))))))) → b.1(a.0(b.1(a.0(b.1(a.0(b.0(b.1(a.0(b.0(b.0(B.1(x))))))))))))
b.0(b.0(b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(B.0(x))))))))))) → b.1(a.0(b.1(a.0(b.1(a.0(b.0(b.1(a.0(b.0(b.0(B.0(x))))))))))))
b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.1(x))))))))) → b.1(a.0(b.0(b.0(b.0(b.1(x))))))
b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(x))))))))) → b.1(a.0(b.0(b.0(b.0(b.0(x))))))
b.0(b.0(b.1(x))) → b.1(a.0(b.1(a.0(b.1(a.0(b.0(b.1(x))))))))
b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(B.1(x))))))))) → b.1(a.0(b.0(B.1(x))))
b.0(b.0(b.0(x))) → b.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(x))))))))
b.0(b.0(b.0(b.0(B.0(x))))) → b.1(a.0(b.1(a.0(b.1(a.0(b.0(b.1(a.0(b.1(a.0(b.1(a.0(b.0(B.0(x)))))))))))))))
b.0(b.1(a.0(b.0(x)))) → b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(x))))))))
b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(B.1(x))))))))) → b.1(a.0(b.0(b.0(b.0(B.1(x))))))
b.0(a.1(a.0(b.1(a.0(b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(B.1(x)))))))))))))) → b.1(a.0(b.0(b.0(b.0(b.0(B.1(x)))))))
b.0(b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(B.1(x)))))))))) → b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(B.1(x))))))))))
b.0(b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(B.0(x)))))))))) → b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(B.0(x))))))))))
b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(b.0(B.0(x))))))))))) → b.1(a.0(b.0(b.0(b.0(b.1(a.0(b.1(a.0(b.1(a.0(b.0(B.0(x)))))))))))))
b.0(b.0(b.0(b.0(B.1(x))))) → b.1(a.0(b.1(a.0(b.1(a.0(b.0(b.1(a.0(b.1(a.0(b.1(a.0(b.0(B.1(x)))))))))))))))
b.0(b.1(a.0(b.1(x)))) → b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.1(x))))))))
b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(b.0(B.1(x))))))))))) → b.1(a.0(b.0(b.0(b.0(b.1(a.0(b.1(a.0(b.1(a.0(b.0(B.1(x)))))))))))))
b.0(a.1(a.0(b.1(a.0(b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(B.0(x)))))))))))))) → b.1(a.0(b.0(b.0(b.0(b.0(B.0(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
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPToSRSProof
↳ QTRS
↳ QTRS Reverse
↳ QTRS
↳ QTRS Reverse
↳ QTRS Reverse
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ SemLabProof
↳ SemLabProof2
↳ QDP
↳ SemLabProof
↳ QDP
↳ DependencyGraphProof
↳ SemLabProof2
Q DP problem:
The TRS P consists of the following rules:
B1.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(B.0(y0))))))))))))))))) → B1.0(b.1(a.0(b.1(a.0(b.1(a.0(b.0(b.1(a.0(b.0(b.0(B.0(y0)))))))))))))
B1.0(b.0(b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(B.1(x))))))))))) → B1.0(b.1(a.0(b.0(b.0(B.1(x))))))
B1.0(b.0(b.0(b.0(B.1(x))))) → B1.0(b.1(a.0(b.1(a.0(b.1(a.0(b.0(B.1(x)))))))))
B1.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(B.0(y0))))))))))))))))) → B1.0(b.0(b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(B.0(y0))))))))))))
B1.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(B.1(x))))))))))))))))) → B1.0(b.1(a.0(b.0(b.0(B.1(x))))))
B1.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(B.1(y0))))))))))))))))) → B1.0(b.1(a.0(b.1(a.0(b.1(a.0(b.0(b.1(a.0(b.0(b.0(B.1(y0)))))))))))))
B1.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(b.0(B.0(x))))))))))) → B1.0(b.1(a.0(b.1(a.0(b.1(a.0(b.0(B.0(x)))))))))
B1.0(b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(B.0(x)))))))))) → B1.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(B.0(x))))))))))
B1.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.1(x))))))))) → B1.0(b.0(b.0(b.1(x))))
B1.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(B.1(y0))))))))))))))))) → B1.0(b.0(b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(B.1(y0))))))))))))
B1.0(b.0(b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(B.0(x))))))))))) → B1.0(b.1(a.0(b.0(b.0(B.0(x))))))
B1.0(b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(B.1(x)))))))))) → B1.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(B.1(x))))))))))
B1.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.1(x))))))))) → B1.0(b.0(b.1(x)))
B1.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(b.0(B.1(x))))))))))) → B1.0(b.1(a.0(b.1(a.0(b.1(a.0(b.0(B.1(x)))))))))
B1.0(b.0(b.0(b.0(B.0(x))))) → B1.0(b.1(a.0(b.1(a.0(b.1(a.0(b.0(B.0(x)))))))))
B1.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(B.0(x))))))))))))))))) → B1.0(b.1(a.0(b.0(b.0(B.0(x))))))
B1.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(x))))))))) → B1.0(b.0(b.0(x)))
B1.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(b.0(B.0(y0))))))))))) → B1.0(b.1(a.0(b.1(a.0(b.1(a.0(b.0(b.1(a.0(b.1(a.0(b.1(a.0(b.0(B.0(y0))))))))))))))))
B1.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(x))))))))) → B1.0(b.0(b.0(b.0(x))))
B1.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(b.0(B.1(y0))))))))))) → B1.0(b.1(a.0(b.1(a.0(b.1(a.0(b.0(b.1(a.0(b.1(a.0(b.1(a.0(b.0(B.1(y0))))))))))))))))
B1.0(b.1(a.0(b.0(x)))) → B1.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(x))))))))
B1.0(b.1(a.0(b.1(x)))) → B1.0(a.1(a.0(b.1(a.0(b.1(a.0(b.1(x))))))))
B1.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(B.1(y0))))))))))))))))) → B1.0(b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(B.1(y0)))))))))))
B1.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(B.0(y0))))))))))))))))) → B1.0(b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(B.0(y0)))))))))))
The TRS R consists of the following rules:
b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(B.0(x))))))))) → b.1(a.0(b.0(b.0(b.0(B.0(x))))))
b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(B.0(x))))))))) → b.1(a.0(b.0(B.0(x))))
b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(B.1(x))))))))))))))))) → b.1(a.0(b.0(b.0(b.0(b.1(a.0(b.0(b.0(B.1(x))))))))))
b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(B.0(x))))))))))))))))) → b.1(a.0(b.0(b.0(b.0(b.1(a.0(b.0(b.0(B.0(x))))))))))
b.0(b.0(b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(B.1(x))))))))))) → b.1(a.0(b.1(a.0(b.1(a.0(b.0(b.1(a.0(b.0(b.0(B.1(x))))))))))))
b.0(b.0(b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(B.0(x))))))))))) → b.1(a.0(b.1(a.0(b.1(a.0(b.0(b.1(a.0(b.0(b.0(B.0(x))))))))))))
b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.1(x))))))))) → b.1(a.0(b.0(b.0(b.0(b.1(x))))))
b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(x))))))))) → b.1(a.0(b.0(b.0(b.0(b.0(x))))))
b.0(b.0(b.1(x))) → b.1(a.0(b.1(a.0(b.1(a.0(b.0(b.1(x))))))))
b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(B.1(x))))))))) → b.1(a.0(b.0(B.1(x))))
b.0(b.0(b.0(x))) → b.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(x))))))))
b.0(b.0(b.0(b.0(B.0(x))))) → b.1(a.0(b.1(a.0(b.1(a.0(b.0(b.1(a.0(b.1(a.0(b.1(a.0(b.0(B.0(x)))))))))))))))
b.0(b.1(a.0(b.0(x)))) → b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(x))))))))
b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(B.1(x))))))))) → b.1(a.0(b.0(b.0(b.0(B.1(x))))))
b.0(a.1(a.0(b.1(a.0(b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(B.1(x)))))))))))))) → b.1(a.0(b.0(b.0(b.0(b.0(B.1(x)))))))
b.0(b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(B.1(x)))))))))) → b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(B.1(x))))))))))
b.0(b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(B.0(x)))))))))) → b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(B.0(x))))))))))
b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(b.0(B.0(x))))))))))) → b.1(a.0(b.0(b.0(b.0(b.1(a.0(b.1(a.0(b.1(a.0(b.0(B.0(x)))))))))))))
b.0(b.0(b.0(b.0(B.1(x))))) → b.1(a.0(b.1(a.0(b.1(a.0(b.0(b.1(a.0(b.1(a.0(b.1(a.0(b.0(B.1(x)))))))))))))))
b.0(b.1(a.0(b.1(x)))) → b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.1(x))))))))
b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(b.0(B.1(x))))))))))) → b.1(a.0(b.0(b.0(b.0(b.1(a.0(b.1(a.0(b.1(a.0(b.0(B.1(x)))))))))))))
b.0(a.1(a.0(b.1(a.0(b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(B.0(x)))))))))))))) → b.1(a.0(b.0(b.0(b.0(b.0(B.0(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 9 less nodes.
↳ QTRS
↳ QTRS Reverse
↳ QTRS Reverse
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPToSRSProof
↳ QTRS
↳ QTRS Reverse
↳ QTRS
↳ QTRS Reverse
↳ QTRS Reverse
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ SemLabProof
↳ SemLabProof2
↳ QDP
↳ SemLabProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ SemLabProof2
Q DP problem:
The TRS P consists of the following rules:
B1.0(b.0(b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(B.1(x))))))))))) → B1.0(b.1(a.0(b.0(b.0(B.1(x))))))
B1.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(B.1(x))))))))))))))))) → B1.0(b.1(a.0(b.0(b.0(B.1(x))))))
B1.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(B.0(y0))))))))))))))))) → B1.0(b.0(b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(B.0(y0))))))))))))
B1.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.1(x))))))))) → B1.0(b.0(b.0(b.1(x))))
B1.0(b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(B.0(x)))))))))) → B1.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(B.0(x))))))))))
B1.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(B.1(y0))))))))))))))))) → B1.0(b.0(b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(B.1(y0))))))))))))
B1.0(b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(B.1(x)))))))))) → B1.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(B.1(x))))))))))
B1.0(b.0(b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(B.0(x))))))))))) → B1.0(b.1(a.0(b.0(b.0(B.0(x))))))
B1.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.1(x))))))))) → B1.0(b.0(b.1(x)))
B1.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(B.0(x))))))))))))))))) → B1.0(b.1(a.0(b.0(b.0(B.0(x))))))
B1.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(x))))))))) → B1.0(b.0(b.0(x)))
B1.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(x))))))))) → B1.0(b.0(b.0(b.0(x))))
B1.0(b.1(a.0(b.0(x)))) → B1.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(x))))))))
B1.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(B.1(y0))))))))))))))))) → B1.0(b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(B.1(y0)))))))))))
B1.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(B.0(y0))))))))))))))))) → B1.0(b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(B.0(y0)))))))))))
The TRS R consists of the following rules:
b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(B.0(x))))))))) → b.1(a.0(b.0(b.0(b.0(B.0(x))))))
b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(B.0(x))))))))) → b.1(a.0(b.0(B.0(x))))
b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(B.1(x))))))))))))))))) → b.1(a.0(b.0(b.0(b.0(b.1(a.0(b.0(b.0(B.1(x))))))))))
b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(B.0(x))))))))))))))))) → b.1(a.0(b.0(b.0(b.0(b.1(a.0(b.0(b.0(B.0(x))))))))))
b.0(b.0(b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(B.1(x))))))))))) → b.1(a.0(b.1(a.0(b.1(a.0(b.0(b.1(a.0(b.0(b.0(B.1(x))))))))))))
b.0(b.0(b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(B.0(x))))))))))) → b.1(a.0(b.1(a.0(b.1(a.0(b.0(b.1(a.0(b.0(b.0(B.0(x))))))))))))
b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.1(x))))))))) → b.1(a.0(b.0(b.0(b.0(b.1(x))))))
b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(x))))))))) → b.1(a.0(b.0(b.0(b.0(b.0(x))))))
b.0(b.0(b.1(x))) → b.1(a.0(b.1(a.0(b.1(a.0(b.0(b.1(x))))))))
b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(B.1(x))))))))) → b.1(a.0(b.0(B.1(x))))
b.0(b.0(b.0(x))) → b.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(x))))))))
b.0(b.0(b.0(b.0(B.0(x))))) → b.1(a.0(b.1(a.0(b.1(a.0(b.0(b.1(a.0(b.1(a.0(b.1(a.0(b.0(B.0(x)))))))))))))))
b.0(b.1(a.0(b.0(x)))) → b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(x))))))))
b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(B.1(x))))))))) → b.1(a.0(b.0(b.0(b.0(B.1(x))))))
b.0(a.1(a.0(b.1(a.0(b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(B.1(x)))))))))))))) → b.1(a.0(b.0(b.0(b.0(b.0(B.1(x)))))))
b.0(b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(B.1(x)))))))))) → b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(B.1(x))))))))))
b.0(b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(B.0(x)))))))))) → b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(B.0(x))))))))))
b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(b.0(B.0(x))))))))))) → b.1(a.0(b.0(b.0(b.0(b.1(a.0(b.1(a.0(b.1(a.0(b.0(B.0(x)))))))))))))
b.0(b.0(b.0(b.0(B.1(x))))) → b.1(a.0(b.1(a.0(b.1(a.0(b.0(b.1(a.0(b.1(a.0(b.1(a.0(b.0(B.1(x)))))))))))))))
b.0(b.1(a.0(b.1(x)))) → b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.1(x))))))))
b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(b.0(B.1(x))))))))))) → b.1(a.0(b.0(b.0(b.0(b.1(a.0(b.1(a.0(b.1(a.0(b.0(B.1(x)))))))))))))
b.0(a.1(a.0(b.1(a.0(b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(B.0(x)))))))))))))) → b.1(a.0(b.0(b.0(b.0(b.0(B.0(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
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPToSRSProof
↳ QTRS
↳ QTRS Reverse
↳ QTRS
↳ QTRS Reverse
↳ QTRS Reverse
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ SemLabProof
↳ SemLabProof2
↳ QDP
↳ SemLabProof
↳ SemLabProof2
↳ QDP
↳ Narrowing
Q DP problem:
The TRS P consists of the following rules:
B1(a(a(b(a(b(a(b(b(x))))))))) → B1(b(b(b(x))))
B1(a(a(b(a(b(a(b(b(x))))))))) → B1(b(b(x)))
B1(a(a(b(a(b(a(b(b(a(a(b(a(b(a(b(B(y0))))))))))))))))) → B1(b(a(a(b(a(b(a(b(b(B(y0)))))))))))
B1(a(a(b(a(b(a(b(b(a(a(b(a(b(a(b(B(x))))))))))))))))) → B1(b(a(b(b(B(x))))))
B1(b(b(a(a(b(a(b(a(b(B(x))))))))))) → B1(b(a(b(b(B(x))))))
B1(a(a(b(a(b(a(b(b(a(a(b(a(b(a(b(B(y0))))))))))))))))) → B1(b(b(a(a(b(a(b(a(b(b(B(y0))))))))))))
B1(b(a(b(x)))) → B1(a(a(b(a(b(a(b(x))))))))
B1(b(a(a(b(a(b(a(b(B(x)))))))))) → B1(a(a(b(a(b(a(b(b(B(x))))))))))
The TRS R consists of the following rules:
b(b(b(x))) → b(a(b(a(b(a(b(b(x))))))))
b(b(a(b(x)))) → b(a(a(b(a(b(a(b(x))))))))
b(a(a(b(a(b(a(b(b(x))))))))) → b(a(b(b(b(b(x))))))
b(b(a(a(b(a(b(a(b(B(x)))))))))) → b(a(a(b(a(b(a(b(b(B(x))))))))))
b(a(a(b(a(b(a(b(b(b(B(x))))))))))) → b(a(b(b(b(b(a(b(a(b(a(b(B(x)))))))))))))
b(a(a(b(a(b(a(b(B(x))))))))) → b(a(b(b(b(B(x))))))
b(b(b(b(B(x))))) → b(a(b(a(b(a(b(b(a(b(a(b(a(b(B(x)))))))))))))))
b(a(a(b(a(b(a(a(b(a(b(a(b(B(x)))))))))))))) → b(a(b(b(b(b(B(x)))))))
b(b(b(a(a(b(a(b(a(b(B(x))))))))))) → b(a(b(a(b(a(b(b(a(b(b(B(x))))))))))))
b(a(a(b(a(b(a(b(B(x))))))))) → b(a(b(B(x))))
b(a(a(b(a(b(a(b(b(a(a(b(a(b(a(b(B(x))))))))))))))))) → b(a(b(b(b(b(a(b(b(B(x))))))))))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule B1(a(a(b(a(b(a(b(b(x))))))))) → B1(b(b(b(x)))) at position [0] we obtained the following new rules:
B1(a(a(b(a(b(a(b(b(a(a(b(a(b(a(a(b(a(b(a(b(B(x0)))))))))))))))))))))) → B1(b(b(b(a(b(b(b(b(B(x0))))))))))
B1(a(a(b(a(b(a(b(b(a(a(b(a(b(a(b(B(x0))))))))))))))))) → B1(b(b(b(a(b(b(b(B(x0)))))))))
B1(a(a(b(a(b(a(b(b(b(b(B(x0)))))))))))) → B1(b(b(a(b(a(b(a(b(b(a(b(a(b(a(b(B(x0)))))))))))))))))
B1(a(a(b(a(b(a(b(b(a(a(b(a(b(a(b(b(a(a(b(a(b(a(b(B(x0))))))))))))))))))))))))) → B1(b(b(b(a(b(b(b(b(a(b(b(B(x0)))))))))))))
B1(a(a(b(a(b(a(b(b(b(b(b(B(x0))))))))))))) → B1(b(b(b(a(b(a(b(a(b(b(a(b(a(b(a(b(B(x0))))))))))))))))))
B1(a(a(b(a(b(a(b(b(b(b(x0))))))))))) → B1(b(b(b(a(b(a(b(a(b(b(x0)))))))))))
B1(a(a(b(a(b(a(b(b(b(b(a(a(b(a(b(a(b(B(x0))))))))))))))))))) → B1(b(b(b(a(b(a(b(a(b(b(a(b(b(B(x0)))))))))))))))
B1(a(a(b(a(b(a(b(b(a(a(b(a(b(a(b(B(x0))))))))))))))))) → B1(b(b(a(a(b(a(b(a(b(b(B(x0))))))))))))
B1(a(a(b(a(b(a(b(b(a(a(b(a(b(a(b(b(x0))))))))))))))))) → B1(b(b(b(a(b(b(b(b(x0)))))))))
B1(a(a(b(a(b(a(b(b(b(a(a(b(a(b(a(b(B(x0)))))))))))))))))) → B1(b(b(b(a(a(b(a(b(a(b(b(B(x0)))))))))))))
B1(a(a(b(a(b(a(b(b(a(a(b(a(b(a(b(B(x0))))))))))))))))) → B1(b(b(b(a(b(B(x0)))))))
B1(a(a(b(a(b(a(b(b(b(B(x0))))))))))) → B1(b(a(b(a(b(a(b(b(a(b(a(b(a(b(B(x0))))))))))))))))
B1(a(a(b(a(b(a(b(b(a(a(b(a(b(a(b(B(x0))))))))))))))))) → B1(b(a(b(a(b(a(b(b(a(b(b(B(x0)))))))))))))
B1(a(a(b(a(b(a(b(b(x0))))))))) → B1(b(a(b(a(b(a(b(b(x0)))))))))
B1(a(a(b(a(b(a(b(b(b(a(b(x0)))))))))))) → B1(b(b(b(a(a(b(a(b(a(b(x0)))))))))))
B1(a(a(b(a(b(a(b(b(a(b(x0))))))))))) → B1(b(b(a(a(b(a(b(a(b(x0))))))))))
B1(a(a(b(a(b(a(b(b(b(x0)))))))))) → B1(b(b(a(b(a(b(a(b(b(x0))))))))))
B1(a(a(b(a(b(a(b(b(b(a(a(b(a(b(a(b(B(x0)))))))))))))))))) → B1(b(b(a(b(a(b(a(b(b(a(b(b(B(x0))))))))))))))
B1(a(a(b(a(b(a(b(b(a(a(b(a(b(a(b(b(b(B(x0))))))))))))))))))) → B1(b(b(b(a(b(b(b(b(a(b(a(b(a(b(B(x0))))))))))))))))
↳ QTRS
↳ QTRS Reverse
↳ QTRS Reverse
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPToSRSProof
↳ QTRS
↳ QTRS Reverse
↳ QTRS
↳ QTRS Reverse
↳ QTRS Reverse
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ SemLabProof
↳ SemLabProof2
↳ QDP
↳ SemLabProof
↳ SemLabProof2
↳ QDP
↳ Narrowing
↳ QDP
↳ SemLabProof
↳ SemLabProof2
Q DP problem:
The TRS P consists of the following rules:
B1(a(a(b(a(b(a(b(b(a(a(b(a(b(a(a(b(a(b(a(b(B(x0)))))))))))))))))))))) → B1(b(b(b(a(b(b(b(b(B(x0))))))))))
B1(a(a(b(a(b(a(b(b(a(a(b(a(b(a(b(B(x0))))))))))))))))) → B1(b(b(b(a(b(b(b(B(x0)))))))))
B1(a(a(b(a(b(a(b(b(a(a(b(a(b(a(b(B(x))))))))))))))))) → B1(b(a(b(b(B(x))))))
B1(a(a(b(a(b(a(b(b(b(b(a(a(b(a(b(a(b(B(x0))))))))))))))))))) → B1(b(b(b(a(b(a(b(a(b(b(a(b(b(B(x0)))))))))))))))
B1(b(b(a(a(b(a(b(a(b(B(x))))))))))) → B1(b(a(b(b(B(x))))))
B1(a(a(b(a(b(a(b(b(b(a(a(b(a(b(a(b(B(x0)))))))))))))))))) → B1(b(b(b(a(a(b(a(b(a(b(b(B(x0)))))))))))))
B1(a(a(b(a(b(a(b(b(a(a(b(a(b(a(b(B(x0))))))))))))))))) → B1(b(b(b(a(b(B(x0)))))))
B1(b(a(b(x)))) → B1(a(a(b(a(b(a(b(x))))))))
B1(a(a(b(a(b(a(b(b(b(B(x0))))))))))) → B1(b(a(b(a(b(a(b(b(a(b(a(b(a(b(B(x0))))))))))))))))
B1(a(a(b(a(b(a(b(b(b(a(b(x0)))))))))))) → B1(b(b(b(a(a(b(a(b(a(b(x0)))))))))))
B1(a(a(b(a(b(a(b(b(a(a(b(a(b(a(b(b(b(B(x0))))))))))))))))))) → B1(b(b(b(a(b(b(b(b(a(b(a(b(a(b(B(x0))))))))))))))))
B1(a(a(b(a(b(a(b(b(b(a(a(b(a(b(a(b(B(x0)))))))))))))))))) → B1(b(b(a(b(a(b(a(b(b(a(b(b(B(x0))))))))))))))
B1(a(a(b(a(b(a(b(b(b(b(B(x0)))))))))))) → B1(b(b(a(b(a(b(a(b(b(a(b(a(b(a(b(B(x0)))))))))))))))))
B1(a(a(b(a(b(a(b(b(a(a(b(a(b(a(b(b(a(a(b(a(b(a(b(B(x0))))))))))))))))))))))))) → B1(b(b(b(a(b(b(b(b(a(b(b(B(x0)))))))))))))
B1(a(a(b(a(b(a(b(b(b(b(b(B(x0))))))))))))) → B1(b(b(b(a(b(a(b(a(b(b(a(b(a(b(a(b(B(x0))))))))))))))))))
B1(a(a(b(a(b(a(b(b(b(b(x0))))))))))) → B1(b(b(b(a(b(a(b(a(b(b(x0)))))))))))
B1(a(a(b(a(b(a(b(b(a(a(b(a(b(a(b(B(y0))))))))))))))))) → B1(b(b(a(a(b(a(b(a(b(b(B(y0))))))))))))
B1(a(a(b(a(b(a(b(b(a(a(b(a(b(a(b(b(x0))))))))))))))))) → B1(b(b(b(a(b(b(b(b(x0)))))))))
B1(a(a(b(a(b(a(b(b(x))))))))) → B1(b(b(x)))
B1(a(a(b(a(b(a(b(b(a(a(b(a(b(a(b(B(x0))))))))))))))))) → B1(b(a(b(a(b(a(b(b(a(b(b(B(x0)))))))))))))
B1(a(a(b(a(b(a(b(b(x0))))))))) → B1(b(a(b(a(b(a(b(b(x0)))))))))
B1(a(a(b(a(b(a(b(b(a(a(b(a(b(a(b(B(y0))))))))))))))))) → B1(b(a(a(b(a(b(a(b(b(B(y0)))))))))))
B1(a(a(b(a(b(a(b(b(a(b(x0))))))))))) → B1(b(b(a(a(b(a(b(a(b(x0))))))))))
B1(a(a(b(a(b(a(b(b(b(x0)))))))))) → B1(b(b(a(b(a(b(a(b(b(x0))))))))))
B1(b(a(a(b(a(b(a(b(B(x)))))))))) → B1(a(a(b(a(b(a(b(b(B(x))))))))))
The TRS R consists of the following rules:
b(b(b(x))) → b(a(b(a(b(a(b(b(x))))))))
b(b(a(b(x)))) → b(a(a(b(a(b(a(b(x))))))))
b(a(a(b(a(b(a(b(b(x))))))))) → b(a(b(b(b(b(x))))))
b(b(a(a(b(a(b(a(b(B(x)))))))))) → b(a(a(b(a(b(a(b(b(B(x))))))))))
b(a(a(b(a(b(a(b(b(b(B(x))))))))))) → b(a(b(b(b(b(a(b(a(b(a(b(B(x)))))))))))))
b(a(a(b(a(b(a(b(B(x))))))))) → b(a(b(b(b(B(x))))))
b(b(b(b(B(x))))) → b(a(b(a(b(a(b(b(a(b(a(b(a(b(B(x)))))))))))))))
b(a(a(b(a(b(a(a(b(a(b(a(b(B(x)))))))))))))) → b(a(b(b(b(b(B(x)))))))
b(b(b(a(a(b(a(b(a(b(B(x))))))))))) → b(a(b(a(b(a(b(b(a(b(b(B(x))))))))))))
b(a(a(b(a(b(a(b(B(x))))))))) → b(a(b(B(x))))
b(a(a(b(a(b(a(b(b(a(a(b(a(b(a(b(B(x))))))))))))))))) → b(a(b(b(b(b(a(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.B: 0
a: 1 + x0
B1: 0
b: 0
By semantic labelling [33] we obtain the following labelled TRS:Q DP problem:
The TRS P consists of the following rules:
B1.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(B.1(x0))))))))))))))))))))))))) → B1.0(b.0(b.0(b.1(a.0(b.0(b.0(b.0(b.1(a.0(b.0(b.0(B.1(x0)))))))))))))
B1.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(b.0(B.1(x0))))))))))))))))))) → B1.0(b.0(b.0(b.1(a.0(b.0(b.0(b.0(b.1(a.0(b.1(a.0(b.1(a.0(b.0(B.1(x0))))))))))))))))
B1.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(B.0(x0))))))))))))))))))))))))) → B1.0(b.0(b.0(b.1(a.0(b.0(b.0(b.0(b.1(a.0(b.0(b.0(B.0(x0)))))))))))))
B1.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(B.0(x0))))))))))))))))) → B1.0(b.0(b.0(b.1(a.0(b.0(b.0(b.0(B.0(x0)))))))))
B1.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(B.1(x))))))))))))))))) → B1.0(b.1(a.0(b.0(b.0(B.1(x))))))
B1.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(x0))))))))))))))))) → B1.0(b.0(b.0(b.1(a.0(b.0(b.0(b.0(b.0(x0)))))))))
B1.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(B.1(x0))))))))))))))))) → B1.0(b.0(b.0(b.1(a.0(b.0(b.0(b.0(B.1(x0)))))))))
B1.0(b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(B.0(x)))))))))) → B1.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(B.0(x))))))))))
B1.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(b.0(b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(B.0(x0))))))))))))))))))) → B1.0(b.0(b.0(b.1(a.0(b.1(a.0(b.1(a.0(b.0(b.1(a.0(b.0(b.0(B.0(x0)))))))))))))))
B1.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(a.1(a.0(b.1(a.0(b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(B.1(x0)))))))))))))))))))))) → B1.0(b.0(b.0(b.1(a.0(b.0(b.0(b.0(b.0(B.1(x0))))))))))
B1.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(b.0(b.0(b.0(B.1(x0))))))))))))) → B1.0(b.0(b.0(b.1(a.0(b.1(a.0(b.1(a.0(b.0(b.1(a.0(b.1(a.0(b.1(a.0(b.0(B.1(x0))))))))))))))))))
B1.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(b.0(b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(B.1(x0))))))))))))))))))) → B1.0(b.0(b.0(b.1(a.0(b.1(a.0(b.1(a.0(b.0(b.1(a.0(b.0(b.0(B.1(x0)))))))))))))))
B1.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(b.0(b.0(x0))))))))))) → B1.0(b.0(b.0(b.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(x0)))))))))))
B1.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(B.0(x0))))))))))))))))) → B1.0(b.1(a.0(b.1(a.0(b.1(a.0(b.0(b.1(a.0(b.0(b.0(B.0(x0)))))))))))))
B1.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(B.0(y0))))))))))))))))) → B1.0(b.0(b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(B.0(y0))))))))))))
B1.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(b.1(a.0(b.1(x0)))))))))))) → B1.0(b.0(b.0(b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.1(x0)))))))))))
B1.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(B.0(x0))))))))))))))))) → B1.0(b.0(b.0(b.1(a.0(b.0(B.0(x0)))))))
B1.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(B.1(y0))))))))))))))))) → B1.0(b.0(b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(B.1(y0))))))))))))
B1.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(b.1(x0)))))))))) → B1.0(b.0(b.1(a.0(b.1(a.0(b.1(a.0(b.0(b.1(x0))))))))))
B1.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.1(x))))))))) → B1.0(b.0(b.1(x)))
B1.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(b.1(a.0(b.0(x0)))))))))))) → B1.0(b.0(b.0(b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(x0)))))))))))
B1.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(B.0(x))))))))))))))))) → B1.0(b.1(a.0(b.0(b.0(B.0(x))))))
B1.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.1(a.0(b.0(x0))))))))))) → B1.0(b.0(b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(x0))))))))))
B1.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(b.0(b.0(B.0(x0)))))))))))) → B1.0(b.0(b.1(a.0(b.1(a.0(b.1(a.0(b.0(b.1(a.0(b.1(a.0(b.1(a.0(b.0(B.0(x0)))))))))))))))))
B1.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(b.0(b.1(x0))))))))))) → B1.0(b.0(b.0(b.1(a.0(b.1(a.0(b.1(a.0(b.0(b.1(x0)))))))))))
B1.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(b.0(b.0(b.0(B.0(x0))))))))))))) → B1.0(b.0(b.0(b.1(a.0(b.1(a.0(b.1(a.0(b.0(b.1(a.0(b.1(a.0(b.1(a.0(b.0(B.0(x0))))))))))))))))))
B1.0(b.1(a.0(b.1(x)))) → B1.0(a.1(a.0(b.1(a.0(b.1(a.0(b.1(x))))))))
B1.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(B.1(x0))))))))))))))))) → B1.0(b.1(a.0(b.1(a.0(b.1(a.0(b.0(b.1(a.0(b.0(b.0(B.1(x0)))))))))))))
B1.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(B.1(x0)))))))))))))))))) → B1.0(b.0(b.1(a.0(b.1(a.0(b.1(a.0(b.0(b.1(a.0(b.0(b.0(B.1(x0))))))))))))))
B1.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(B.1(x0))))))))))))))))) → B1.0(b.0(b.0(b.1(a.0(b.0(B.1(x0)))))))
B1.0(b.0(b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(B.0(x))))))))))) → B1.0(b.1(a.0(b.0(b.0(B.0(x))))))
B1.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(b.0(b.0(B.1(x0)))))))))))) → B1.0(b.0(b.1(a.0(b.1(a.0(b.1(a.0(b.0(b.1(a.0(b.1(a.0(b.1(a.0(b.0(B.1(x0)))))))))))))))))
B1.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(x))))))))) → B1.0(b.0(b.0(x)))
B1.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(b.0(B.0(x0))))))))))))))))))) → B1.0(b.0(b.0(b.1(a.0(b.0(b.0(b.0(b.1(a.0(b.1(a.0(b.1(a.0(b.0(B.0(x0))))))))))))))))
B1.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(B.1(x0)))))))))))))))))) → B1.0(b.0(b.0(b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(B.1(x0)))))))))))))
B1.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(x0))))))))) → B1.0(b.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(x0)))))))))
B1.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(b.0(B.1(x0))))))))))) → B1.0(b.1(a.0(b.1(a.0(b.1(a.0(b.0(b.1(a.0(b.1(a.0(b.1(a.0(b.0(B.1(x0))))))))))))))))
B1.0(b.0(b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(B.1(x))))))))))) → B1.0(b.1(a.0(b.0(b.0(B.1(x))))))
B1.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.1(x0))))))))))))))))) → B1.0(b.0(b.0(b.1(a.0(b.0(b.0(b.0(b.1(x0)))))))))
B1.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(b.0(x0)))))))))) → B1.0(b.0(b.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(x0))))))))))
B1.0(b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(B.1(x)))))))))) → B1.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(B.1(x))))))))))
B1.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(b.0(B.0(x0))))))))))) → B1.0(b.1(a.0(b.1(a.0(b.1(a.0(b.0(b.1(a.0(b.1(a.0(b.1(a.0(b.0(B.0(x0))))))))))))))))
B1.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(B.0(x0)))))))))))))))))) → B1.0(b.0(b.1(a.0(b.1(a.0(b.1(a.0(b.0(b.1(a.0(b.0(b.0(B.0(x0))))))))))))))
B1.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.1(x0))))))))) → B1.0(b.1(a.0(b.1(a.0(b.1(a.0(b.0(b.1(x0)))))))))
B1.0(b.1(a.0(b.0(x)))) → B1.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(x))))))))
B1.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(B.0(x0)))))))))))))))))) → B1.0(b.0(b.0(b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(B.0(x0)))))))))))))
B1.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.1(a.0(b.1(x0))))))))))) → B1.0(b.0(b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.1(x0))))))))))
B1.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(a.1(a.0(b.1(a.0(b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(B.0(x0)))))))))))))))))))))) → B1.0(b.0(b.0(b.1(a.0(b.0(b.0(b.0(b.0(B.0(x0))))))))))
B1.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(B.1(y0))))))))))))))))) → B1.0(b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(B.1(y0)))))))))))
B1.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(B.0(y0))))))))))))))))) → B1.0(b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(B.0(y0)))))))))))
The TRS R consists of the following rules:
b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(B.0(x))))))))) → b.1(a.0(b.0(b.0(b.0(B.0(x))))))
b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(B.0(x))))))))) → b.1(a.0(b.0(B.0(x))))
b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(B.1(x))))))))))))))))) → b.1(a.0(b.0(b.0(b.0(b.1(a.0(b.0(b.0(B.1(x))))))))))
b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(B.0(x))))))))))))))))) → b.1(a.0(b.0(b.0(b.0(b.1(a.0(b.0(b.0(B.0(x))))))))))
b.0(b.0(b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(B.1(x))))))))))) → b.1(a.0(b.1(a.0(b.1(a.0(b.0(b.1(a.0(b.0(b.0(B.1(x))))))))))))
b.0(b.0(b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(B.0(x))))))))))) → b.1(a.0(b.1(a.0(b.1(a.0(b.0(b.1(a.0(b.0(b.0(B.0(x))))))))))))
b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.1(x))))))))) → b.1(a.0(b.0(b.0(b.0(b.1(x))))))
b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(x))))))))) → b.1(a.0(b.0(b.0(b.0(b.0(x))))))
b.0(b.0(b.1(x))) → b.1(a.0(b.1(a.0(b.1(a.0(b.0(b.1(x))))))))
b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(B.1(x))))))))) → b.1(a.0(b.0(B.1(x))))
b.0(b.0(b.0(x))) → b.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(x))))))))
b.0(b.0(b.0(b.0(B.0(x))))) → b.1(a.0(b.1(a.0(b.1(a.0(b.0(b.1(a.0(b.1(a.0(b.1(a.0(b.0(B.0(x)))))))))))))))
b.0(b.1(a.0(b.0(x)))) → b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(x))))))))
b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(B.1(x))))))))) → b.1(a.0(b.0(b.0(b.0(B.1(x))))))
b.0(a.1(a.0(b.1(a.0(b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(B.1(x)))))))))))))) → b.1(a.0(b.0(b.0(b.0(b.0(B.1(x)))))))
b.0(b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(B.1(x)))))))))) → b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(B.1(x))))))))))
b.0(b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(B.0(x)))))))))) → b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(B.0(x))))))))))
b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(b.0(B.0(x))))))))))) → b.1(a.0(b.0(b.0(b.0(b.1(a.0(b.1(a.0(b.1(a.0(b.0(B.0(x)))))))))))))
b.0(b.0(b.0(b.0(B.1(x))))) → b.1(a.0(b.1(a.0(b.1(a.0(b.0(b.1(a.0(b.1(a.0(b.1(a.0(b.0(B.1(x)))))))))))))))
b.0(b.1(a.0(b.1(x)))) → b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.1(x))))))))
b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(b.0(B.1(x))))))))))) → b.1(a.0(b.0(b.0(b.0(b.1(a.0(b.1(a.0(b.1(a.0(b.0(B.1(x)))))))))))))
b.0(a.1(a.0(b.1(a.0(b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(B.0(x)))))))))))))) → b.1(a.0(b.0(b.0(b.0(b.0(B.0(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
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPToSRSProof
↳ QTRS
↳ QTRS Reverse
↳ QTRS
↳ QTRS Reverse
↳ QTRS Reverse
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ SemLabProof
↳ SemLabProof2
↳ QDP
↳ SemLabProof
↳ SemLabProof2
↳ QDP
↳ Narrowing
↳ QDP
↳ SemLabProof
↳ QDP
↳ DependencyGraphProof
↳ SemLabProof2
Q DP problem:
The TRS P consists of the following rules:
B1.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(B.1(x0))))))))))))))))))))))))) → B1.0(b.0(b.0(b.1(a.0(b.0(b.0(b.0(b.1(a.0(b.0(b.0(B.1(x0)))))))))))))
B1.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(b.0(B.1(x0))))))))))))))))))) → B1.0(b.0(b.0(b.1(a.0(b.0(b.0(b.0(b.1(a.0(b.1(a.0(b.1(a.0(b.0(B.1(x0))))))))))))))))
B1.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(B.0(x0))))))))))))))))))))))))) → B1.0(b.0(b.0(b.1(a.0(b.0(b.0(b.0(b.1(a.0(b.0(b.0(B.0(x0)))))))))))))
B1.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(B.0(x0))))))))))))))))) → B1.0(b.0(b.0(b.1(a.0(b.0(b.0(b.0(B.0(x0)))))))))
B1.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(B.1(x))))))))))))))))) → B1.0(b.1(a.0(b.0(b.0(B.1(x))))))
B1.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(x0))))))))))))))))) → B1.0(b.0(b.0(b.1(a.0(b.0(b.0(b.0(b.0(x0)))))))))
B1.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(B.1(x0))))))))))))))))) → B1.0(b.0(b.0(b.1(a.0(b.0(b.0(b.0(B.1(x0)))))))))
B1.0(b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(B.0(x)))))))))) → B1.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(B.0(x))))))))))
B1.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(b.0(b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(B.0(x0))))))))))))))))))) → B1.0(b.0(b.0(b.1(a.0(b.1(a.0(b.1(a.0(b.0(b.1(a.0(b.0(b.0(B.0(x0)))))))))))))))
B1.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(a.1(a.0(b.1(a.0(b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(B.1(x0)))))))))))))))))))))) → B1.0(b.0(b.0(b.1(a.0(b.0(b.0(b.0(b.0(B.1(x0))))))))))
B1.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(b.0(b.0(b.0(B.1(x0))))))))))))) → B1.0(b.0(b.0(b.1(a.0(b.1(a.0(b.1(a.0(b.0(b.1(a.0(b.1(a.0(b.1(a.0(b.0(B.1(x0))))))))))))))))))
B1.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(b.0(b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(B.1(x0))))))))))))))))))) → B1.0(b.0(b.0(b.1(a.0(b.1(a.0(b.1(a.0(b.0(b.1(a.0(b.0(b.0(B.1(x0)))))))))))))))
B1.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(b.0(b.0(x0))))))))))) → B1.0(b.0(b.0(b.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(x0)))))))))))
B1.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(B.0(x0))))))))))))))))) → B1.0(b.1(a.0(b.1(a.0(b.1(a.0(b.0(b.1(a.0(b.0(b.0(B.0(x0)))))))))))))
B1.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(B.0(y0))))))))))))))))) → B1.0(b.0(b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(B.0(y0))))))))))))
B1.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(b.1(a.0(b.1(x0)))))))))))) → B1.0(b.0(b.0(b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.1(x0)))))))))))
B1.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(B.0(x0))))))))))))))))) → B1.0(b.0(b.0(b.1(a.0(b.0(B.0(x0)))))))
B1.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(B.1(y0))))))))))))))))) → B1.0(b.0(b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(B.1(y0))))))))))))
B1.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(b.1(x0)))))))))) → B1.0(b.0(b.1(a.0(b.1(a.0(b.1(a.0(b.0(b.1(x0))))))))))
B1.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.1(x))))))))) → B1.0(b.0(b.1(x)))
B1.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(b.1(a.0(b.0(x0)))))))))))) → B1.0(b.0(b.0(b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(x0)))))))))))
B1.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(B.0(x))))))))))))))))) → B1.0(b.1(a.0(b.0(b.0(B.0(x))))))
B1.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.1(a.0(b.0(x0))))))))))) → B1.0(b.0(b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(x0))))))))))
B1.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(b.0(b.0(B.0(x0)))))))))))) → B1.0(b.0(b.1(a.0(b.1(a.0(b.1(a.0(b.0(b.1(a.0(b.1(a.0(b.1(a.0(b.0(B.0(x0)))))))))))))))))
B1.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(b.0(b.1(x0))))))))))) → B1.0(b.0(b.0(b.1(a.0(b.1(a.0(b.1(a.0(b.0(b.1(x0)))))))))))
B1.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(b.0(b.0(b.0(B.0(x0))))))))))))) → B1.0(b.0(b.0(b.1(a.0(b.1(a.0(b.1(a.0(b.0(b.1(a.0(b.1(a.0(b.1(a.0(b.0(B.0(x0))))))))))))))))))
B1.0(b.1(a.0(b.1(x)))) → B1.0(a.1(a.0(b.1(a.0(b.1(a.0(b.1(x))))))))
B1.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(B.1(x0))))))))))))))))) → B1.0(b.1(a.0(b.1(a.0(b.1(a.0(b.0(b.1(a.0(b.0(b.0(B.1(x0)))))))))))))
B1.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(B.1(x0)))))))))))))))))) → B1.0(b.0(b.1(a.0(b.1(a.0(b.1(a.0(b.0(b.1(a.0(b.0(b.0(B.1(x0))))))))))))))
B1.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(B.1(x0))))))))))))))))) → B1.0(b.0(b.0(b.1(a.0(b.0(B.1(x0)))))))
B1.0(b.0(b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(B.0(x))))))))))) → B1.0(b.1(a.0(b.0(b.0(B.0(x))))))
B1.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(b.0(b.0(B.1(x0)))))))))))) → B1.0(b.0(b.1(a.0(b.1(a.0(b.1(a.0(b.0(b.1(a.0(b.1(a.0(b.1(a.0(b.0(B.1(x0)))))))))))))))))
B1.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(x))))))))) → B1.0(b.0(b.0(x)))
B1.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(b.0(B.0(x0))))))))))))))))))) → B1.0(b.0(b.0(b.1(a.0(b.0(b.0(b.0(b.1(a.0(b.1(a.0(b.1(a.0(b.0(B.0(x0))))))))))))))))
B1.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(B.1(x0)))))))))))))))))) → B1.0(b.0(b.0(b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(B.1(x0)))))))))))))
B1.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(x0))))))))) → B1.0(b.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(x0)))))))))
B1.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(b.0(B.1(x0))))))))))) → B1.0(b.1(a.0(b.1(a.0(b.1(a.0(b.0(b.1(a.0(b.1(a.0(b.1(a.0(b.0(B.1(x0))))))))))))))))
B1.0(b.0(b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(B.1(x))))))))))) → B1.0(b.1(a.0(b.0(b.0(B.1(x))))))
B1.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.1(x0))))))))))))))))) → B1.0(b.0(b.0(b.1(a.0(b.0(b.0(b.0(b.1(x0)))))))))
B1.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(b.0(x0)))))))))) → B1.0(b.0(b.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(x0))))))))))
B1.0(b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(B.1(x)))))))))) → B1.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(B.1(x))))))))))
B1.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(b.0(B.0(x0))))))))))) → B1.0(b.1(a.0(b.1(a.0(b.1(a.0(b.0(b.1(a.0(b.1(a.0(b.1(a.0(b.0(B.0(x0))))))))))))))))
B1.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(B.0(x0)))))))))))))))))) → B1.0(b.0(b.1(a.0(b.1(a.0(b.1(a.0(b.0(b.1(a.0(b.0(b.0(B.0(x0))))))))))))))
B1.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.1(x0))))))))) → B1.0(b.1(a.0(b.1(a.0(b.1(a.0(b.0(b.1(x0)))))))))
B1.0(b.1(a.0(b.0(x)))) → B1.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(x))))))))
B1.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(B.0(x0)))))))))))))))))) → B1.0(b.0(b.0(b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(B.0(x0)))))))))))))
B1.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.1(a.0(b.1(x0))))))))))) → B1.0(b.0(b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.1(x0))))))))))
B1.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(a.1(a.0(b.1(a.0(b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(B.0(x0)))))))))))))))))))))) → B1.0(b.0(b.0(b.1(a.0(b.0(b.0(b.0(b.0(B.0(x0))))))))))
B1.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(B.1(y0))))))))))))))))) → B1.0(b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(B.1(y0)))))))))))
B1.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(B.0(y0))))))))))))))))) → B1.0(b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(B.0(y0)))))))))))
The TRS R consists of the following rules:
b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(B.0(x))))))))) → b.1(a.0(b.0(b.0(b.0(B.0(x))))))
b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(B.0(x))))))))) → b.1(a.0(b.0(B.0(x))))
b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(B.1(x))))))))))))))))) → b.1(a.0(b.0(b.0(b.0(b.1(a.0(b.0(b.0(B.1(x))))))))))
b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(B.0(x))))))))))))))))) → b.1(a.0(b.0(b.0(b.0(b.1(a.0(b.0(b.0(B.0(x))))))))))
b.0(b.0(b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(B.1(x))))))))))) → b.1(a.0(b.1(a.0(b.1(a.0(b.0(b.1(a.0(b.0(b.0(B.1(x))))))))))))
b.0(b.0(b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(B.0(x))))))))))) → b.1(a.0(b.1(a.0(b.1(a.0(b.0(b.1(a.0(b.0(b.0(B.0(x))))))))))))
b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.1(x))))))))) → b.1(a.0(b.0(b.0(b.0(b.1(x))))))
b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(x))))))))) → b.1(a.0(b.0(b.0(b.0(b.0(x))))))
b.0(b.0(b.1(x))) → b.1(a.0(b.1(a.0(b.1(a.0(b.0(b.1(x))))))))
b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(B.1(x))))))))) → b.1(a.0(b.0(B.1(x))))
b.0(b.0(b.0(x))) → b.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(x))))))))
b.0(b.0(b.0(b.0(B.0(x))))) → b.1(a.0(b.1(a.0(b.1(a.0(b.0(b.1(a.0(b.1(a.0(b.1(a.0(b.0(B.0(x)))))))))))))))
b.0(b.1(a.0(b.0(x)))) → b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(x))))))))
b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(B.1(x))))))))) → b.1(a.0(b.0(b.0(b.0(B.1(x))))))
b.0(a.1(a.0(b.1(a.0(b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(B.1(x)))))))))))))) → b.1(a.0(b.0(b.0(b.0(b.0(B.1(x)))))))
b.0(b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(B.1(x)))))))))) → b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(B.1(x))))))))))
b.0(b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(B.0(x)))))))))) → b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(B.0(x))))))))))
b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(b.0(B.0(x))))))))))) → b.1(a.0(b.0(b.0(b.0(b.1(a.0(b.1(a.0(b.1(a.0(b.0(B.0(x)))))))))))))
b.0(b.0(b.0(b.0(B.1(x))))) → b.1(a.0(b.1(a.0(b.1(a.0(b.0(b.1(a.0(b.1(a.0(b.1(a.0(b.0(B.1(x)))))))))))))))
b.0(b.1(a.0(b.1(x)))) → b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.1(x))))))))
b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(b.0(B.1(x))))))))))) → b.1(a.0(b.0(b.0(b.0(b.1(a.0(b.1(a.0(b.1(a.0(b.0(B.1(x)))))))))))))
b.0(a.1(a.0(b.1(a.0(b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(B.0(x)))))))))))))) → b.1(a.0(b.0(b.0(b.0(b.0(B.0(x)))))))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 1 SCC with 8 less nodes.
↳ QTRS
↳ QTRS Reverse
↳ QTRS Reverse
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPToSRSProof
↳ QTRS
↳ QTRS Reverse
↳ QTRS
↳ QTRS Reverse
↳ QTRS Reverse
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ SemLabProof
↳ SemLabProof2
↳ QDP
↳ SemLabProof
↳ SemLabProof2
↳ QDP
↳ Narrowing
↳ QDP
↳ SemLabProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ SemLabProof2
Q DP problem:
The TRS P consists of the following rules:
B1.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(B.1(x0))))))))))))))))))))))))) → B1.0(b.0(b.0(b.1(a.0(b.0(b.0(b.0(b.1(a.0(b.0(b.0(B.1(x0)))))))))))))
B1.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(b.0(B.1(x0))))))))))))))))))) → B1.0(b.0(b.0(b.1(a.0(b.0(b.0(b.0(b.1(a.0(b.1(a.0(b.1(a.0(b.0(B.1(x0))))))))))))))))
B1.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(B.0(x0))))))))))))))))))))))))) → B1.0(b.0(b.0(b.1(a.0(b.0(b.0(b.0(b.1(a.0(b.0(b.0(B.0(x0)))))))))))))
B1.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(B.1(x))))))))))))))))) → B1.0(b.1(a.0(b.0(b.0(B.1(x))))))
B1.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(B.0(x0))))))))))))))))) → B1.0(b.0(b.0(b.1(a.0(b.0(b.0(b.0(B.0(x0)))))))))
B1.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(x0))))))))))))))))) → B1.0(b.0(b.0(b.1(a.0(b.0(b.0(b.0(b.0(x0)))))))))
B1.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(B.1(x0))))))))))))))))) → B1.0(b.0(b.0(b.1(a.0(b.0(b.0(b.0(B.1(x0)))))))))
B1.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(B.1(x0)))))))))))))))))) → B1.0(b.0(b.1(a.0(b.1(a.0(b.1(a.0(b.0(b.1(a.0(b.0(b.0(B.1(x0))))))))))))))
B1.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(B.1(x0))))))))))))))))) → B1.0(b.0(b.0(b.1(a.0(b.0(B.1(x0)))))))
B1.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(b.0(b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(B.0(x0))))))))))))))))))) → B1.0(b.0(b.0(b.1(a.0(b.1(a.0(b.1(a.0(b.0(b.1(a.0(b.0(b.0(B.0(x0)))))))))))))))
B1.0(b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(B.0(x)))))))))) → B1.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(B.0(x))))))))))
B1.0(b.0(b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(B.0(x))))))))))) → B1.0(b.1(a.0(b.0(b.0(B.0(x))))))
B1.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(b.0(b.0(B.1(x0)))))))))))) → B1.0(b.0(b.1(a.0(b.1(a.0(b.1(a.0(b.0(b.1(a.0(b.1(a.0(b.1(a.0(b.0(B.1(x0)))))))))))))))))
B1.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(a.1(a.0(b.1(a.0(b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(B.1(x0)))))))))))))))))))))) → B1.0(b.0(b.0(b.1(a.0(b.0(b.0(b.0(b.0(B.1(x0))))))))))
B1.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(b.0(b.0(b.0(B.1(x0))))))))))))) → B1.0(b.0(b.0(b.1(a.0(b.1(a.0(b.1(a.0(b.0(b.1(a.0(b.1(a.0(b.1(a.0(b.0(B.1(x0))))))))))))))))))
B1.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(b.0(b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(B.1(x0))))))))))))))))))) → B1.0(b.0(b.0(b.1(a.0(b.1(a.0(b.1(a.0(b.0(b.1(a.0(b.0(b.0(B.1(x0)))))))))))))))
B1.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(x))))))))) → B1.0(b.0(b.0(x)))
B1.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(b.0(B.0(x0))))))))))))))))))) → B1.0(b.0(b.0(b.1(a.0(b.0(b.0(b.0(b.1(a.0(b.1(a.0(b.1(a.0(b.0(B.0(x0))))))))))))))))
B1.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(B.1(x0)))))))))))))))))) → B1.0(b.0(b.0(b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(B.1(x0)))))))))))))
B1.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(b.0(b.0(x0))))))))))) → B1.0(b.0(b.0(b.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(x0)))))))))))
B1.0(b.0(b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(B.1(x))))))))))) → B1.0(b.1(a.0(b.0(b.0(B.1(x))))))
B1.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.1(x0))))))))))))))))) → B1.0(b.0(b.0(b.1(a.0(b.0(b.0(b.0(b.1(x0)))))))))
B1.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(B.0(y0))))))))))))))))) → B1.0(b.0(b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(B.0(y0))))))))))))
B1.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(b.0(x0)))))))))) → B1.0(b.0(b.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(x0))))))))))
B1.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(b.1(a.0(b.1(x0)))))))))))) → B1.0(b.0(b.0(b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.1(x0)))))))))))
B1.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(B.0(x0))))))))))))))))) → B1.0(b.0(b.0(b.1(a.0(b.0(B.0(x0)))))))
B1.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(b.1(x0)))))))))) → B1.0(b.0(b.1(a.0(b.1(a.0(b.1(a.0(b.0(b.1(x0))))))))))
B1.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(B.1(y0))))))))))))))))) → B1.0(b.0(b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(B.1(y0))))))))))))
B1.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.1(x))))))))) → B1.0(b.0(b.1(x)))
B1.0(b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(B.1(x)))))))))) → B1.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(B.1(x))))))))))
B1.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(b.1(a.0(b.0(x0)))))))))))) → B1.0(b.0(b.0(b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(x0)))))))))))
B1.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(B.0(x))))))))))))))))) → B1.0(b.1(a.0(b.0(b.0(B.0(x))))))
B1.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.1(a.0(b.0(x0))))))))))) → B1.0(b.0(b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(x0))))))))))
B1.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(b.0(b.0(B.0(x0)))))))))))) → B1.0(b.0(b.1(a.0(b.1(a.0(b.1(a.0(b.0(b.1(a.0(b.1(a.0(b.1(a.0(b.0(B.0(x0)))))))))))))))))
B1.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(b.0(b.1(x0))))))))))) → B1.0(b.0(b.0(b.1(a.0(b.1(a.0(b.1(a.0(b.0(b.1(x0)))))))))))
B1.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(B.0(x0)))))))))))))))))) → B1.0(b.0(b.1(a.0(b.1(a.0(b.1(a.0(b.0(b.1(a.0(b.0(b.0(B.0(x0))))))))))))))
B1.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(B.0(x0)))))))))))))))))) → B1.0(b.0(b.0(b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(B.0(x0)))))))))))))
B1.0(b.1(a.0(b.0(x)))) → B1.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(x))))))))
B1.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(b.0(b.0(b.0(B.0(x0))))))))))))) → B1.0(b.0(b.0(b.1(a.0(b.1(a.0(b.1(a.0(b.0(b.1(a.0(b.1(a.0(b.1(a.0(b.0(B.0(x0))))))))))))))))))
B1.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(B.1(y0))))))))))))))))) → B1.0(b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(B.1(y0)))))))))))
B1.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(B.0(y0))))))))))))))))) → B1.0(b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(B.0(y0)))))))))))
B1.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(a.1(a.0(b.1(a.0(b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(B.0(x0)))))))))))))))))))))) → B1.0(b.0(b.0(b.1(a.0(b.0(b.0(b.0(b.0(B.0(x0))))))))))
The TRS R consists of the following rules:
b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(B.0(x))))))))) → b.1(a.0(b.0(b.0(b.0(B.0(x))))))
b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(B.0(x))))))))) → b.1(a.0(b.0(B.0(x))))
b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(B.1(x))))))))))))))))) → b.1(a.0(b.0(b.0(b.0(b.1(a.0(b.0(b.0(B.1(x))))))))))
b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(B.0(x))))))))))))))))) → b.1(a.0(b.0(b.0(b.0(b.1(a.0(b.0(b.0(B.0(x))))))))))
b.0(b.0(b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(B.1(x))))))))))) → b.1(a.0(b.1(a.0(b.1(a.0(b.0(b.1(a.0(b.0(b.0(B.1(x))))))))))))
b.0(b.0(b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(B.0(x))))))))))) → b.1(a.0(b.1(a.0(b.1(a.0(b.0(b.1(a.0(b.0(b.0(B.0(x))))))))))))
b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.1(x))))))))) → b.1(a.0(b.0(b.0(b.0(b.1(x))))))
b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(x))))))))) → b.1(a.0(b.0(b.0(b.0(b.0(x))))))
b.0(b.0(b.1(x))) → b.1(a.0(b.1(a.0(b.1(a.0(b.0(b.1(x))))))))
b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(B.1(x))))))))) → b.1(a.0(b.0(B.1(x))))
b.0(b.0(b.0(x))) → b.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(x))))))))
b.0(b.0(b.0(b.0(B.0(x))))) → b.1(a.0(b.1(a.0(b.1(a.0(b.0(b.1(a.0(b.1(a.0(b.1(a.0(b.0(B.0(x)))))))))))))))
b.0(b.1(a.0(b.0(x)))) → b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(x))))))))
b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(B.1(x))))))))) → b.1(a.0(b.0(b.0(b.0(B.1(x))))))
b.0(a.1(a.0(b.1(a.0(b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(B.1(x)))))))))))))) → b.1(a.0(b.0(b.0(b.0(b.0(B.1(x)))))))
b.0(b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(B.1(x)))))))))) → b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(B.1(x))))))))))
b.0(b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(B.0(x)))))))))) → b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(B.0(x))))))))))
b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(b.0(B.0(x))))))))))) → b.1(a.0(b.0(b.0(b.0(b.1(a.0(b.1(a.0(b.1(a.0(b.0(B.0(x)))))))))))))
b.0(b.0(b.0(b.0(B.1(x))))) → b.1(a.0(b.1(a.0(b.1(a.0(b.0(b.1(a.0(b.1(a.0(b.1(a.0(b.0(B.1(x)))))))))))))))
b.0(b.1(a.0(b.1(x)))) → b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.1(x))))))))
b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(b.0(B.1(x))))))))))) → b.1(a.0(b.0(b.0(b.0(b.1(a.0(b.1(a.0(b.1(a.0(b.0(B.1(x)))))))))))))
b.0(a.1(a.0(b.1(a.0(b.0(a.1(a.0(b.1(a.0(b.1(a.0(b.0(B.0(x)))))))))))))) → b.1(a.0(b.0(b.0(b.0(b.0(B.0(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
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ QDPToSRSProof
↳ QTRS
↳ QTRS Reverse
↳ QTRS
↳ QTRS Reverse
↳ QTRS Reverse
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ SemLabProof
↳ SemLabProof2
↳ QDP
↳ SemLabProof
↳ SemLabProof2
↳ QDP
↳ Narrowing
↳ QDP
↳ SemLabProof
↳ SemLabProof2
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
B1(a(a(b(a(b(a(b(b(a(a(b(a(b(a(a(b(a(b(a(b(B(x0)))))))))))))))))))))) → B1(b(b(b(a(b(b(b(b(B(x0))))))))))
B1(a(a(b(a(b(a(b(b(a(a(b(a(b(a(b(B(x0))))))))))))))))) → B1(b(b(b(a(b(b(b(B(x0)))))))))
B1(a(a(b(a(b(a(b(b(b(b(B(x0)))))))))))) → B1(b(b(a(b(a(b(a(b(b(a(b(a(b(a(b(B(x0)))))))))))))))))
B1(a(a(b(a(b(a(b(b(a(a(b(a(b(a(b(b(a(a(b(a(b(a(b(B(x0))))))))))))))))))))))))) → B1(b(b(b(a(b(b(b(b(a(b(b(B(x0)))))))))))))
B1(a(a(b(a(b(a(b(b(b(b(b(B(x0))))))))))))) → B1(b(b(b(a(b(a(b(a(b(b(a(b(a(b(a(b(B(x0))))))))))))))))))
B1(a(a(b(a(b(a(b(b(b(b(x0))))))))))) → B1(b(b(b(a(b(a(b(a(b(b(x0)))))))))))
B1(a(a(b(a(b(a(b(b(a(a(b(a(b(a(b(B(x))))))))))))))))) → B1(b(a(b(b(B(x))))))
B1(a(a(b(a(b(a(b(b(b(b(a(a(b(a(b(a(b(B(x0))))))))))))))))))) → B1(b(b(b(a(b(a(b(a(b(b(a(b(b(B(x0)))))))))))))))
B1(b(b(a(a(b(a(b(a(b(B(x))))))))))) → B1(b(a(b(b(B(x))))))
B1(a(a(b(a(b(a(b(b(a(a(b(a(b(a(b(B(y0))))))))))))))))) → B1(b(b(a(a(b(a(b(a(b(b(B(y0))))))))))))
B1(a(a(b(a(b(a(b(b(a(a(b(a(b(a(b(b(x0))))))))))))))))) → B1(b(b(b(a(b(b(b(b(x0)))))))))
B1(a(a(b(a(b(a(b(b(b(a(a(b(a(b(a(b(B(x0)))))))))))))))))) → B1(b(b(b(a(a(b(a(b(a(b(b(B(x0)))))))))))))
B1(a(a(b(a(b(a(b(b(a(a(b(a(b(a(b(B(x0))))))))))))))))) → B1(b(b(b(a(b(B(x0)))))))
B1(b(a(b(x)))) → B1(a(a(b(a(b(a(b(x))))))))
B1(a(a(b(a(b(a(b(b(x))))))))) → B1(b(b(x)))
B1(a(a(b(a(b(a(b(b(b(a(b(x0)))))))))))) → B1(b(b(b(a(a(b(a(b(a(b(x0)))))))))))
B1(a(a(b(a(b(a(b(b(a(a(b(a(b(a(b(B(y0))))))))))))))))) → B1(b(a(a(b(a(b(a(b(b(B(y0)))))))))))
B1(a(a(b(a(b(a(b(b(a(b(x0))))))))))) → B1(b(b(a(a(b(a(b(a(b(x0))))))))))
B1(a(a(b(a(b(a(b(b(b(x0)))))))))) → B1(b(b(a(b(a(b(a(b(b(x0))))))))))
B1(a(a(b(a(b(a(b(b(a(a(b(a(b(a(b(b(b(B(x0))))))))))))))))))) → B1(b(b(b(a(b(b(b(b(a(b(a(b(a(b(B(x0))))))))))))))))
B1(a(a(b(a(b(a(b(b(b(a(a(b(a(b(a(b(B(x0)))))))))))))))))) → B1(b(b(a(b(a(b(a(b(b(a(b(b(B(x0))))))))))))))
B1(b(a(a(b(a(b(a(b(B(x)))))))))) → B1(a(a(b(a(b(a(b(b(B(x))))))))))
The TRS R consists of the following rules:
b(b(b(x))) → b(a(b(a(b(a(b(b(x))))))))
b(b(a(b(x)))) → b(a(a(b(a(b(a(b(x))))))))
b(a(a(b(a(b(a(b(b(x))))))))) → b(a(b(b(b(b(x))))))
b(b(a(a(b(a(b(a(b(B(x)))))))))) → b(a(a(b(a(b(a(b(b(B(x))))))))))
b(a(a(b(a(b(a(b(b(b(B(x))))))))))) → b(a(b(b(b(b(a(b(a(b(a(b(B(x)))))))))))))
b(a(a(b(a(b(a(b(B(x))))))))) → b(a(b(b(b(B(x))))))
b(b(b(b(B(x))))) → b(a(b(a(b(a(b(b(a(b(a(b(a(b(B(x)))))))))))))))
b(a(a(b(a(b(a(a(b(a(b(a(b(B(x)))))))))))))) → b(a(b(b(b(b(B(x)))))))
b(b(b(a(a(b(a(b(a(b(B(x))))))))))) → b(a(b(a(b(a(b(b(a(b(b(B(x))))))))))))
b(a(a(b(a(b(a(b(B(x))))))))) → b(a(b(B(x))))
b(a(a(b(a(b(a(b(b(a(a(b(a(b(a(b(B(x))))))))))))))))) → b(a(b(b(b(b(a(b(b(B(x))))))))))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.