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(a(a(a(b(x1)))))) → b(a(a(a(b(a(b(x1)))))))
b(b(x1)) → b(a(b(a(b(a(b(x1)))))))
b(a(b(a(a(a(b(a(b(x1))))))))) → b(b(a(a(b(x1)))))
b(a(a(b(x1)))) → b(a(a(a(b(x1)))))
Q is empty.
↳ QTRS
↳ DependencyPairsProof
↳ QTRS Reverse
↳ QTRS Reverse
Q restricted rewrite system:
The TRS R consists of the following rules:
b(b(a(a(a(b(x1)))))) → b(a(a(a(b(a(b(x1)))))))
b(b(x1)) → b(a(b(a(b(a(b(x1)))))))
b(a(b(a(a(a(b(a(b(x1))))))))) → b(b(a(a(b(x1)))))
b(a(a(b(x1)))) → b(a(a(a(b(x1)))))
Q is empty.
Using Dependency Pairs [1,15] we result in the following initial DP problem:
Q DP problem:
The TRS P consists of the following rules:
B(b(x1)) → B(a(b(a(b(a(b(x1)))))))
B(b(x1)) → B(a(b(x1)))
B(a(b(a(a(a(b(a(b(x1))))))))) → B(b(a(a(b(x1)))))
B(b(a(a(a(b(x1)))))) → B(a(a(a(b(a(b(x1)))))))
B(b(a(a(a(b(x1)))))) → B(a(b(x1)))
B(a(b(a(a(a(b(a(b(x1))))))))) → B(a(a(b(x1))))
B(b(x1)) → B(a(b(a(b(x1)))))
B(a(a(b(x1)))) → B(a(a(a(b(x1)))))
The TRS R consists of the following rules:
b(b(a(a(a(b(x1)))))) → b(a(a(a(b(a(b(x1)))))))
b(b(x1)) → b(a(b(a(b(a(b(x1)))))))
b(a(b(a(a(a(b(a(b(x1))))))))) → b(b(a(a(b(x1)))))
b(a(a(b(x1)))) → b(a(a(a(b(x1)))))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QTRS Reverse
↳ QTRS Reverse
Q DP problem:
The TRS P consists of the following rules:
B(b(x1)) → B(a(b(a(b(a(b(x1)))))))
B(b(x1)) → B(a(b(x1)))
B(a(b(a(a(a(b(a(b(x1))))))))) → B(b(a(a(b(x1)))))
B(b(a(a(a(b(x1)))))) → B(a(a(a(b(a(b(x1)))))))
B(b(a(a(a(b(x1)))))) → B(a(b(x1)))
B(a(b(a(a(a(b(a(b(x1))))))))) → B(a(a(b(x1))))
B(b(x1)) → B(a(b(a(b(x1)))))
B(a(a(b(x1)))) → B(a(a(a(b(x1)))))
The TRS R consists of the following rules:
b(b(a(a(a(b(x1)))))) → b(a(a(a(b(a(b(x1)))))))
b(b(x1)) → b(a(b(a(b(a(b(x1)))))))
b(a(b(a(a(a(b(a(b(x1))))))))) → b(b(a(a(b(x1)))))
b(a(a(b(x1)))) → b(a(a(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 3 less nodes.
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QTRS Reverse
↳ QTRS Reverse
Q DP problem:
The TRS P consists of the following rules:
B(b(x1)) → B(a(b(a(b(a(b(x1)))))))
B(b(x1)) → B(a(b(x1)))
B(a(b(a(a(a(b(a(b(x1))))))))) → B(b(a(a(b(x1)))))
B(b(a(a(a(b(x1)))))) → B(a(b(x1)))
B(b(x1)) → B(a(b(a(b(x1)))))
The TRS R consists of the following rules:
b(b(a(a(a(b(x1)))))) → b(a(a(a(b(a(b(x1)))))))
b(b(x1)) → b(a(b(a(b(a(b(x1)))))))
b(a(b(a(a(a(b(a(b(x1))))))))) → b(b(a(a(b(x1)))))
b(a(a(b(x1)))) → b(a(a(a(b(x1)))))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule B(b(x1)) → B(a(b(a(b(a(b(x1))))))) at position [0,0] we obtained the following new rules:
B(b(a(a(b(x0))))) → B(a(b(a(b(a(b(a(a(a(b(x0)))))))))))
B(b(b(x0))) → B(a(b(a(b(a(b(a(b(a(b(a(b(x0)))))))))))))
B(b(a(b(a(a(a(b(a(b(x0)))))))))) → B(a(b(a(b(a(b(b(a(a(b(x0)))))))))))
B(b(a(a(a(b(a(b(x0)))))))) → B(a(b(a(b(b(a(a(b(x0)))))))))
B(b(b(a(a(a(b(x0))))))) → B(a(b(a(b(a(b(a(a(a(b(a(b(x0)))))))))))))
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QTRS Reverse
↳ QTRS Reverse
Q DP problem:
The TRS P consists of the following rules:
B(b(b(x0))) → B(a(b(a(b(a(b(a(b(a(b(a(b(x0)))))))))))))
B(b(a(a(b(x0))))) → B(a(b(a(b(a(b(a(a(a(b(x0)))))))))))
B(b(x1)) → B(a(b(x1)))
B(b(b(a(a(a(b(x0))))))) → B(a(b(a(b(a(b(a(a(a(b(a(b(x0)))))))))))))
B(a(b(a(a(a(b(a(b(x1))))))))) → B(b(a(a(b(x1)))))
B(b(a(a(a(b(x1)))))) → B(a(b(x1)))
B(b(x1)) → B(a(b(a(b(x1)))))
B(b(a(b(a(a(a(b(a(b(x0)))))))))) → B(a(b(a(b(a(b(b(a(a(b(x0)))))))))))
B(b(a(a(a(b(a(b(x0)))))))) → B(a(b(a(b(b(a(a(b(x0)))))))))
The TRS R consists of the following rules:
b(b(a(a(a(b(x1)))))) → b(a(a(a(b(a(b(x1)))))))
b(b(x1)) → b(a(b(a(b(a(b(x1)))))))
b(a(b(a(a(a(b(a(b(x1))))))))) → b(b(a(a(b(x1)))))
b(a(a(b(x1)))) → b(a(a(a(b(x1)))))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule B(b(x1)) → B(a(b(a(b(x1))))) at position [0,0] we obtained the following new rules:
B(b(a(a(b(x0))))) → B(a(b(a(b(a(a(a(b(x0)))))))))
B(b(a(b(a(a(a(b(a(b(x0)))))))))) → B(a(b(a(b(b(a(a(b(x0)))))))))
B(b(b(x0))) → B(a(b(a(b(a(b(a(b(a(b(x0)))))))))))
B(b(b(a(a(a(b(x0))))))) → B(a(b(a(b(a(a(a(b(a(b(x0)))))))))))
B(b(a(a(a(b(a(b(x0)))))))) → B(a(b(b(a(a(b(x0)))))))
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ SemLabProof
↳ SemLabProof2
↳ QTRS Reverse
↳ QTRS Reverse
Q DP problem:
The TRS P consists of the following rules:
B(b(b(x0))) → B(a(b(a(b(a(b(a(b(a(b(a(b(x0)))))))))))))
B(b(a(a(b(x0))))) → B(a(b(a(b(a(b(a(a(a(b(x0)))))))))))
B(b(a(b(a(a(a(b(a(b(x0)))))))))) → B(a(b(a(b(b(a(a(b(x0)))))))))
B(b(b(a(a(a(b(x0))))))) → B(a(b(a(b(a(b(a(a(a(b(a(b(x0)))))))))))))
B(a(b(a(a(a(b(a(b(x1))))))))) → B(b(a(a(b(x1)))))
B(b(b(a(a(a(b(x0))))))) → B(a(b(a(b(a(a(a(b(a(b(x0)))))))))))
B(b(a(a(a(b(a(b(x0)))))))) → B(a(b(b(a(a(b(x0)))))))
B(b(a(a(b(x0))))) → B(a(b(a(b(a(a(a(b(x0)))))))))
B(b(b(x0))) → B(a(b(a(b(a(b(a(b(a(b(x0)))))))))))
B(b(x1)) → B(a(b(x1)))
B(b(a(a(a(b(x1)))))) → B(a(b(x1)))
B(b(a(b(a(a(a(b(a(b(x0)))))))))) → B(a(b(a(b(a(b(b(a(a(b(x0)))))))))))
B(b(a(a(a(b(a(b(x0)))))))) → B(a(b(a(b(b(a(a(b(x0)))))))))
The TRS R consists of the following rules:
b(b(a(a(a(b(x1)))))) → b(a(a(a(b(a(b(x1)))))))
b(b(x1)) → b(a(b(a(b(a(b(x1)))))))
b(a(b(a(a(a(b(a(b(x1))))))))) → b(b(a(a(b(x1)))))
b(a(a(b(x1)))) → b(a(a(a(b(x1)))))
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
b: 0
By semantic labelling [33] we obtain the following labelled TRS:Q DP problem:
The TRS P consists of the following rules:
B.0(b.0(a.1(a.0(b.1(x0))))) → B.1(a.0(b.1(a.0(b.1(a.0(a.1(a.0(b.1(x0)))))))))
B.0(b.1(a.0(a.1(a.0(b.0(x1)))))) → B.1(a.0(b.0(x1)))
B.0(b.0(b.1(x0))) → B.1(a.0(b.1(a.0(b.1(a.0(b.1(a.0(b.1(a.0(b.1(a.0(b.1(x0)))))))))))))
B.0(b.0(b.1(a.0(a.1(a.0(b.1(x0))))))) → B.1(a.0(b.1(a.0(b.1(a.0(a.1(a.0(b.1(a.0(b.1(x0)))))))))))
B.1(a.0(b.1(a.0(a.1(a.0(b.1(a.0(b.0(x1))))))))) → B.0(b.0(a.1(a.0(b.0(x1)))))
B.0(b.1(a.0(a.1(a.0(b.1(a.0(b.1(x0)))))))) → B.1(a.0(b.0(b.0(a.1(a.0(b.1(x0)))))))
B.0(b.1(a.0(a.1(a.0(b.1(a.0(b.0(x0)))))))) → B.1(a.0(b.0(b.0(a.1(a.0(b.0(x0)))))))
B.0(b.1(a.0(a.1(a.0(b.1(a.0(b.0(x0)))))))) → B.1(a.0(b.1(a.0(b.0(b.0(a.1(a.0(b.0(x0)))))))))
B.0(b.0(b.1(a.0(a.1(a.0(b.1(x0))))))) → B.1(a.0(b.1(a.0(b.1(a.0(b.1(a.0(a.1(a.0(b.1(a.0(b.1(x0)))))))))))))
B.0(b.1(a.0(b.1(a.0(a.1(a.0(b.1(a.0(b.0(x0)))))))))) → B.1(a.0(b.1(a.0(b.0(b.0(a.1(a.0(b.0(x0)))))))))
B.0(b.0(x1)) → B.1(a.0(b.0(x1)))
B.0(b.0(a.1(a.0(b.0(x0))))) → B.1(a.0(b.1(a.0(b.1(a.0(b.1(a.0(a.1(a.0(b.0(x0)))))))))))
B.0(b.1(a.0(b.1(a.0(a.1(a.0(b.1(a.0(b.0(x0)))))))))) → B.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(a.1(a.0(b.0(x0)))))))))))
B.0(b.0(a.1(a.0(b.1(x0))))) → B.1(a.0(b.1(a.0(b.1(a.0(b.1(a.0(a.1(a.0(b.1(x0)))))))))))
B.0(b.0(b.1(a.0(a.1(a.0(b.0(x0))))))) → B.1(a.0(b.1(a.0(b.1(a.0(b.1(a.0(a.1(a.0(b.1(a.0(b.0(x0)))))))))))))
B.0(b.1(a.0(a.1(a.0(b.1(a.0(b.1(x0)))))))) → B.1(a.0(b.1(a.0(b.0(b.0(a.1(a.0(b.1(x0)))))))))
B.0(b.0(b.0(x0))) → B.1(a.0(b.1(a.0(b.1(a.0(b.1(a.0(b.1(a.0(b.0(x0)))))))))))
B.0(b.0(b.1(x0))) → B.1(a.0(b.1(a.0(b.1(a.0(b.1(a.0(b.1(a.0(b.1(x0)))))))))))
B.0(b.0(b.0(x0))) → B.1(a.0(b.1(a.0(b.1(a.0(b.1(a.0(b.1(a.0(b.1(a.0(b.0(x0)))))))))))))
B.0(b.1(x1)) → B.1(a.0(b.1(x1)))
B.0(b.1(a.0(a.1(a.0(b.1(x1)))))) → B.1(a.0(b.1(x1)))
B.1(a.0(b.1(a.0(a.1(a.0(b.1(a.0(b.1(x1))))))))) → B.0(b.0(a.1(a.0(b.1(x1)))))
B.0(b.0(a.1(a.0(b.0(x0))))) → B.1(a.0(b.1(a.0(b.1(a.0(a.1(a.0(b.0(x0)))))))))
B.0(b.0(b.1(a.0(a.1(a.0(b.0(x0))))))) → B.1(a.0(b.1(a.0(b.1(a.0(a.1(a.0(b.1(a.0(b.0(x0)))))))))))
B.0(b.1(a.0(b.1(a.0(a.1(a.0(b.1(a.0(b.1(x0)))))))))) → B.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(a.1(a.0(b.1(x0)))))))))))
B.0(b.1(a.0(b.1(a.0(a.1(a.0(b.1(a.0(b.1(x0)))))))))) → B.1(a.0(b.1(a.0(b.0(b.0(a.1(a.0(b.1(x0)))))))))
The TRS R consists of the following rules:
b.1(a.0(b.1(a.0(a.1(a.0(b.1(a.0(b.0(x1))))))))) → b.0(b.0(a.1(a.0(b.0(x1)))))
b.0(a.1(a.0(b.1(x1)))) → b.1(a.0(a.1(a.0(b.1(x1)))))
b.1(a.0(b.1(a.0(a.1(a.0(b.1(a.0(b.1(x1))))))))) → b.0(b.0(a.1(a.0(b.1(x1)))))
b.0(b.1(a.0(a.1(a.0(b.0(x1)))))) → b.1(a.0(a.1(a.0(b.1(a.0(b.0(x1)))))))
b.0(b.1(x1)) → b.1(a.0(b.1(a.0(b.1(a.0(b.1(x1)))))))
b.0(b.1(a.0(a.1(a.0(b.1(x1)))))) → b.1(a.0(a.1(a.0(b.1(a.0(b.1(x1)))))))
b.0(a.1(a.0(b.0(x1)))) → b.1(a.0(a.1(a.0(b.0(x1)))))
b.0(b.0(x1)) → b.1(a.0(b.1(a.0(b.1(a.0(b.0(x1)))))))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ SemLabProof
↳ QDP
↳ RuleRemovalProof
↳ SemLabProof2
↳ QTRS Reverse
↳ QTRS Reverse
Q DP problem:
The TRS P consists of the following rules:
B.0(b.0(a.1(a.0(b.1(x0))))) → B.1(a.0(b.1(a.0(b.1(a.0(a.1(a.0(b.1(x0)))))))))
B.0(b.1(a.0(a.1(a.0(b.0(x1)))))) → B.1(a.0(b.0(x1)))
B.0(b.0(b.1(x0))) → B.1(a.0(b.1(a.0(b.1(a.0(b.1(a.0(b.1(a.0(b.1(a.0(b.1(x0)))))))))))))
B.0(b.0(b.1(a.0(a.1(a.0(b.1(x0))))))) → B.1(a.0(b.1(a.0(b.1(a.0(a.1(a.0(b.1(a.0(b.1(x0)))))))))))
B.1(a.0(b.1(a.0(a.1(a.0(b.1(a.0(b.0(x1))))))))) → B.0(b.0(a.1(a.0(b.0(x1)))))
B.0(b.1(a.0(a.1(a.0(b.1(a.0(b.1(x0)))))))) → B.1(a.0(b.0(b.0(a.1(a.0(b.1(x0)))))))
B.0(b.1(a.0(a.1(a.0(b.1(a.0(b.0(x0)))))))) → B.1(a.0(b.0(b.0(a.1(a.0(b.0(x0)))))))
B.0(b.1(a.0(a.1(a.0(b.1(a.0(b.0(x0)))))))) → B.1(a.0(b.1(a.0(b.0(b.0(a.1(a.0(b.0(x0)))))))))
B.0(b.0(b.1(a.0(a.1(a.0(b.1(x0))))))) → B.1(a.0(b.1(a.0(b.1(a.0(b.1(a.0(a.1(a.0(b.1(a.0(b.1(x0)))))))))))))
B.0(b.1(a.0(b.1(a.0(a.1(a.0(b.1(a.0(b.0(x0)))))))))) → B.1(a.0(b.1(a.0(b.0(b.0(a.1(a.0(b.0(x0)))))))))
B.0(b.0(x1)) → B.1(a.0(b.0(x1)))
B.0(b.0(a.1(a.0(b.0(x0))))) → B.1(a.0(b.1(a.0(b.1(a.0(b.1(a.0(a.1(a.0(b.0(x0)))))))))))
B.0(b.1(a.0(b.1(a.0(a.1(a.0(b.1(a.0(b.0(x0)))))))))) → B.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(a.1(a.0(b.0(x0)))))))))))
B.0(b.0(a.1(a.0(b.1(x0))))) → B.1(a.0(b.1(a.0(b.1(a.0(b.1(a.0(a.1(a.0(b.1(x0)))))))))))
B.0(b.0(b.1(a.0(a.1(a.0(b.0(x0))))))) → B.1(a.0(b.1(a.0(b.1(a.0(b.1(a.0(a.1(a.0(b.1(a.0(b.0(x0)))))))))))))
B.0(b.1(a.0(a.1(a.0(b.1(a.0(b.1(x0)))))))) → B.1(a.0(b.1(a.0(b.0(b.0(a.1(a.0(b.1(x0)))))))))
B.0(b.0(b.0(x0))) → B.1(a.0(b.1(a.0(b.1(a.0(b.1(a.0(b.1(a.0(b.0(x0)))))))))))
B.0(b.0(b.1(x0))) → B.1(a.0(b.1(a.0(b.1(a.0(b.1(a.0(b.1(a.0(b.1(x0)))))))))))
B.0(b.0(b.0(x0))) → B.1(a.0(b.1(a.0(b.1(a.0(b.1(a.0(b.1(a.0(b.1(a.0(b.0(x0)))))))))))))
B.0(b.1(x1)) → B.1(a.0(b.1(x1)))
B.0(b.1(a.0(a.1(a.0(b.1(x1)))))) → B.1(a.0(b.1(x1)))
B.1(a.0(b.1(a.0(a.1(a.0(b.1(a.0(b.1(x1))))))))) → B.0(b.0(a.1(a.0(b.1(x1)))))
B.0(b.0(a.1(a.0(b.0(x0))))) → B.1(a.0(b.1(a.0(b.1(a.0(a.1(a.0(b.0(x0)))))))))
B.0(b.0(b.1(a.0(a.1(a.0(b.0(x0))))))) → B.1(a.0(b.1(a.0(b.1(a.0(a.1(a.0(b.1(a.0(b.0(x0)))))))))))
B.0(b.1(a.0(b.1(a.0(a.1(a.0(b.1(a.0(b.1(x0)))))))))) → B.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(a.1(a.0(b.1(x0)))))))))))
B.0(b.1(a.0(b.1(a.0(a.1(a.0(b.1(a.0(b.1(x0)))))))))) → B.1(a.0(b.1(a.0(b.0(b.0(a.1(a.0(b.1(x0)))))))))
The TRS R consists of the following rules:
b.1(a.0(b.1(a.0(a.1(a.0(b.1(a.0(b.0(x1))))))))) → b.0(b.0(a.1(a.0(b.0(x1)))))
b.0(a.1(a.0(b.1(x1)))) → b.1(a.0(a.1(a.0(b.1(x1)))))
b.1(a.0(b.1(a.0(a.1(a.0(b.1(a.0(b.1(x1))))))))) → b.0(b.0(a.1(a.0(b.1(x1)))))
b.0(b.1(a.0(a.1(a.0(b.0(x1)))))) → b.1(a.0(a.1(a.0(b.1(a.0(b.0(x1)))))))
b.0(b.1(x1)) → b.1(a.0(b.1(a.0(b.1(a.0(b.1(x1)))))))
b.0(b.1(a.0(a.1(a.0(b.1(x1)))))) → b.1(a.0(a.1(a.0(b.1(a.0(b.1(x1)))))))
b.0(a.1(a.0(b.0(x1)))) → b.1(a.0(a.1(a.0(b.0(x1)))))
b.0(b.0(x1)) → b.1(a.0(b.1(a.0(b.1(a.0(b.0(x1)))))))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By using the rule removal processor [15] with the following polynomial ordering [25], at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented.
Strictly oriented dependency pairs:
B.0(b.1(a.0(a.1(a.0(b.0(x1)))))) → B.1(a.0(b.0(x1)))
B.0(b.1(a.0(a.1(a.0(b.1(x1)))))) → B.1(a.0(b.1(x1)))
Used ordering: POLO with Polynomial interpretation [25]:
POL(B.0(x1)) = x1
POL(B.1(x1)) = x1
POL(a.0(x1)) = x1
POL(a.1(x1)) = 1 + x1
POL(b.0(x1)) = x1
POL(b.1(x1)) = x1
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ SemLabProof
↳ QDP
↳ RuleRemovalProof
↳ QDP
↳ SemLabProof2
↳ QTRS Reverse
↳ QTRS Reverse
Q DP problem:
The TRS P consists of the following rules:
B.0(b.1(a.0(a.1(a.0(b.1(a.0(b.0(x0)))))))) → B.1(a.0(b.1(a.0(b.0(b.0(a.1(a.0(b.0(x0)))))))))
B.0(b.0(a.1(a.0(b.1(x0))))) → B.1(a.0(b.1(a.0(b.1(a.0(a.1(a.0(b.1(x0)))))))))
B.0(b.0(b.1(a.0(a.1(a.0(b.1(x0))))))) → B.1(a.0(b.1(a.0(b.1(a.0(b.1(a.0(a.1(a.0(b.1(a.0(b.1(x0)))))))))))))
B.0(b.1(a.0(b.1(a.0(a.1(a.0(b.1(a.0(b.0(x0)))))))))) → B.1(a.0(b.1(a.0(b.0(b.0(a.1(a.0(b.0(x0)))))))))
B.0(b.0(x1)) → B.1(a.0(b.0(x1)))
B.0(b.0(b.1(x0))) → B.1(a.0(b.1(a.0(b.1(a.0(b.1(a.0(b.1(a.0(b.1(a.0(b.1(x0)))))))))))))
B.0(b.0(a.1(a.0(b.0(x0))))) → B.1(a.0(b.1(a.0(b.1(a.0(b.1(a.0(a.1(a.0(b.0(x0)))))))))))
B.0(b.0(b.1(a.0(a.1(a.0(b.1(x0))))))) → B.1(a.0(b.1(a.0(b.1(a.0(a.1(a.0(b.1(a.0(b.1(x0)))))))))))
B.0(b.1(a.0(b.1(a.0(a.1(a.0(b.1(a.0(b.0(x0)))))))))) → B.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(a.1(a.0(b.0(x0)))))))))))
B.0(b.0(a.1(a.0(b.1(x0))))) → B.1(a.0(b.1(a.0(b.1(a.0(b.1(a.0(a.1(a.0(b.1(x0)))))))))))
B.0(b.0(b.1(a.0(a.1(a.0(b.0(x0))))))) → B.1(a.0(b.1(a.0(b.1(a.0(b.1(a.0(a.1(a.0(b.1(a.0(b.0(x0)))))))))))))
B.0(b.0(b.1(x0))) → B.1(a.0(b.1(a.0(b.1(a.0(b.1(a.0(b.1(a.0(b.1(x0)))))))))))
B.0(b.0(b.0(x0))) → B.1(a.0(b.1(a.0(b.1(a.0(b.1(a.0(b.1(a.0(b.0(x0)))))))))))
B.0(b.1(a.0(a.1(a.0(b.1(a.0(b.1(x0)))))))) → B.1(a.0(b.1(a.0(b.0(b.0(a.1(a.0(b.1(x0)))))))))
B.0(b.1(x1)) → B.1(a.0(b.1(x1)))
B.0(b.0(b.0(x0))) → B.1(a.0(b.1(a.0(b.1(a.0(b.1(a.0(b.1(a.0(b.1(a.0(b.0(x0)))))))))))))
B.1(a.0(b.1(a.0(a.1(a.0(b.1(a.0(b.1(x1))))))))) → B.0(b.0(a.1(a.0(b.1(x1)))))
B.1(a.0(b.1(a.0(a.1(a.0(b.1(a.0(b.0(x1))))))))) → B.0(b.0(a.1(a.0(b.0(x1)))))
B.0(b.0(b.1(a.0(a.1(a.0(b.0(x0))))))) → B.1(a.0(b.1(a.0(b.1(a.0(a.1(a.0(b.1(a.0(b.0(x0)))))))))))
B.0(b.0(a.1(a.0(b.0(x0))))) → B.1(a.0(b.1(a.0(b.1(a.0(a.1(a.0(b.0(x0)))))))))
B.0(b.1(a.0(b.1(a.0(a.1(a.0(b.1(a.0(b.1(x0)))))))))) → B.1(a.0(b.1(a.0(b.0(b.0(a.1(a.0(b.1(x0)))))))))
B.0(b.1(a.0(b.1(a.0(a.1(a.0(b.1(a.0(b.1(x0)))))))))) → B.1(a.0(b.1(a.0(b.1(a.0(b.0(b.0(a.1(a.0(b.1(x0)))))))))))
B.0(b.1(a.0(a.1(a.0(b.1(a.0(b.1(x0)))))))) → B.1(a.0(b.0(b.0(a.1(a.0(b.1(x0)))))))
B.0(b.1(a.0(a.1(a.0(b.1(a.0(b.0(x0)))))))) → B.1(a.0(b.0(b.0(a.1(a.0(b.0(x0)))))))
The TRS R consists of the following rules:
b.1(a.0(b.1(a.0(a.1(a.0(b.1(a.0(b.0(x1))))))))) → b.0(b.0(a.1(a.0(b.0(x1)))))
b.0(a.1(a.0(b.1(x1)))) → b.1(a.0(a.1(a.0(b.1(x1)))))
b.1(a.0(b.1(a.0(a.1(a.0(b.1(a.0(b.1(x1))))))))) → b.0(b.0(a.1(a.0(b.1(x1)))))
b.0(b.1(a.0(a.1(a.0(b.0(x1)))))) → b.1(a.0(a.1(a.0(b.1(a.0(b.0(x1)))))))
b.0(b.1(x1)) → b.1(a.0(b.1(a.0(b.1(a.0(b.1(x1)))))))
b.0(b.1(a.0(a.1(a.0(b.1(x1)))))) → b.1(a.0(a.1(a.0(b.1(a.0(b.1(x1)))))))
b.0(a.1(a.0(b.0(x1)))) → b.1(a.0(a.1(a.0(b.0(x1)))))
b.0(b.0(x1)) → b.1(a.0(b.1(a.0(b.1(a.0(b.0(x1)))))))
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
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ SemLabProof
↳ SemLabProof2
↳ QDP
↳ QDPToSRSProof
↳ QTRS Reverse
↳ QTRS Reverse
Q DP problem:
The TRS P consists of the following rules:
B(b(a(a(b(x0))))) → B(a(b(a(b(a(a(a(b(x0)))))))))
B(b(a(a(b(x0))))) → B(a(b(a(b(a(b(a(a(a(b(x0)))))))))))
B(b(b(x0))) → B(a(b(a(b(a(b(a(b(a(b(a(b(x0)))))))))))))
B(b(b(x0))) → B(a(b(a(b(a(b(a(b(a(b(x0)))))))))))
B(b(a(b(a(a(a(b(a(b(x0)))))))))) → B(a(b(a(b(b(a(a(b(x0)))))))))
B(b(x1)) → B(a(b(x1)))
B(a(b(a(a(a(b(a(b(x1))))))))) → B(b(a(a(b(x1)))))
B(b(b(a(a(a(b(x0))))))) → B(a(b(a(b(a(b(a(a(a(b(a(b(x0)))))))))))))
B(b(b(a(a(a(b(x0))))))) → B(a(b(a(b(a(a(a(b(a(b(x0)))))))))))
B(b(a(a(a(b(a(b(x0)))))))) → B(a(b(b(a(a(b(x0)))))))
B(b(a(b(a(a(a(b(a(b(x0)))))))))) → B(a(b(a(b(a(b(b(a(a(b(x0)))))))))))
B(b(a(a(a(b(a(b(x0)))))))) → B(a(b(a(b(b(a(a(b(x0)))))))))
The TRS R consists of the following rules:
b(b(a(a(a(b(x1)))))) → b(a(a(a(b(a(b(x1)))))))
b(b(x1)) → b(a(b(a(b(a(b(x1)))))))
b(a(b(a(a(a(b(a(b(x1))))))))) → b(b(a(a(b(x1)))))
b(a(a(b(x1)))) → b(a(a(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
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ SemLabProof
↳ SemLabProof2
↳ QDP
↳ QDPToSRSProof
↳ QTRS
↳ QTRS Reverse
↳ QTRS Reverse
↳ QTRS Reverse
Q restricted rewrite system:
The TRS R consists of the following rules:
b(b(a(a(a(b(x1)))))) → b(a(a(a(b(a(b(x1)))))))
b(b(x1)) → b(a(b(a(b(a(b(x1)))))))
b(a(b(a(a(a(b(a(b(x1))))))))) → b(b(a(a(b(x1)))))
b(a(a(b(x1)))) → b(a(a(a(b(x1)))))
B(b(a(a(b(x0))))) → B(a(b(a(b(a(a(a(b(x0)))))))))
B(b(a(a(b(x0))))) → B(a(b(a(b(a(b(a(a(a(b(x0)))))))))))
B(b(b(x0))) → B(a(b(a(b(a(b(a(b(a(b(a(b(x0)))))))))))))
B(b(b(x0))) → B(a(b(a(b(a(b(a(b(a(b(x0)))))))))))
B(b(a(b(a(a(a(b(a(b(x0)))))))))) → B(a(b(a(b(b(a(a(b(x0)))))))))
B(b(x1)) → B(a(b(x1)))
B(a(b(a(a(a(b(a(b(x1))))))))) → B(b(a(a(b(x1)))))
B(b(b(a(a(a(b(x0))))))) → B(a(b(a(b(a(b(a(a(a(b(a(b(x0)))))))))))))
B(b(b(a(a(a(b(x0))))))) → B(a(b(a(b(a(a(a(b(a(b(x0)))))))))))
B(b(a(a(a(b(a(b(x0)))))))) → B(a(b(b(a(a(b(x0)))))))
B(b(a(b(a(a(a(b(a(b(x0)))))))))) → B(a(b(a(b(a(b(b(a(a(b(x0)))))))))))
B(b(a(a(a(b(a(b(x0)))))))) → B(a(b(a(b(b(a(a(b(x0)))))))))
Q is empty.
We have reversed the following QTRS:
The set of rules R is
b(b(a(a(a(b(x1)))))) → b(a(a(a(b(a(b(x1)))))))
b(b(x1)) → b(a(b(a(b(a(b(x1)))))))
b(a(b(a(a(a(b(a(b(x1))))))))) → b(b(a(a(b(x1)))))
b(a(a(b(x1)))) → b(a(a(a(b(x1)))))
B(b(a(a(b(x0))))) → B(a(b(a(b(a(a(a(b(x0)))))))))
B(b(a(a(b(x0))))) → B(a(b(a(b(a(b(a(a(a(b(x0)))))))))))
B(b(b(x0))) → B(a(b(a(b(a(b(a(b(a(b(a(b(x0)))))))))))))
B(b(b(x0))) → B(a(b(a(b(a(b(a(b(a(b(x0)))))))))))
B(b(a(b(a(a(a(b(a(b(x0)))))))))) → B(a(b(a(b(b(a(a(b(x0)))))))))
B(b(x1)) → B(a(b(x1)))
B(a(b(a(a(a(b(a(b(x1))))))))) → B(b(a(a(b(x1)))))
B(b(b(a(a(a(b(x0))))))) → B(a(b(a(b(a(b(a(a(a(b(a(b(x0)))))))))))))
B(b(b(a(a(a(b(x0))))))) → B(a(b(a(b(a(a(a(b(a(b(x0)))))))))))
B(b(a(a(a(b(a(b(x0)))))))) → B(a(b(b(a(a(b(x0)))))))
B(b(a(b(a(a(a(b(a(b(x0)))))))))) → B(a(b(a(b(a(b(b(a(a(b(x0)))))))))))
B(b(a(a(a(b(a(b(x0)))))))) → B(a(b(a(b(b(a(a(b(x0)))))))))
The set Q is empty.
We have obtained the following QTRS:
b(a(a(a(b(b(x)))))) → b(a(b(a(a(a(b(x)))))))
b(b(x)) → b(a(b(a(b(a(b(x)))))))
b(a(b(a(a(a(b(a(b(x))))))))) → b(a(a(b(b(x)))))
b(a(a(b(x)))) → b(a(a(a(b(x)))))
b(a(a(b(B(x))))) → b(a(a(a(b(a(b(a(B(x)))))))))
b(a(a(b(B(x))))) → b(a(a(a(b(a(b(a(b(a(B(x)))))))))))
b(b(B(x))) → b(a(b(a(b(a(b(a(b(a(b(a(B(x)))))))))))))
b(b(B(x))) → b(a(b(a(b(a(b(a(b(a(B(x)))))))))))
b(a(b(a(a(a(b(a(b(B(x)))))))))) → b(a(a(b(b(a(b(a(B(x)))))))))
b(B(x)) → b(a(B(x)))
b(a(b(a(a(a(b(a(B(x))))))))) → b(a(a(b(B(x)))))
b(a(a(a(b(b(B(x))))))) → b(a(b(a(a(a(b(a(b(a(b(a(B(x)))))))))))))
b(a(a(a(b(b(B(x))))))) → b(a(b(a(a(a(b(a(b(a(B(x)))))))))))
b(a(b(a(a(a(b(B(x)))))))) → b(a(a(b(b(a(B(x)))))))
b(a(b(a(a(a(b(a(b(B(x)))))))))) → b(a(a(b(b(a(b(a(b(a(B(x)))))))))))
b(a(b(a(a(a(b(B(x)))))))) → b(a(a(b(b(a(b(a(B(x)))))))))
The set Q is empty.
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ SemLabProof
↳ SemLabProof2
↳ QDP
↳ QDPToSRSProof
↳ QTRS
↳ QTRS Reverse
↳ QTRS
↳ QTRS Reverse
↳ QTRS Reverse
↳ DependencyPairsProof
↳ QTRS Reverse
↳ QTRS Reverse
Q restricted rewrite system:
The TRS R consists of the following rules:
b(a(a(a(b(b(x)))))) → b(a(b(a(a(a(b(x)))))))
b(b(x)) → b(a(b(a(b(a(b(x)))))))
b(a(b(a(a(a(b(a(b(x))))))))) → b(a(a(b(b(x)))))
b(a(a(b(x)))) → b(a(a(a(b(x)))))
b(a(a(b(B(x))))) → b(a(a(a(b(a(b(a(B(x)))))))))
b(a(a(b(B(x))))) → b(a(a(a(b(a(b(a(b(a(B(x)))))))))))
b(b(B(x))) → b(a(b(a(b(a(b(a(b(a(b(a(B(x)))))))))))))
b(b(B(x))) → b(a(b(a(b(a(b(a(b(a(B(x)))))))))))
b(a(b(a(a(a(b(a(b(B(x)))))))))) → b(a(a(b(b(a(b(a(B(x)))))))))
b(B(x)) → b(a(B(x)))
b(a(b(a(a(a(b(a(B(x))))))))) → b(a(a(b(B(x)))))
b(a(a(a(b(b(B(x))))))) → b(a(b(a(a(a(b(a(b(a(b(a(B(x)))))))))))))
b(a(a(a(b(b(B(x))))))) → b(a(b(a(a(a(b(a(b(a(B(x)))))))))))
b(a(b(a(a(a(b(B(x)))))))) → b(a(a(b(b(a(B(x)))))))
b(a(b(a(a(a(b(a(b(B(x)))))))))) → b(a(a(b(b(a(b(a(b(a(B(x)))))))))))
b(a(b(a(a(a(b(B(x)))))))) → b(a(a(b(b(a(b(a(B(x)))))))))
Q is empty.
We have reversed the following QTRS:
The set of rules R is
b(a(a(a(b(b(x)))))) → b(a(b(a(a(a(b(x)))))))
b(b(x)) → b(a(b(a(b(a(b(x)))))))
b(a(b(a(a(a(b(a(b(x))))))))) → b(a(a(b(b(x)))))
b(a(a(b(x)))) → b(a(a(a(b(x)))))
b(a(a(b(B(x))))) → b(a(a(a(b(a(b(a(B(x)))))))))
b(a(a(b(B(x))))) → b(a(a(a(b(a(b(a(b(a(B(x)))))))))))
b(b(B(x))) → b(a(b(a(b(a(b(a(b(a(b(a(B(x)))))))))))))
b(b(B(x))) → b(a(b(a(b(a(b(a(b(a(B(x)))))))))))
b(a(b(a(a(a(b(a(b(B(x)))))))))) → b(a(a(b(b(a(b(a(B(x)))))))))
b(B(x)) → b(a(B(x)))
b(a(b(a(a(a(b(a(B(x))))))))) → b(a(a(b(B(x)))))
b(a(a(a(b(b(B(x))))))) → b(a(b(a(a(a(b(a(b(a(b(a(B(x)))))))))))))
b(a(a(a(b(b(B(x))))))) → b(a(b(a(a(a(b(a(b(a(B(x)))))))))))
b(a(b(a(a(a(b(B(x)))))))) → b(a(a(b(b(a(B(x)))))))
b(a(b(a(a(a(b(a(b(B(x)))))))))) → b(a(a(b(b(a(b(a(b(a(B(x)))))))))))
b(a(b(a(a(a(b(B(x)))))))) → b(a(a(b(b(a(b(a(B(x)))))))))
The set Q is empty.
We have obtained the following QTRS:
b(b(a(a(a(b(x)))))) → b(a(a(a(b(a(b(x)))))))
b(b(x)) → b(a(b(a(b(a(b(x)))))))
b(a(b(a(a(a(b(a(b(x))))))))) → b(b(a(a(b(x)))))
b(a(a(b(x)))) → b(a(a(a(b(x)))))
B(b(a(a(b(x))))) → B(a(b(a(b(a(a(a(b(x)))))))))
B(b(a(a(b(x))))) → B(a(b(a(b(a(b(a(a(a(b(x)))))))))))
B(b(b(x))) → B(a(b(a(b(a(b(a(b(a(b(a(b(x)))))))))))))
B(b(b(x))) → B(a(b(a(b(a(b(a(b(a(b(x)))))))))))
B(b(a(b(a(a(a(b(a(b(x)))))))))) → B(a(b(a(b(b(a(a(b(x)))))))))
B(b(x)) → B(a(b(x)))
B(a(b(a(a(a(b(a(b(x))))))))) → B(b(a(a(b(x)))))
B(b(b(a(a(a(b(x))))))) → B(a(b(a(b(a(b(a(a(a(b(a(b(x)))))))))))))
B(b(b(a(a(a(b(x))))))) → B(a(b(a(b(a(a(a(b(a(b(x)))))))))))
B(b(a(a(a(b(a(b(x)))))))) → B(a(b(b(a(a(b(x)))))))
B(b(a(b(a(a(a(b(a(b(x)))))))))) → B(a(b(a(b(a(b(b(a(a(b(x)))))))))))
B(b(a(a(a(b(a(b(x)))))))) → B(a(b(a(b(b(a(a(b(x)))))))))
The set Q is empty.
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ SemLabProof
↳ SemLabProof2
↳ QDP
↳ QDPToSRSProof
↳ QTRS
↳ QTRS Reverse
↳ QTRS
↳ QTRS Reverse
↳ QTRS
↳ QTRS Reverse
↳ DependencyPairsProof
↳ QTRS Reverse
↳ QTRS Reverse
Q restricted rewrite system:
The TRS R consists of the following rules:
b(b(a(a(a(b(x)))))) → b(a(a(a(b(a(b(x)))))))
b(b(x)) → b(a(b(a(b(a(b(x)))))))
b(a(b(a(a(a(b(a(b(x))))))))) → b(b(a(a(b(x)))))
b(a(a(b(x)))) → b(a(a(a(b(x)))))
B(b(a(a(b(x))))) → B(a(b(a(b(a(a(a(b(x)))))))))
B(b(a(a(b(x))))) → B(a(b(a(b(a(b(a(a(a(b(x)))))))))))
B(b(b(x))) → B(a(b(a(b(a(b(a(b(a(b(a(b(x)))))))))))))
B(b(b(x))) → B(a(b(a(b(a(b(a(b(a(b(x)))))))))))
B(b(a(b(a(a(a(b(a(b(x)))))))))) → B(a(b(a(b(b(a(a(b(x)))))))))
B(b(x)) → B(a(b(x)))
B(a(b(a(a(a(b(a(b(x))))))))) → B(b(a(a(b(x)))))
B(b(b(a(a(a(b(x))))))) → B(a(b(a(b(a(b(a(a(a(b(a(b(x)))))))))))))
B(b(b(a(a(a(b(x))))))) → B(a(b(a(b(a(a(a(b(a(b(x)))))))))))
B(b(a(a(a(b(a(b(x)))))))) → B(a(b(b(a(a(b(x)))))))
B(b(a(b(a(a(a(b(a(b(x)))))))))) → B(a(b(a(b(a(b(b(a(a(b(x)))))))))))
B(b(a(a(a(b(a(b(x)))))))) → B(a(b(a(b(b(a(a(b(x)))))))))
Q is empty.
We have reversed the following QTRS:
The set of rules R is
b(a(a(a(b(b(x)))))) → b(a(b(a(a(a(b(x)))))))
b(b(x)) → b(a(b(a(b(a(b(x)))))))
b(a(b(a(a(a(b(a(b(x))))))))) → b(a(a(b(b(x)))))
b(a(a(b(x)))) → b(a(a(a(b(x)))))
b(a(a(b(B(x))))) → b(a(a(a(b(a(b(a(B(x)))))))))
b(a(a(b(B(x))))) → b(a(a(a(b(a(b(a(b(a(B(x)))))))))))
b(b(B(x))) → b(a(b(a(b(a(b(a(b(a(b(a(B(x)))))))))))))
b(b(B(x))) → b(a(b(a(b(a(b(a(b(a(B(x)))))))))))
b(a(b(a(a(a(b(a(b(B(x)))))))))) → b(a(a(b(b(a(b(a(B(x)))))))))
b(B(x)) → b(a(B(x)))
b(a(b(a(a(a(b(a(B(x))))))))) → b(a(a(b(B(x)))))
b(a(a(a(b(b(B(x))))))) → b(a(b(a(a(a(b(a(b(a(b(a(B(x)))))))))))))
b(a(a(a(b(b(B(x))))))) → b(a(b(a(a(a(b(a(b(a(B(x)))))))))))
b(a(b(a(a(a(b(B(x)))))))) → b(a(a(b(b(a(B(x)))))))
b(a(b(a(a(a(b(a(b(B(x)))))))))) → b(a(a(b(b(a(b(a(b(a(B(x)))))))))))
b(a(b(a(a(a(b(B(x)))))))) → b(a(a(b(b(a(b(a(B(x)))))))))
The set Q is empty.
We have obtained the following QTRS:
b(b(a(a(a(b(x)))))) → b(a(a(a(b(a(b(x)))))))
b(b(x)) → b(a(b(a(b(a(b(x)))))))
b(a(b(a(a(a(b(a(b(x))))))))) → b(b(a(a(b(x)))))
b(a(a(b(x)))) → b(a(a(a(b(x)))))
B(b(a(a(b(x))))) → B(a(b(a(b(a(a(a(b(x)))))))))
B(b(a(a(b(x))))) → B(a(b(a(b(a(b(a(a(a(b(x)))))))))))
B(b(b(x))) → B(a(b(a(b(a(b(a(b(a(b(a(b(x)))))))))))))
B(b(b(x))) → B(a(b(a(b(a(b(a(b(a(b(x)))))))))))
B(b(a(b(a(a(a(b(a(b(x)))))))))) → B(a(b(a(b(b(a(a(b(x)))))))))
B(b(x)) → B(a(b(x)))
B(a(b(a(a(a(b(a(b(x))))))))) → B(b(a(a(b(x)))))
B(b(b(a(a(a(b(x))))))) → B(a(b(a(b(a(b(a(a(a(b(a(b(x)))))))))))))
B(b(b(a(a(a(b(x))))))) → B(a(b(a(b(a(a(a(b(a(b(x)))))))))))
B(b(a(a(a(b(a(b(x)))))))) → B(a(b(b(a(a(b(x)))))))
B(b(a(b(a(a(a(b(a(b(x)))))))))) → B(a(b(a(b(a(b(b(a(a(b(x)))))))))))
B(b(a(a(a(b(a(b(x)))))))) → B(a(b(a(b(b(a(a(b(x)))))))))
The set Q is empty.
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ SemLabProof
↳ SemLabProof2
↳ QDP
↳ QDPToSRSProof
↳ QTRS
↳ QTRS Reverse
↳ QTRS
↳ QTRS Reverse
↳ QTRS Reverse
↳ QTRS
↳ DependencyPairsProof
↳ QTRS Reverse
↳ QTRS Reverse
Q restricted rewrite system:
The TRS R consists of the following rules:
b(b(a(a(a(b(x)))))) → b(a(a(a(b(a(b(x)))))))
b(b(x)) → b(a(b(a(b(a(b(x)))))))
b(a(b(a(a(a(b(a(b(x))))))))) → b(b(a(a(b(x)))))
b(a(a(b(x)))) → b(a(a(a(b(x)))))
B(b(a(a(b(x))))) → B(a(b(a(b(a(a(a(b(x)))))))))
B(b(a(a(b(x))))) → B(a(b(a(b(a(b(a(a(a(b(x)))))))))))
B(b(b(x))) → B(a(b(a(b(a(b(a(b(a(b(a(b(x)))))))))))))
B(b(b(x))) → B(a(b(a(b(a(b(a(b(a(b(x)))))))))))
B(b(a(b(a(a(a(b(a(b(x)))))))))) → B(a(b(a(b(b(a(a(b(x)))))))))
B(b(x)) → B(a(b(x)))
B(a(b(a(a(a(b(a(b(x))))))))) → B(b(a(a(b(x)))))
B(b(b(a(a(a(b(x))))))) → B(a(b(a(b(a(b(a(a(a(b(a(b(x)))))))))))))
B(b(b(a(a(a(b(x))))))) → B(a(b(a(b(a(a(a(b(a(b(x)))))))))))
B(b(a(a(a(b(a(b(x)))))))) → B(a(b(b(a(a(b(x)))))))
B(b(a(b(a(a(a(b(a(b(x)))))))))) → B(a(b(a(b(a(b(b(a(a(b(x)))))))))))
B(b(a(a(a(b(a(b(x)))))))) → B(a(b(a(b(b(a(a(b(x)))))))))
Q is empty.
Using Dependency Pairs [1,15] we result in the following initial DP problem:
Q DP problem:
The TRS P consists of the following rules:
B1(a(a(b(B(x))))) → B1(a(b(a(B(x)))))
B1(b(x)) → B1(a(b(x)))
B1(a(a(a(b(b(B(x))))))) → B1(a(b(a(a(a(b(a(b(a(B(x)))))))))))
B1(a(a(a(b(b(x)))))) → B1(a(a(a(b(x)))))
B1(a(b(a(a(a(b(B(x)))))))) → B1(a(b(a(B(x)))))
B1(a(a(a(b(b(B(x))))))) → B1(a(a(a(b(a(b(a(b(a(B(x)))))))))))
B1(a(a(b(B(x))))) → B1(a(b(a(b(a(B(x)))))))
B1(a(b(a(a(a(b(a(b(B(x)))))))))) → B1(b(a(b(a(b(a(B(x))))))))
B1(a(a(b(x)))) → B1(a(a(a(b(x)))))
B1(B(x)) → B1(a(B(x)))
B1(a(a(a(b(b(B(x))))))) → B1(a(B(x)))
B1(a(a(b(B(x))))) → B1(a(a(a(b(a(b(a(b(a(B(x)))))))))))
B1(b(B(x))) → B1(a(b(a(b(a(b(a(b(a(b(a(B(x)))))))))))))
B1(a(b(a(a(a(b(B(x)))))))) → B1(a(a(b(b(a(B(x)))))))
B1(a(a(b(B(x))))) → B1(a(B(x)))
B1(b(B(x))) → B1(a(b(a(b(a(b(a(b(a(B(x)))))))))))
B1(b(x)) → B1(a(b(a(b(a(b(x)))))))
B1(b(B(x))) → B1(a(b(a(b(a(b(a(B(x)))))))))
B1(a(b(a(a(a(b(a(b(B(x)))))))))) → B1(b(a(b(a(B(x))))))
B1(a(b(a(a(a(b(a(b(B(x)))))))))) → B1(a(a(b(b(a(b(a(B(x)))))))))
B1(a(b(a(a(a(b(a(b(B(x)))))))))) → B1(a(b(a(B(x)))))
B1(a(b(a(a(a(b(B(x)))))))) → B1(a(B(x)))
B1(a(a(a(b(b(B(x))))))) → B1(a(b(a(B(x)))))
B1(b(x)) → B1(a(b(a(b(x)))))
B1(a(a(a(b(b(B(x))))))) → B1(a(b(a(b(a(B(x)))))))
B1(a(a(a(b(b(x)))))) → B1(a(b(a(a(a(b(x)))))))
B1(a(b(a(a(a(b(B(x)))))))) → B1(a(a(b(b(a(b(a(B(x)))))))))
B1(b(B(x))) → B1(a(b(a(b(a(B(x)))))))
B1(b(B(x))) → B1(a(B(x)))
B1(a(b(a(a(a(b(a(b(B(x)))))))))) → B1(a(b(a(b(a(B(x)))))))
B1(a(a(a(b(b(B(x))))))) → B1(a(b(a(a(a(b(a(b(a(b(a(B(x)))))))))))))
B1(b(B(x))) → B1(a(b(a(B(x)))))
B1(a(b(a(a(a(b(a(b(x))))))))) → B1(a(a(b(b(x)))))
B1(a(b(a(a(a(b(a(b(B(x)))))))))) → B1(a(B(x)))
B1(a(b(a(a(a(b(a(B(x))))))))) → B1(B(x))
B1(a(a(b(B(x))))) → B1(a(a(a(b(a(b(a(B(x)))))))))
B1(a(b(a(a(a(b(B(x)))))))) → B1(b(a(B(x))))
B1(a(b(a(a(a(b(a(b(B(x)))))))))) → B1(a(a(b(b(a(b(a(b(a(B(x)))))))))))
B1(a(b(a(a(a(b(a(b(x))))))))) → B1(b(x))
B1(a(b(a(a(a(b(B(x)))))))) → B1(b(a(b(a(B(x))))))
B1(a(b(a(a(a(b(a(B(x))))))))) → B1(a(a(b(B(x)))))
B1(a(a(a(b(b(B(x))))))) → B1(a(a(a(b(a(b(a(B(x)))))))))
The TRS R consists of the following rules:
b(a(a(a(b(b(x)))))) → b(a(b(a(a(a(b(x)))))))
b(b(x)) → b(a(b(a(b(a(b(x)))))))
b(a(b(a(a(a(b(a(b(x))))))))) → b(a(a(b(b(x)))))
b(a(a(b(x)))) → b(a(a(a(b(x)))))
b(a(a(b(B(x))))) → b(a(a(a(b(a(b(a(B(x)))))))))
b(a(a(b(B(x))))) → b(a(a(a(b(a(b(a(b(a(B(x)))))))))))
b(b(B(x))) → b(a(b(a(b(a(b(a(b(a(b(a(B(x)))))))))))))
b(b(B(x))) → b(a(b(a(b(a(b(a(b(a(B(x)))))))))))
b(a(b(a(a(a(b(a(b(B(x)))))))))) → b(a(a(b(b(a(b(a(B(x)))))))))
b(B(x)) → b(a(B(x)))
b(a(b(a(a(a(b(a(B(x))))))))) → b(a(a(b(B(x)))))
b(a(a(a(b(b(B(x))))))) → b(a(b(a(a(a(b(a(b(a(b(a(B(x)))))))))))))
b(a(a(a(b(b(B(x))))))) → b(a(b(a(a(a(b(a(b(a(B(x)))))))))))
b(a(b(a(a(a(b(B(x)))))))) → b(a(a(b(b(a(B(x)))))))
b(a(b(a(a(a(b(a(b(B(x)))))))))) → b(a(a(b(b(a(b(a(b(a(B(x)))))))))))
b(a(b(a(a(a(b(B(x)))))))) → b(a(a(b(b(a(b(a(B(x)))))))))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ SemLabProof
↳ SemLabProof2
↳ QDP
↳ QDPToSRSProof
↳ QTRS
↳ QTRS Reverse
↳ QTRS
↳ QTRS Reverse
↳ QTRS Reverse
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QTRS Reverse
↳ QTRS Reverse
Q DP problem:
The TRS P consists of the following rules:
B1(a(a(b(B(x))))) → B1(a(b(a(B(x)))))
B1(b(x)) → B1(a(b(x)))
B1(a(a(a(b(b(B(x))))))) → B1(a(b(a(a(a(b(a(b(a(B(x)))))))))))
B1(a(a(a(b(b(x)))))) → B1(a(a(a(b(x)))))
B1(a(b(a(a(a(b(B(x)))))))) → B1(a(b(a(B(x)))))
B1(a(a(a(b(b(B(x))))))) → B1(a(a(a(b(a(b(a(b(a(B(x)))))))))))
B1(a(a(b(B(x))))) → B1(a(b(a(b(a(B(x)))))))
B1(a(b(a(a(a(b(a(b(B(x)))))))))) → B1(b(a(b(a(b(a(B(x))))))))
B1(a(a(b(x)))) → B1(a(a(a(b(x)))))
B1(B(x)) → B1(a(B(x)))
B1(a(a(a(b(b(B(x))))))) → B1(a(B(x)))
B1(a(a(b(B(x))))) → B1(a(a(a(b(a(b(a(b(a(B(x)))))))))))
B1(b(B(x))) → B1(a(b(a(b(a(b(a(b(a(b(a(B(x)))))))))))))
B1(a(b(a(a(a(b(B(x)))))))) → B1(a(a(b(b(a(B(x)))))))
B1(a(a(b(B(x))))) → B1(a(B(x)))
B1(b(B(x))) → B1(a(b(a(b(a(b(a(b(a(B(x)))))))))))
B1(b(x)) → B1(a(b(a(b(a(b(x)))))))
B1(b(B(x))) → B1(a(b(a(b(a(b(a(B(x)))))))))
B1(a(b(a(a(a(b(a(b(B(x)))))))))) → B1(b(a(b(a(B(x))))))
B1(a(b(a(a(a(b(a(b(B(x)))))))))) → B1(a(a(b(b(a(b(a(B(x)))))))))
B1(a(b(a(a(a(b(a(b(B(x)))))))))) → B1(a(b(a(B(x)))))
B1(a(b(a(a(a(b(B(x)))))))) → B1(a(B(x)))
B1(a(a(a(b(b(B(x))))))) → B1(a(b(a(B(x)))))
B1(b(x)) → B1(a(b(a(b(x)))))
B1(a(a(a(b(b(B(x))))))) → B1(a(b(a(b(a(B(x)))))))
B1(a(a(a(b(b(x)))))) → B1(a(b(a(a(a(b(x)))))))
B1(a(b(a(a(a(b(B(x)))))))) → B1(a(a(b(b(a(b(a(B(x)))))))))
B1(b(B(x))) → B1(a(b(a(b(a(B(x)))))))
B1(b(B(x))) → B1(a(B(x)))
B1(a(b(a(a(a(b(a(b(B(x)))))))))) → B1(a(b(a(b(a(B(x)))))))
B1(a(a(a(b(b(B(x))))))) → B1(a(b(a(a(a(b(a(b(a(b(a(B(x)))))))))))))
B1(b(B(x))) → B1(a(b(a(B(x)))))
B1(a(b(a(a(a(b(a(b(x))))))))) → B1(a(a(b(b(x)))))
B1(a(b(a(a(a(b(a(b(B(x)))))))))) → B1(a(B(x)))
B1(a(b(a(a(a(b(a(B(x))))))))) → B1(B(x))
B1(a(a(b(B(x))))) → B1(a(a(a(b(a(b(a(B(x)))))))))
B1(a(b(a(a(a(b(B(x)))))))) → B1(b(a(B(x))))
B1(a(b(a(a(a(b(a(b(B(x)))))))))) → B1(a(a(b(b(a(b(a(b(a(B(x)))))))))))
B1(a(b(a(a(a(b(a(b(x))))))))) → B1(b(x))
B1(a(b(a(a(a(b(B(x)))))))) → B1(b(a(b(a(B(x))))))
B1(a(b(a(a(a(b(a(B(x))))))))) → B1(a(a(b(B(x)))))
B1(a(a(a(b(b(B(x))))))) → B1(a(a(a(b(a(b(a(B(x)))))))))
The TRS R consists of the following rules:
b(a(a(a(b(b(x)))))) → b(a(b(a(a(a(b(x)))))))
b(b(x)) → b(a(b(a(b(a(b(x)))))))
b(a(b(a(a(a(b(a(b(x))))))))) → b(a(a(b(b(x)))))
b(a(a(b(x)))) → b(a(a(a(b(x)))))
b(a(a(b(B(x))))) → b(a(a(a(b(a(b(a(B(x)))))))))
b(a(a(b(B(x))))) → b(a(a(a(b(a(b(a(b(a(B(x)))))))))))
b(b(B(x))) → b(a(b(a(b(a(b(a(b(a(b(a(B(x)))))))))))))
b(b(B(x))) → b(a(b(a(b(a(b(a(b(a(B(x)))))))))))
b(a(b(a(a(a(b(a(b(B(x)))))))))) → b(a(a(b(b(a(b(a(B(x)))))))))
b(B(x)) → b(a(B(x)))
b(a(b(a(a(a(b(a(B(x))))))))) → b(a(a(b(B(x)))))
b(a(a(a(b(b(B(x))))))) → b(a(b(a(a(a(b(a(b(a(b(a(B(x)))))))))))))
b(a(a(a(b(b(B(x))))))) → b(a(b(a(a(a(b(a(b(a(B(x)))))))))))
b(a(b(a(a(a(b(B(x)))))))) → b(a(a(b(b(a(B(x)))))))
b(a(b(a(a(a(b(a(b(B(x)))))))))) → b(a(a(b(b(a(b(a(b(a(B(x)))))))))))
b(a(b(a(a(a(b(B(x)))))))) → b(a(a(b(b(a(b(a(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 23 less nodes.
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ SemLabProof
↳ SemLabProof2
↳ QDP
↳ QDPToSRSProof
↳ QTRS
↳ QTRS Reverse
↳ QTRS
↳ QTRS Reverse
↳ QTRS Reverse
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QTRS Reverse
↳ QTRS Reverse
Q DP problem:
The TRS P consists of the following rules:
B1(a(a(a(b(b(x)))))) → B1(a(b(a(a(a(b(x)))))))
B1(a(b(a(a(a(b(B(x)))))))) → B1(a(a(b(b(a(b(a(B(x)))))))))
B1(b(x)) → B1(a(b(x)))
B1(a(a(a(b(b(B(x))))))) → B1(a(b(a(a(a(b(a(b(a(B(x)))))))))))
B1(a(a(a(b(b(x)))))) → B1(a(a(a(b(x)))))
B1(a(a(a(b(b(B(x))))))) → B1(a(b(a(a(a(b(a(b(a(b(a(B(x)))))))))))))
B1(a(b(a(a(a(b(a(b(x))))))))) → B1(a(a(b(b(x)))))
B1(a(b(a(a(a(b(a(b(B(x)))))))))) → B1(b(a(b(a(b(a(B(x))))))))
B1(a(a(b(x)))) → B1(a(a(a(b(x)))))
B1(a(b(a(a(a(b(B(x)))))))) → B1(b(a(B(x))))
B1(a(b(a(a(a(b(a(b(x))))))))) → B1(b(x))
B1(a(b(a(a(a(b(a(b(B(x)))))))))) → B1(a(a(b(b(a(b(a(b(a(B(x)))))))))))
B1(a(b(a(a(a(b(B(x)))))))) → B1(a(a(b(b(a(B(x)))))))
B1(a(b(a(a(a(b(B(x)))))))) → B1(b(a(b(a(B(x))))))
B1(b(x)) → B1(a(b(a(b(a(b(x)))))))
B1(a(b(a(a(a(b(a(b(B(x)))))))))) → B1(b(a(b(a(B(x))))))
B1(a(b(a(a(a(b(a(b(B(x)))))))))) → B1(a(a(b(b(a(b(a(B(x)))))))))
B1(a(b(a(a(a(b(a(B(x))))))))) → B1(a(a(b(B(x)))))
B1(b(x)) → B1(a(b(a(b(x)))))
The TRS R consists of the following rules:
b(a(a(a(b(b(x)))))) → b(a(b(a(a(a(b(x)))))))
b(b(x)) → b(a(b(a(b(a(b(x)))))))
b(a(b(a(a(a(b(a(b(x))))))))) → b(a(a(b(b(x)))))
b(a(a(b(x)))) → b(a(a(a(b(x)))))
b(a(a(b(B(x))))) → b(a(a(a(b(a(b(a(B(x)))))))))
b(a(a(b(B(x))))) → b(a(a(a(b(a(b(a(b(a(B(x)))))))))))
b(b(B(x))) → b(a(b(a(b(a(b(a(b(a(b(a(B(x)))))))))))))
b(b(B(x))) → b(a(b(a(b(a(b(a(b(a(B(x)))))))))))
b(a(b(a(a(a(b(a(b(B(x)))))))))) → b(a(a(b(b(a(b(a(B(x)))))))))
b(B(x)) → b(a(B(x)))
b(a(b(a(a(a(b(a(B(x))))))))) → b(a(a(b(B(x)))))
b(a(a(a(b(b(B(x))))))) → b(a(b(a(a(a(b(a(b(a(b(a(B(x)))))))))))))
b(a(a(a(b(b(B(x))))))) → b(a(b(a(a(a(b(a(b(a(B(x)))))))))))
b(a(b(a(a(a(b(B(x)))))))) → b(a(a(b(b(a(B(x)))))))
b(a(b(a(a(a(b(a(b(B(x)))))))))) → b(a(a(b(b(a(b(a(b(a(B(x)))))))))))
b(a(b(a(a(a(b(B(x)))))))) → b(a(a(b(b(a(b(a(B(x)))))))))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule B1(b(x)) → B1(a(b(a(b(a(b(x))))))) at position [0,0] we obtained the following new rules:
B1(b(a(a(a(b(B(x0))))))) → B1(a(b(a(b(a(a(b(b(a(B(x0)))))))))))
B1(b(b(B(x0)))) → B1(a(b(a(b(a(b(a(b(a(b(a(b(a(b(a(b(a(B(x0)))))))))))))))))))
B1(b(a(a(a(b(a(b(x0)))))))) → B1(a(b(a(b(a(a(b(b(x0)))))))))
B1(b(b(x0))) → B1(a(b(a(b(a(b(a(b(a(b(a(b(x0)))))))))))))
B1(b(a(a(a(b(a(b(B(x0))))))))) → B1(a(b(a(b(a(a(b(b(a(b(a(B(x0)))))))))))))
B1(b(a(a(b(B(x0)))))) → B1(a(b(a(b(a(b(a(a(a(b(a(b(a(b(a(B(x0)))))))))))))))))
B1(b(a(a(a(b(b(B(x0)))))))) → B1(a(b(a(b(a(b(a(b(a(a(a(b(a(b(a(b(a(B(x0)))))))))))))))))))
B1(b(a(b(a(a(a(b(B(x0))))))))) → B1(a(b(a(b(a(b(a(a(b(b(a(b(a(B(x0)))))))))))))))
B1(b(a(b(a(a(a(b(B(x0))))))))) → B1(a(b(a(b(a(b(a(a(b(b(a(B(x0)))))))))))))
B1(b(a(a(a(b(a(B(x0)))))))) → B1(a(b(a(b(a(a(b(B(x0)))))))))
B1(b(a(b(a(a(a(b(a(B(x0)))))))))) → B1(a(b(a(b(a(b(a(a(b(B(x0)))))))))))
B1(b(B(x0))) → B1(a(b(a(b(a(b(a(B(x0)))))))))
B1(b(a(b(a(a(a(b(a(b(B(x0))))))))))) → B1(a(b(a(b(a(b(a(a(b(b(a(b(a(B(x0)))))))))))))))
B1(b(a(a(b(x0))))) → B1(a(b(a(b(a(b(a(a(a(b(x0)))))))))))
B1(b(a(b(a(a(a(b(a(b(B(x0))))))))))) → B1(a(b(a(b(a(b(a(a(b(b(a(b(a(b(a(B(x0)))))))))))))))))
B1(b(a(a(a(b(B(x0))))))) → B1(a(b(a(b(a(a(b(b(a(b(a(B(x0)))))))))))))
B1(b(a(a(a(b(b(B(x0)))))))) → B1(a(b(a(b(a(b(a(b(a(a(a(b(a(b(a(B(x0)))))))))))))))))
B1(b(a(a(a(b(b(x0))))))) → B1(a(b(a(b(a(b(a(b(a(a(a(b(x0)))))))))))))
B1(b(a(a(a(b(a(b(B(x0))))))))) → B1(a(b(a(b(a(a(b(b(a(b(a(b(a(B(x0)))))))))))))))
B1(b(b(B(x0)))) → B1(a(b(a(b(a(b(a(b(a(b(a(b(a(b(a(B(x0)))))))))))))))))
B1(b(a(b(a(a(a(b(a(b(x0)))))))))) → B1(a(b(a(b(a(b(a(a(b(b(x0)))))))))))
B1(b(a(a(b(B(x0)))))) → B1(a(b(a(b(a(b(a(a(a(b(a(b(a(B(x0)))))))))))))))
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ SemLabProof
↳ SemLabProof2
↳ QDP
↳ QDPToSRSProof
↳ QTRS
↳ QTRS Reverse
↳ QTRS
↳ QTRS Reverse
↳ QTRS Reverse
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QTRS Reverse
↳ QTRS Reverse
Q DP problem:
The TRS P consists of the following rules:
B1(b(a(a(a(b(B(x0))))))) → B1(a(b(a(b(a(a(b(b(a(B(x0)))))))))))
B1(b(x)) → B1(a(b(x)))
B1(b(b(B(x0)))) → B1(a(b(a(b(a(b(a(b(a(b(a(b(a(b(a(b(a(B(x0)))))))))))))))))))
B1(a(a(a(b(b(B(x))))))) → B1(a(b(a(a(a(b(a(b(a(B(x)))))))))))
B1(b(b(x0))) → B1(a(b(a(b(a(b(a(b(a(b(a(b(x0)))))))))))))
B1(a(a(a(b(b(x)))))) → B1(a(a(a(b(x)))))
B1(b(a(a(a(b(a(b(B(x0))))))))) → B1(a(b(a(b(a(a(b(b(a(b(a(B(x0)))))))))))))
B1(b(a(a(b(B(x0)))))) → B1(a(b(a(b(a(b(a(a(a(b(a(b(a(b(a(B(x0)))))))))))))))))
B1(a(b(a(a(a(b(a(b(B(x)))))))))) → B1(b(a(b(a(b(a(B(x))))))))
B1(a(a(b(x)))) → B1(a(a(a(b(x)))))
B1(b(a(b(a(a(a(b(B(x0))))))))) → B1(a(b(a(b(a(b(a(a(b(b(a(b(a(B(x0)))))))))))))))
B1(b(a(a(a(b(a(B(x0)))))))) → B1(a(b(a(b(a(a(b(B(x0)))))))))
B1(a(b(a(a(a(b(B(x)))))))) → B1(a(a(b(b(a(B(x)))))))
B1(b(B(x0))) → B1(a(b(a(b(a(b(a(B(x0)))))))))
B1(a(b(a(a(a(b(a(b(B(x)))))))))) → B1(b(a(b(a(B(x))))))
B1(b(a(a(b(x0))))) → B1(a(b(a(b(a(b(a(a(a(b(x0)))))))))))
B1(b(a(b(a(a(a(b(a(b(B(x0))))))))))) → B1(a(b(a(b(a(b(a(a(b(b(a(b(a(B(x0)))))))))))))))
B1(a(b(a(a(a(b(a(b(B(x)))))))))) → B1(a(a(b(b(a(b(a(B(x)))))))))
B1(b(a(b(a(a(a(b(a(b(B(x0))))))))))) → B1(a(b(a(b(a(b(a(a(b(b(a(b(a(b(a(B(x0)))))))))))))))))
B1(b(x)) → B1(a(b(a(b(x)))))
B1(b(a(a(a(b(b(B(x0)))))))) → B1(a(b(a(b(a(b(a(b(a(a(a(b(a(b(a(B(x0)))))))))))))))))
B1(b(a(a(b(B(x0)))))) → B1(a(b(a(b(a(b(a(a(a(b(a(b(a(B(x0)))))))))))))))
B1(b(b(B(x0)))) → B1(a(b(a(b(a(b(a(b(a(b(a(b(a(b(a(B(x0)))))))))))))))))
B1(a(a(a(b(b(x)))))) → B1(a(b(a(a(a(b(x)))))))
B1(a(b(a(a(a(b(B(x)))))))) → B1(a(a(b(b(a(b(a(B(x)))))))))
B1(b(a(a(a(b(a(b(x0)))))))) → B1(a(b(a(b(a(a(b(b(x0)))))))))
B1(a(a(a(b(b(B(x))))))) → B1(a(b(a(a(a(b(a(b(a(b(a(B(x)))))))))))))
B1(b(a(a(a(b(b(B(x0)))))))) → B1(a(b(a(b(a(b(a(b(a(a(a(b(a(b(a(b(a(B(x0)))))))))))))))))))
B1(a(b(a(a(a(b(a(b(x))))))))) → B1(a(a(b(b(x)))))
B1(b(a(b(a(a(a(b(B(x0))))))))) → B1(a(b(a(b(a(b(a(a(b(b(a(B(x0)))))))))))))
B1(a(b(a(a(a(b(B(x)))))))) → B1(b(a(B(x))))
B1(a(b(a(a(a(b(a(b(x))))))))) → B1(b(x))
B1(a(b(a(a(a(b(a(b(B(x)))))))))) → B1(a(a(b(b(a(b(a(b(a(B(x)))))))))))
B1(b(a(b(a(a(a(b(a(B(x0)))))))))) → B1(a(b(a(b(a(b(a(a(b(B(x0)))))))))))
B1(a(b(a(a(a(b(B(x)))))))) → B1(b(a(b(a(B(x))))))
B1(a(b(a(a(a(b(a(B(x))))))))) → B1(a(a(b(B(x)))))
B1(b(a(a(a(b(B(x0))))))) → B1(a(b(a(b(a(a(b(b(a(b(a(B(x0)))))))))))))
B1(b(a(a(a(b(a(b(B(x0))))))))) → B1(a(b(a(b(a(a(b(b(a(b(a(b(a(B(x0)))))))))))))))
B1(b(a(a(a(b(b(x0))))))) → B1(a(b(a(b(a(b(a(b(a(a(a(b(x0)))))))))))))
B1(b(a(b(a(a(a(b(a(b(x0)))))))))) → B1(a(b(a(b(a(b(a(a(b(b(x0)))))))))))
The TRS R consists of the following rules:
b(a(a(a(b(b(x)))))) → b(a(b(a(a(a(b(x)))))))
b(b(x)) → b(a(b(a(b(a(b(x)))))))
b(a(b(a(a(a(b(a(b(x))))))))) → b(a(a(b(b(x)))))
b(a(a(b(x)))) → b(a(a(a(b(x)))))
b(a(a(b(B(x))))) → b(a(a(a(b(a(b(a(B(x)))))))))
b(a(a(b(B(x))))) → b(a(a(a(b(a(b(a(b(a(B(x)))))))))))
b(b(B(x))) → b(a(b(a(b(a(b(a(b(a(b(a(B(x)))))))))))))
b(b(B(x))) → b(a(b(a(b(a(b(a(b(a(B(x)))))))))))
b(a(b(a(a(a(b(a(b(B(x)))))))))) → b(a(a(b(b(a(b(a(B(x)))))))))
b(B(x)) → b(a(B(x)))
b(a(b(a(a(a(b(a(B(x))))))))) → b(a(a(b(B(x)))))
b(a(a(a(b(b(B(x))))))) → b(a(b(a(a(a(b(a(b(a(b(a(B(x)))))))))))))
b(a(a(a(b(b(B(x))))))) → b(a(b(a(a(a(b(a(b(a(B(x)))))))))))
b(a(b(a(a(a(b(B(x)))))))) → b(a(a(b(b(a(B(x)))))))
b(a(b(a(a(a(b(a(b(B(x)))))))))) → b(a(a(b(b(a(b(a(b(a(B(x)))))))))))
b(a(b(a(a(a(b(B(x)))))))) → b(a(a(b(b(a(b(a(B(x)))))))))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 1 SCC with 3 less nodes.
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ SemLabProof
↳ SemLabProof2
↳ QDP
↳ QDPToSRSProof
↳ QTRS
↳ QTRS Reverse
↳ QTRS
↳ QTRS Reverse
↳ QTRS Reverse
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QTRS Reverse
↳ QTRS Reverse
Q DP problem:
The TRS P consists of the following rules:
B1(b(a(a(a(b(B(x0))))))) → B1(a(b(a(b(a(a(b(b(a(B(x0)))))))))))
B1(b(x)) → B1(a(b(x)))
B1(a(a(a(b(b(B(x))))))) → B1(a(b(a(a(a(b(a(b(a(B(x)))))))))))
B1(b(b(x0))) → B1(a(b(a(b(a(b(a(b(a(b(a(b(x0)))))))))))))
B1(a(a(a(b(b(x)))))) → B1(a(a(a(b(x)))))
B1(b(a(a(a(b(a(b(B(x0))))))))) → B1(a(b(a(b(a(a(b(b(a(b(a(B(x0)))))))))))))
B1(b(a(a(b(B(x0)))))) → B1(a(b(a(b(a(b(a(a(a(b(a(b(a(b(a(B(x0)))))))))))))))))
B1(a(b(a(a(a(b(a(b(B(x)))))))))) → B1(b(a(b(a(b(a(B(x))))))))
B1(a(a(b(x)))) → B1(a(a(a(b(x)))))
B1(b(a(b(a(a(a(b(B(x0))))))))) → B1(a(b(a(b(a(b(a(a(b(b(a(b(a(B(x0)))))))))))))))
B1(b(a(a(a(b(a(B(x0)))))))) → B1(a(b(a(b(a(a(b(B(x0)))))))))
B1(a(b(a(a(a(b(B(x)))))))) → B1(a(a(b(b(a(B(x)))))))
B1(a(b(a(a(a(b(a(b(B(x)))))))))) → B1(b(a(b(a(B(x))))))
B1(b(a(a(b(x0))))) → B1(a(b(a(b(a(b(a(a(a(b(x0)))))))))))
B1(b(a(b(a(a(a(b(a(b(B(x0))))))))))) → B1(a(b(a(b(a(b(a(a(b(b(a(b(a(B(x0)))))))))))))))
B1(a(b(a(a(a(b(a(b(B(x)))))))))) → B1(a(a(b(b(a(b(a(B(x)))))))))
B1(b(a(b(a(a(a(b(a(b(B(x0))))))))))) → B1(a(b(a(b(a(b(a(a(b(b(a(b(a(b(a(B(x0)))))))))))))))))
B1(b(x)) → B1(a(b(a(b(x)))))
B1(b(a(a(a(b(b(B(x0)))))))) → B1(a(b(a(b(a(b(a(b(a(a(a(b(a(b(a(B(x0)))))))))))))))))
B1(b(a(a(b(B(x0)))))) → B1(a(b(a(b(a(b(a(a(a(b(a(b(a(B(x0)))))))))))))))
B1(a(a(a(b(b(x)))))) → B1(a(b(a(a(a(b(x)))))))
B1(a(b(a(a(a(b(B(x)))))))) → B1(a(a(b(b(a(b(a(B(x)))))))))
B1(b(a(a(a(b(a(b(x0)))))))) → B1(a(b(a(b(a(a(b(b(x0)))))))))
B1(a(a(a(b(b(B(x))))))) → B1(a(b(a(a(a(b(a(b(a(b(a(B(x)))))))))))))
B1(a(b(a(a(a(b(a(b(x))))))))) → B1(a(a(b(b(x)))))
B1(b(a(a(a(b(b(B(x0)))))))) → B1(a(b(a(b(a(b(a(b(a(a(a(b(a(b(a(b(a(B(x0)))))))))))))))))))
B1(b(a(b(a(a(a(b(B(x0))))))))) → B1(a(b(a(b(a(b(a(a(b(b(a(B(x0)))))))))))))
B1(a(b(a(a(a(b(B(x)))))))) → B1(b(a(B(x))))
B1(a(b(a(a(a(b(a(b(B(x)))))))))) → B1(a(a(b(b(a(b(a(b(a(B(x)))))))))))
B1(a(b(a(a(a(b(a(b(x))))))))) → B1(b(x))
B1(b(a(b(a(a(a(b(a(B(x0)))))))))) → B1(a(b(a(b(a(b(a(a(b(B(x0)))))))))))
B1(a(b(a(a(a(b(B(x)))))))) → B1(b(a(b(a(B(x))))))
B1(a(b(a(a(a(b(a(B(x))))))))) → B1(a(a(b(B(x)))))
B1(b(a(a(a(b(B(x0))))))) → B1(a(b(a(b(a(a(b(b(a(b(a(B(x0)))))))))))))
B1(b(a(a(a(b(a(b(B(x0))))))))) → B1(a(b(a(b(a(a(b(b(a(b(a(b(a(B(x0)))))))))))))))
B1(b(a(a(a(b(b(x0))))))) → B1(a(b(a(b(a(b(a(b(a(a(a(b(x0)))))))))))))
B1(b(a(b(a(a(a(b(a(b(x0)))))))))) → B1(a(b(a(b(a(b(a(a(b(b(x0)))))))))))
The TRS R consists of the following rules:
b(a(a(a(b(b(x)))))) → b(a(b(a(a(a(b(x)))))))
b(b(x)) → b(a(b(a(b(a(b(x)))))))
b(a(b(a(a(a(b(a(b(x))))))))) → b(a(a(b(b(x)))))
b(a(a(b(x)))) → b(a(a(a(b(x)))))
b(a(a(b(B(x))))) → b(a(a(a(b(a(b(a(B(x)))))))))
b(a(a(b(B(x))))) → b(a(a(a(b(a(b(a(b(a(B(x)))))))))))
b(b(B(x))) → b(a(b(a(b(a(b(a(b(a(b(a(B(x)))))))))))))
b(b(B(x))) → b(a(b(a(b(a(b(a(b(a(B(x)))))))))))
b(a(b(a(a(a(b(a(b(B(x)))))))))) → b(a(a(b(b(a(b(a(B(x)))))))))
b(B(x)) → b(a(B(x)))
b(a(b(a(a(a(b(a(B(x))))))))) → b(a(a(b(B(x)))))
b(a(a(a(b(b(B(x))))))) → b(a(b(a(a(a(b(a(b(a(b(a(B(x)))))))))))))
b(a(a(a(b(b(B(x))))))) → b(a(b(a(a(a(b(a(b(a(B(x)))))))))))
b(a(b(a(a(a(b(B(x)))))))) → b(a(a(b(b(a(B(x)))))))
b(a(b(a(a(a(b(a(b(B(x)))))))))) → b(a(a(b(b(a(b(a(b(a(B(x)))))))))))
b(a(b(a(a(a(b(B(x)))))))) → b(a(a(b(b(a(b(a(B(x)))))))))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule B1(b(x)) → B1(a(b(a(b(x))))) at position [0,0] we obtained the following new rules:
B1(b(B(x0))) → B1(a(b(a(b(a(B(x0)))))))
B1(b(a(a(b(x0))))) → B1(a(b(a(b(a(a(a(b(x0)))))))))
B1(b(a(a(a(b(B(x0))))))) → B1(a(b(a(a(b(b(a(b(a(B(x0)))))))))))
B1(b(a(b(a(a(a(b(a(b(x0)))))))))) → B1(a(b(a(b(a(a(b(b(x0)))))))))
B1(b(a(b(a(a(a(b(a(b(B(x0))))))))))) → B1(a(b(a(b(a(a(b(b(a(b(a(b(a(B(x0)))))))))))))))
B1(b(a(a(b(B(x0)))))) → B1(a(b(a(b(a(a(a(b(a(b(a(b(a(B(x0)))))))))))))))
B1(b(a(b(a(a(a(b(a(B(x0)))))))))) → B1(a(b(a(b(a(a(b(B(x0)))))))))
B1(b(a(a(a(b(a(B(x0)))))))) → B1(a(b(a(a(b(B(x0)))))))
B1(b(a(b(a(a(a(b(B(x0))))))))) → B1(a(b(a(b(a(a(b(b(a(B(x0)))))))))))
B1(b(a(b(a(a(a(b(a(b(B(x0))))))))))) → B1(a(b(a(b(a(a(b(b(a(b(a(B(x0)))))))))))))
B1(b(a(b(a(a(a(b(B(x0))))))))) → B1(a(b(a(b(a(a(b(b(a(b(a(B(x0)))))))))))))
B1(b(b(B(x0)))) → B1(a(b(a(b(a(b(a(b(a(b(a(b(a(B(x0)))))))))))))))
B1(b(a(a(a(b(a(b(B(x0))))))))) → B1(a(b(a(a(b(b(a(b(a(B(x0)))))))))))
B1(b(a(a(a(b(a(b(x0)))))))) → B1(a(b(a(a(b(b(x0)))))))
B1(b(a(a(a(b(a(b(B(x0))))))))) → B1(a(b(a(a(b(b(a(b(a(b(a(B(x0)))))))))))))
B1(b(a(a(b(B(x0)))))) → B1(a(b(a(b(a(a(a(b(a(b(a(B(x0)))))))))))))
B1(b(a(a(a(b(b(x0))))))) → B1(a(b(a(b(a(b(a(a(a(b(x0)))))))))))
B1(b(b(x0))) → B1(a(b(a(b(a(b(a(b(a(b(x0)))))))))))
B1(b(a(a(a(b(B(x0))))))) → B1(a(b(a(a(b(b(a(B(x0)))))))))
B1(b(b(B(x0)))) → B1(a(b(a(b(a(b(a(b(a(b(a(b(a(b(a(B(x0)))))))))))))))))
B1(b(a(a(a(b(b(B(x0)))))))) → B1(a(b(a(b(a(b(a(a(a(b(a(b(a(B(x0)))))))))))))))
B1(b(a(a(a(b(b(B(x0)))))))) → B1(a(b(a(b(a(b(a(a(a(b(a(b(a(b(a(B(x0)))))))))))))))))
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ SemLabProof
↳ SemLabProof2
↳ QDP
↳ QDPToSRSProof
↳ QTRS
↳ QTRS Reverse
↳ QTRS
↳ QTRS Reverse
↳ QTRS Reverse
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QTRS Reverse
↳ QTRS Reverse
Q DP problem:
The TRS P consists of the following rules:
B1(b(a(a(b(x0))))) → B1(a(b(a(b(a(a(a(b(x0)))))))))
B1(b(a(a(a(b(B(x0))))))) → B1(a(b(a(a(b(b(a(b(a(B(x0)))))))))))
B1(b(a(b(a(a(a(b(B(x0))))))))) → B1(a(b(a(b(a(a(b(b(a(B(x0)))))))))))
B1(b(a(b(a(a(a(b(B(x0))))))))) → B1(a(b(a(b(a(b(a(a(b(b(a(b(a(B(x0)))))))))))))))
B1(a(b(a(a(a(b(B(x)))))))) → B1(a(a(b(b(a(B(x)))))))
B1(a(b(a(a(a(b(a(b(B(x)))))))))) → B1(b(a(b(a(B(x))))))
B1(b(a(a(b(x0))))) → B1(a(b(a(b(a(b(a(a(a(b(x0)))))))))))
B1(b(b(B(x0)))) → B1(a(b(a(b(a(b(a(b(a(b(a(b(a(b(a(B(x0)))))))))))))))))
B1(b(a(a(a(b(B(x0))))))) → B1(a(b(a(a(b(b(a(B(x0)))))))))
B1(b(a(a(b(B(x0)))))) → B1(a(b(a(b(a(b(a(a(a(b(a(b(a(B(x0)))))))))))))))
B1(b(a(a(a(b(b(B(x0)))))))) → B1(a(b(a(b(a(b(a(a(a(b(a(b(a(b(a(B(x0)))))))))))))))))
B1(a(a(a(b(b(x)))))) → B1(a(b(a(a(a(b(x)))))))
B1(a(b(a(a(a(b(B(x)))))))) → B1(a(a(b(b(a(b(a(B(x)))))))))
B1(b(a(b(a(a(a(b(a(b(x0)))))))))) → B1(a(b(a(b(a(a(b(b(x0)))))))))
B1(b(a(a(b(B(x0)))))) → B1(a(b(a(b(a(a(a(b(a(b(a(b(a(B(x0)))))))))))))))
B1(b(a(b(a(a(a(b(a(B(x0)))))))))) → B1(a(b(a(b(a(a(b(B(x0)))))))))
B1(a(a(a(b(b(B(x))))))) → B1(a(b(a(a(a(b(a(b(a(b(a(B(x)))))))))))))
B1(b(a(b(a(a(a(b(B(x0))))))))) → B1(a(b(a(b(a(b(a(a(b(b(a(B(x0)))))))))))))
B1(a(b(a(a(a(b(a(b(x))))))))) → B1(b(x))
B1(b(b(B(x0)))) → B1(a(b(a(b(a(b(a(b(a(b(a(b(a(B(x0)))))))))))))))
B1(b(a(b(a(a(a(b(a(B(x0)))))))))) → B1(a(b(a(b(a(b(a(a(b(B(x0)))))))))))
B1(b(a(a(a(b(a(b(B(x0))))))))) → B1(a(b(a(a(b(b(a(b(a(b(a(B(x0)))))))))))))
B1(a(b(a(a(a(b(B(x)))))))) → B1(b(a(b(a(B(x))))))
B1(b(a(a(a(b(b(x0))))))) → B1(a(b(a(b(a(b(a(a(a(b(x0)))))))))))
B1(b(a(a(a(b(B(x0))))))) → B1(a(b(a(b(a(a(b(b(a(b(a(B(x0)))))))))))))
B1(b(a(a(a(b(a(b(B(x0))))))))) → B1(a(b(a(b(a(a(b(b(a(b(a(b(a(B(x0)))))))))))))))
B1(b(a(a(a(b(B(x0))))))) → B1(a(b(a(b(a(a(b(b(a(B(x0)))))))))))
B1(b(x)) → B1(a(b(x)))
B1(a(a(a(b(b(B(x))))))) → B1(a(b(a(a(a(b(a(b(a(B(x)))))))))))
B1(b(b(x0))) → B1(a(b(a(b(a(b(a(b(a(b(a(b(x0)))))))))))))
B1(a(a(a(b(b(x)))))) → B1(a(a(a(b(x)))))
B1(b(a(a(a(b(a(B(x0)))))))) → B1(a(b(a(a(b(B(x0)))))))
B1(b(a(a(a(b(a(b(B(x0))))))))) → B1(a(b(a(b(a(a(b(b(a(b(a(B(x0)))))))))))))
B1(b(a(a(b(B(x0)))))) → B1(a(b(a(b(a(b(a(a(a(b(a(b(a(b(a(B(x0)))))))))))))))))
B1(a(b(a(a(a(b(a(b(B(x)))))))))) → B1(b(a(b(a(b(a(B(x))))))))
B1(a(a(b(x)))) → B1(a(a(a(b(x)))))
B1(b(a(a(a(b(a(B(x0)))))))) → B1(a(b(a(b(a(a(b(B(x0)))))))))
B1(b(a(b(a(a(a(b(a(b(B(x0))))))))))) → B1(a(b(a(b(a(a(b(b(a(b(a(B(x0)))))))))))))
B1(b(a(a(a(b(a(b(B(x0))))))))) → B1(a(b(a(a(b(b(a(b(a(B(x0)))))))))))
B1(b(a(a(a(b(a(b(x0)))))))) → B1(a(b(a(a(b(b(x0)))))))
B1(b(a(b(a(a(a(b(a(b(B(x0))))))))))) → B1(a(b(a(b(a(b(a(a(b(b(a(b(a(B(x0)))))))))))))))
B1(a(b(a(a(a(b(a(b(B(x)))))))))) → B1(a(a(b(b(a(b(a(B(x)))))))))
B1(b(a(b(a(a(a(b(a(b(B(x0))))))))))) → B1(a(b(a(b(a(b(a(a(b(b(a(b(a(b(a(B(x0)))))))))))))))))
B1(b(a(a(a(b(b(B(x0)))))))) → B1(a(b(a(b(a(b(a(b(a(a(a(b(a(b(a(B(x0)))))))))))))))))
B1(b(a(a(a(b(b(B(x0)))))))) → B1(a(b(a(b(a(b(a(a(a(b(a(b(a(B(x0)))))))))))))))
B1(b(B(x0))) → B1(a(b(a(b(a(B(x0)))))))
B1(b(a(a(a(b(a(b(x0)))))))) → B1(a(b(a(b(a(a(b(b(x0)))))))))
B1(b(a(b(a(a(a(b(a(b(B(x0))))))))))) → B1(a(b(a(b(a(a(b(b(a(b(a(b(a(B(x0)))))))))))))))
B1(a(b(a(a(a(b(a(b(x))))))))) → B1(a(a(b(b(x)))))
B1(b(a(a(a(b(b(B(x0)))))))) → B1(a(b(a(b(a(b(a(b(a(a(a(b(a(b(a(b(a(B(x0)))))))))))))))))))
B1(a(b(a(a(a(b(B(x)))))))) → B1(b(a(B(x))))
B1(a(b(a(a(a(b(a(b(B(x)))))))))) → B1(a(a(b(b(a(b(a(b(a(B(x)))))))))))
B1(b(a(b(a(a(a(b(B(x0))))))))) → B1(a(b(a(b(a(a(b(b(a(b(a(B(x0)))))))))))))
B1(b(a(a(b(B(x0)))))) → B1(a(b(a(b(a(a(a(b(a(b(a(B(x0)))))))))))))
B1(a(b(a(a(a(b(a(B(x))))))))) → B1(a(a(b(B(x)))))
B1(b(b(x0))) → B1(a(b(a(b(a(b(a(b(a(b(x0)))))))))))
B1(b(a(a(a(b(b(x0))))))) → B1(a(b(a(b(a(b(a(b(a(a(a(b(x0)))))))))))))
B1(b(a(b(a(a(a(b(a(b(x0)))))))))) → B1(a(b(a(b(a(b(a(a(b(b(x0)))))))))))
The TRS R consists of the following rules:
b(a(a(a(b(b(x)))))) → b(a(b(a(a(a(b(x)))))))
b(b(x)) → b(a(b(a(b(a(b(x)))))))
b(a(b(a(a(a(b(a(b(x))))))))) → b(a(a(b(b(x)))))
b(a(a(b(x)))) → b(a(a(a(b(x)))))
b(a(a(b(B(x))))) → b(a(a(a(b(a(b(a(B(x)))))))))
b(a(a(b(B(x))))) → b(a(a(a(b(a(b(a(b(a(B(x)))))))))))
b(b(B(x))) → b(a(b(a(b(a(b(a(b(a(b(a(B(x)))))))))))))
b(b(B(x))) → b(a(b(a(b(a(b(a(b(a(B(x)))))))))))
b(a(b(a(a(a(b(a(b(B(x)))))))))) → b(a(a(b(b(a(b(a(B(x)))))))))
b(B(x)) → b(a(B(x)))
b(a(b(a(a(a(b(a(B(x))))))))) → b(a(a(b(B(x)))))
b(a(a(a(b(b(B(x))))))) → b(a(b(a(a(a(b(a(b(a(b(a(B(x)))))))))))))
b(a(a(a(b(b(B(x))))))) → b(a(b(a(a(a(b(a(b(a(B(x)))))))))))
b(a(b(a(a(a(b(B(x)))))))) → b(a(a(b(b(a(B(x)))))))
b(a(b(a(a(a(b(a(b(B(x)))))))))) → b(a(a(b(b(a(b(a(b(a(B(x)))))))))))
b(a(b(a(a(a(b(B(x)))))))) → b(a(a(b(b(a(b(a(B(x)))))))))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 1 SCC with 3 less nodes.
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ SemLabProof
↳ SemLabProof2
↳ QDP
↳ QDPToSRSProof
↳ QTRS
↳ QTRS Reverse
↳ QTRS
↳ QTRS Reverse
↳ QTRS Reverse
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ SemLabProof
↳ SemLabProof2
↳ QTRS Reverse
↳ QTRS Reverse
Q DP problem:
The TRS P consists of the following rules:
B1(b(a(a(b(x0))))) → B1(a(b(a(b(a(a(a(b(x0)))))))))
B1(b(a(a(a(b(B(x0))))))) → B1(a(b(a(a(b(b(a(b(a(B(x0)))))))))))
B1(b(a(b(a(a(a(b(B(x0))))))))) → B1(a(b(a(b(a(a(b(b(a(B(x0)))))))))))
B1(b(a(b(a(a(a(b(B(x0))))))))) → B1(a(b(a(b(a(b(a(a(b(b(a(b(a(B(x0)))))))))))))))
B1(a(b(a(a(a(b(B(x)))))))) → B1(a(a(b(b(a(B(x)))))))
B1(a(b(a(a(a(b(a(b(B(x)))))))))) → B1(b(a(b(a(B(x))))))
B1(b(a(a(b(x0))))) → B1(a(b(a(b(a(b(a(a(a(b(x0)))))))))))
B1(b(a(a(a(b(B(x0))))))) → B1(a(b(a(a(b(b(a(B(x0)))))))))
B1(b(a(a(b(B(x0)))))) → B1(a(b(a(b(a(b(a(a(a(b(a(b(a(B(x0)))))))))))))))
B1(b(a(a(a(b(b(B(x0)))))))) → B1(a(b(a(b(a(b(a(a(a(b(a(b(a(b(a(B(x0)))))))))))))))))
B1(a(a(a(b(b(x)))))) → B1(a(b(a(a(a(b(x)))))))
B1(a(b(a(a(a(b(B(x)))))))) → B1(a(a(b(b(a(b(a(B(x)))))))))
B1(b(a(b(a(a(a(b(a(b(x0)))))))))) → B1(a(b(a(b(a(a(b(b(x0)))))))))
B1(b(a(a(b(B(x0)))))) → B1(a(b(a(b(a(a(a(b(a(b(a(b(a(B(x0)))))))))))))))
B1(b(a(b(a(a(a(b(a(B(x0)))))))))) → B1(a(b(a(b(a(a(b(B(x0)))))))))
B1(a(a(a(b(b(B(x))))))) → B1(a(b(a(a(a(b(a(b(a(b(a(B(x)))))))))))))
B1(b(a(b(a(a(a(b(B(x0))))))))) → B1(a(b(a(b(a(b(a(a(b(b(a(B(x0)))))))))))))
B1(a(b(a(a(a(b(a(b(x))))))))) → B1(b(x))
B1(b(a(b(a(a(a(b(a(B(x0)))))))))) → B1(a(b(a(b(a(b(a(a(b(B(x0)))))))))))
B1(a(b(a(a(a(b(B(x)))))))) → B1(b(a(b(a(B(x))))))
B1(b(a(a(a(b(a(b(B(x0))))))))) → B1(a(b(a(a(b(b(a(b(a(b(a(B(x0)))))))))))))
B1(b(a(a(a(b(b(x0))))))) → B1(a(b(a(b(a(b(a(a(a(b(x0)))))))))))
B1(b(a(a(a(b(B(x0))))))) → B1(a(b(a(b(a(a(b(b(a(b(a(B(x0)))))))))))))
B1(b(a(a(a(b(a(b(B(x0))))))))) → B1(a(b(a(b(a(a(b(b(a(b(a(b(a(B(x0)))))))))))))))
B1(b(a(a(a(b(B(x0))))))) → B1(a(b(a(b(a(a(b(b(a(B(x0)))))))))))
B1(b(x)) → B1(a(b(x)))
B1(a(a(a(b(b(B(x))))))) → B1(a(b(a(a(a(b(a(b(a(B(x)))))))))))
B1(b(b(x0))) → B1(a(b(a(b(a(b(a(b(a(b(a(b(x0)))))))))))))
B1(a(a(a(b(b(x)))))) → B1(a(a(a(b(x)))))
B1(b(a(a(a(b(a(B(x0)))))))) → B1(a(b(a(a(b(B(x0)))))))
B1(b(a(a(a(b(a(b(B(x0))))))))) → B1(a(b(a(b(a(a(b(b(a(b(a(B(x0)))))))))))))
B1(b(a(a(b(B(x0)))))) → B1(a(b(a(b(a(b(a(a(a(b(a(b(a(b(a(B(x0)))))))))))))))))
B1(a(b(a(a(a(b(a(b(B(x)))))))))) → B1(b(a(b(a(b(a(B(x))))))))
B1(a(a(b(x)))) → B1(a(a(a(b(x)))))
B1(b(a(a(a(b(a(B(x0)))))))) → B1(a(b(a(b(a(a(b(B(x0)))))))))
B1(b(a(b(a(a(a(b(a(b(B(x0))))))))))) → B1(a(b(a(b(a(a(b(b(a(b(a(B(x0)))))))))))))
B1(b(a(a(a(b(a(b(B(x0))))))))) → B1(a(b(a(a(b(b(a(b(a(B(x0)))))))))))
B1(b(a(a(a(b(a(b(x0)))))))) → B1(a(b(a(a(b(b(x0)))))))
B1(a(b(a(a(a(b(a(b(B(x)))))))))) → B1(a(a(b(b(a(b(a(B(x)))))))))
B1(b(a(b(a(a(a(b(a(b(B(x0))))))))))) → B1(a(b(a(b(a(b(a(a(b(b(a(b(a(B(x0)))))))))))))))
B1(b(a(b(a(a(a(b(a(b(B(x0))))))))))) → B1(a(b(a(b(a(b(a(a(b(b(a(b(a(b(a(B(x0)))))))))))))))))
B1(b(a(a(a(b(b(B(x0)))))))) → B1(a(b(a(b(a(b(a(b(a(a(a(b(a(b(a(B(x0)))))))))))))))))
B1(b(a(a(a(b(b(B(x0)))))))) → B1(a(b(a(b(a(b(a(a(a(b(a(b(a(B(x0)))))))))))))))
B1(b(a(a(a(b(a(b(x0)))))))) → B1(a(b(a(b(a(a(b(b(x0)))))))))
B1(b(a(b(a(a(a(b(a(b(B(x0))))))))))) → B1(a(b(a(b(a(a(b(b(a(b(a(b(a(B(x0)))))))))))))))
B1(b(a(a(a(b(b(B(x0)))))))) → B1(a(b(a(b(a(b(a(b(a(a(a(b(a(b(a(b(a(B(x0)))))))))))))))))))
B1(a(b(a(a(a(b(a(b(x))))))))) → B1(a(a(b(b(x)))))
B1(a(b(a(a(a(b(B(x)))))))) → B1(b(a(B(x))))
B1(a(b(a(a(a(b(a(b(B(x)))))))))) → B1(a(a(b(b(a(b(a(b(a(B(x)))))))))))
B1(b(a(b(a(a(a(b(B(x0))))))))) → B1(a(b(a(b(a(a(b(b(a(b(a(B(x0)))))))))))))
B1(a(b(a(a(a(b(a(B(x))))))))) → B1(a(a(b(B(x)))))
B1(b(a(a(b(B(x0)))))) → B1(a(b(a(b(a(a(a(b(a(b(a(B(x0)))))))))))))
B1(b(b(x0))) → B1(a(b(a(b(a(b(a(b(a(b(x0)))))))))))
B1(b(a(a(a(b(b(x0))))))) → B1(a(b(a(b(a(b(a(b(a(a(a(b(x0)))))))))))))
B1(b(a(b(a(a(a(b(a(b(x0)))))))))) → B1(a(b(a(b(a(b(a(a(b(b(x0)))))))))))
The TRS R consists of the following rules:
b(a(a(a(b(b(x)))))) → b(a(b(a(a(a(b(x)))))))
b(b(x)) → b(a(b(a(b(a(b(x)))))))
b(a(b(a(a(a(b(a(b(x))))))))) → b(a(a(b(b(x)))))
b(a(a(b(x)))) → b(a(a(a(b(x)))))
b(a(a(b(B(x))))) → b(a(a(a(b(a(b(a(B(x)))))))))
b(a(a(b(B(x))))) → b(a(a(a(b(a(b(a(b(a(B(x)))))))))))
b(b(B(x))) → b(a(b(a(b(a(b(a(b(a(b(a(B(x)))))))))))))
b(b(B(x))) → b(a(b(a(b(a(b(a(b(a(B(x)))))))))))
b(a(b(a(a(a(b(a(b(B(x)))))))))) → b(a(a(b(b(a(b(a(B(x)))))))))
b(B(x)) → b(a(B(x)))
b(a(b(a(a(a(b(a(B(x))))))))) → b(a(a(b(B(x)))))
b(a(a(a(b(b(B(x))))))) → b(a(b(a(a(a(b(a(b(a(b(a(B(x)))))))))))))
b(a(a(a(b(b(B(x))))))) → b(a(b(a(a(a(b(a(b(a(B(x)))))))))))
b(a(b(a(a(a(b(B(x)))))))) → b(a(a(b(b(a(B(x)))))))
b(a(b(a(a(a(b(a(b(B(x)))))))))) → b(a(a(b(b(a(b(a(b(a(B(x)))))))))))
b(a(b(a(a(a(b(B(x)))))))) → b(a(a(b(b(a(b(a(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:
B1.1(a.0(a.1(a.0(b.0(b.0(x)))))) → B1.1(a.0(b.1(a.0(a.1(a.0(b.0(x)))))))
B1.1(a.0(b.1(a.0(a.1(a.0(b.0(B.1(x)))))))) → B1.0(b.1(a.0(B.1(x))))
B1.0(b.1(a.0(a.1(a.0(b.0(B.0(x0))))))) → B1.1(a.0(b.1(a.0(b.0(a.1(a.0(b.0(b.1(a.0(b.1(a.0(B.0(x0)))))))))))))
B1.0(b.1(a.0(a.1(a.0(b.1(a.0(b.0(B.0(x0))))))))) → B1.1(a.0(b.0(a.1(a.0(b.0(b.1(a.0(b.1(a.0(b.1(a.0(B.0(x0)))))))))))))
B1.1(a.0(b.1(a.0(a.1(a.0(b.1(a.0(b.0(B.0(x)))))))))) → B1.0(a.1(a.0(b.0(b.1(a.0(b.1(a.0(B.0(x)))))))))
B1.1(a.0(b.1(a.0(a.1(a.0(b.1(a.0(b.0(B.1(x)))))))))) → B1.0(a.1(a.0(b.0(b.1(a.0(b.1(a.0(b.1(a.0(B.1(x)))))))))))
B1.0(b.1(a.0(b.1(a.0(a.1(a.0(b.0(B.1(x0))))))))) → B1.1(a.0(b.1(a.0(b.1(a.0(b.0(a.1(a.0(b.0(b.1(a.0(B.1(x0)))))))))))))
B1.0(b.1(a.0(b.1(a.0(a.1(a.0(b.1(a.0(b.0(B.0(x0))))))))))) → B1.1(a.0(b.1(a.0(b.1(a.0(b.0(a.1(a.0(b.0(b.1(a.0(b.1(a.0(b.1(a.0(B.0(x0)))))))))))))))))
B1.0(b.0(a.1(a.0(b.0(B.0(x0)))))) → B1.1(a.0(b.1(a.0(b.1(a.0(a.1(a.0(b.1(a.0(b.1(a.0(b.1(a.0(B.0(x0)))))))))))))))
B1.1(a.0(a.1(a.0(b.0(b.0(B.1(x))))))) → B1.1(a.0(b.1(a.0(a.1(a.0(b.1(a.0(b.1(a.0(b.1(a.0(B.1(x)))))))))))))
B1.0(b.0(a.1(a.0(b.0(B.0(x0)))))) → B1.1(a.0(b.1(a.0(b.1(a.0(a.1(a.0(b.1(a.0(b.1(a.0(B.0(x0)))))))))))))
B1.0(b.1(a.0(a.1(a.0(b.1(a.0(B.0(x0)))))))) → B1.1(a.0(b.0(a.1(a.0(b.0(B.0(x0)))))))
B1.0(b.1(a.0(a.1(a.0(b.0(b.0(B.1(x0)))))))) → B1.1(a.0(b.1(a.0(b.1(a.0(b.1(a.0(a.1(a.0(b.1(a.0(b.1(a.0(B.1(x0)))))))))))))))
B1.0(b.1(x)) → B1.1(a.0(b.1(x)))
B1.0(b.1(a.0(b.1(a.0(a.1(a.0(b.0(B.0(x0))))))))) → B1.1(a.0(b.1(a.0(b.0(a.1(a.0(b.0(b.1(a.0(b.1(a.0(B.0(x0)))))))))))))
B1.0(b.0(b.1(x0))) → B1.1(a.0(b.1(a.0(b.1(a.0(b.1(a.0(b.1(a.0(b.1(x0)))))))))))
B1.1(a.0(a.1(a.0(b.0(b.1(x)))))) → B1.1(a.0(a.1(a.0(b.1(x)))))
B1.0(b.1(a.0(a.1(a.0(b.1(a.0(b.1(x0)))))))) → B1.1(a.0(b.1(a.0(b.0(a.1(a.0(b.0(b.1(x0)))))))))
B1.0(b.1(a.0(a.1(a.0(b.1(a.0(b.0(x0)))))))) → B1.1(a.0(b.1(a.0(b.0(a.1(a.0(b.0(b.0(x0)))))))))
B1.0(b.1(a.0(a.1(a.0(b.0(b.0(B.0(x0)))))))) → B1.1(a.0(b.1(a.0(b.1(a.0(b.1(a.0(a.1(a.0(b.1(a.0(b.1(a.0(b.1(a.0(B.0(x0)))))))))))))))))
B1.1(a.0(a.1(a.0(b.0(b.1(x)))))) → B1.1(a.0(b.1(a.0(a.1(a.0(b.1(x)))))))
B1.0(b.0(b.0(x0))) → B1.1(a.0(b.1(a.0(b.1(a.0(b.1(a.0(b.1(a.0(b.0(x0)))))))))))
B1.1(a.0(b.1(a.0(a.1(a.0(b.1(a.0(b.0(B.0(x)))))))))) → B1.0(a.1(a.0(b.0(b.1(a.0(b.1(a.0(b.1(a.0(B.0(x)))))))))))
B1.0(b.0(a.1(a.0(b.0(B.1(x0)))))) → B1.1(a.0(b.1(a.0(b.1(a.0(a.1(a.0(b.1(a.0(b.1(a.0(b.1(a.0(B.1(x0)))))))))))))))
B1.1(a.0(b.1(a.0(a.1(a.0(b.1(a.0(b.0(B.0(x)))))))))) → B1.0(b.1(a.0(b.1(a.0(B.0(x))))))
B1.1(a.0(b.1(a.0(a.1(a.0(b.0(B.1(x)))))))) → B1.0(a.1(a.0(b.0(b.1(a.0(B.1(x)))))))
B1.0(b.0(a.1(a.0(b.0(B.0(x0)))))) → B1.1(a.0(b.1(a.0(b.1(a.0(b.1(a.0(a.1(a.0(b.1(a.0(b.1(a.0(B.0(x0)))))))))))))))
B1.0(b.1(a.0(b.1(a.0(a.1(a.0(b.1(a.0(b.0(B.1(x0))))))))))) → B1.1(a.0(b.1(a.0(b.1(a.0(b.0(a.1(a.0(b.0(b.1(a.0(b.1(a.0(B.1(x0)))))))))))))))
B1.1(a.0(b.1(a.0(a.1(a.0(b.0(B.1(x)))))))) → B1.0(b.1(a.0(b.1(a.0(B.1(x))))))
B1.0(b.0(b.1(x0))) → B1.1(a.0(b.1(a.0(b.1(a.0(b.1(a.0(b.1(a.0(b.1(a.0(b.1(x0)))))))))))))
B1.1(a.0(a.1(a.0(b.0(b.0(B.0(x))))))) → B1.1(a.0(b.1(a.0(a.1(a.0(b.1(a.0(b.1(a.0(B.0(x)))))))))))
B1.1(a.0(b.1(a.0(a.1(a.0(b.1(a.0(b.0(B.1(x)))))))))) → B1.0(b.1(a.0(b.1(a.0(B.1(x))))))
B1.0(b.0(a.1(a.0(b.0(B.1(x0)))))) → B1.1(a.0(b.1(a.0(b.1(a.0(a.1(a.0(b.1(a.0(b.1(a.0(B.1(x0)))))))))))))
B1.1(a.0(b.1(a.0(a.1(a.0(b.0(B.0(x)))))))) → B1.0(b.1(a.0(b.1(a.0(B.0(x))))))
B1.1(a.0(b.1(a.0(a.1(a.0(b.1(a.0(b.0(x))))))))) → B1.0(a.1(a.0(b.0(b.0(x)))))
B1.0(b.1(a.0(b.1(a.0(a.1(a.0(b.1(a.0(B.0(x0)))))))))) → B1.1(a.0(b.1(a.0(b.1(a.0(b.0(a.1(a.0(b.0(B.0(x0)))))))))))
B1.0(b.1(a.0(b.1(a.0(a.1(a.0(b.1(a.0(b.0(x0)))))))))) → B1.1(a.0(b.1(a.0(b.0(a.1(a.0(b.0(b.0(x0)))))))))
B1.1(a.0(b.1(a.0(a.1(a.0(b.1(a.0(b.1(x))))))))) → B1.0(b.1(x))
B1.0(b.1(a.0(a.1(a.0(b.0(b.0(B.0(x0)))))))) → B1.1(a.0(b.1(a.0(b.1(a.0(b.1(a.0(a.1(a.0(b.1(a.0(b.1(a.0(B.0(x0)))))))))))))))
B1.0(b.1(a.0(a.1(a.0(b.1(a.0(b.0(B.1(x0))))))))) → B1.1(a.0(b.1(a.0(b.0(a.1(a.0(b.0(b.1(a.0(b.1(a.0(B.1(x0)))))))))))))
B1.0(b.1(a.0(a.1(a.0(b.1(a.0(b.0(B.1(x0))))))))) → B1.1(a.0(b.0(a.1(a.0(b.0(b.1(a.0(b.1(a.0(B.1(x0)))))))))))
B1.1(a.0(b.1(a.0(a.1(a.0(b.1(a.0(B.1(x))))))))) → B1.0(a.1(a.0(b.0(B.1(x)))))
B1.0(b.0(a.1(a.0(b.1(x0))))) → B1.1(a.0(b.1(a.0(b.1(a.0(a.1(a.0(b.1(x0)))))))))
B1.0(b.1(a.0(a.1(a.0(b.1(a.0(b.0(B.1(x0))))))))) → B1.1(a.0(b.1(a.0(b.0(a.1(a.0(b.0(b.1(a.0(b.1(a.0(b.1(a.0(B.1(x0)))))))))))))))
B1.0(b.0(a.1(a.0(b.0(x0))))) → B1.1(a.0(b.1(a.0(b.1(a.0(a.1(a.0(b.0(x0)))))))))
B1.0(b.1(a.0(b.1(a.0(a.1(a.0(b.0(B.1(x0))))))))) → B1.1(a.0(b.1(a.0(b.1(a.0(b.0(a.1(a.0(b.0(b.1(a.0(b.1(a.0(B.1(x0)))))))))))))))
B1.0(b.1(a.0(a.1(a.0(b.0(b.0(B.1(x0)))))))) → B1.1(a.0(b.1(a.0(b.1(a.0(b.1(a.0(b.1(a.0(a.1(a.0(b.1(a.0(b.1(a.0(b.1(a.0(B.1(x0)))))))))))))))))))
B1.0(b.1(a.0(b.1(a.0(a.1(a.0(b.1(a.0(b.1(x0)))))))))) → B1.1(a.0(b.1(a.0(b.1(a.0(b.0(a.1(a.0(b.0(b.1(x0)))))))))))
B1.0(b.1(a.0(a.1(a.0(b.0(b.0(B.0(x0)))))))) → B1.1(a.0(b.1(a.0(b.1(a.0(b.1(a.0(b.1(a.0(a.1(a.0(b.1(a.0(b.1(a.0(b.1(a.0(B.0(x0)))))))))))))))))))
B1.0(b.0(a.1(a.0(b.0(B.0(x0)))))) → B1.1(a.0(b.1(a.0(b.1(a.0(b.1(a.0(a.1(a.0(b.1(a.0(b.1(a.0(b.1(a.0(B.0(x0)))))))))))))))))
B1.0(b.1(a.0(b.1(a.0(a.1(a.0(b.1(a.0(B.0(x0)))))))))) → B1.1(a.0(b.1(a.0(b.0(a.1(a.0(b.0(B.0(x0)))))))))
B1.1(a.0(b.1(a.0(a.1(a.0(b.1(a.0(b.0(B.1(x)))))))))) → B1.0(a.1(a.0(b.0(b.1(a.0(b.1(a.0(B.1(x)))))))))
B1.0(b.1(a.0(a.1(a.0(b.0(B.0(x0))))))) → B1.1(a.0(b.1(a.0(b.0(a.1(a.0(b.0(b.1(a.0(B.0(x0)))))))))))
B1.1(a.0(b.1(a.0(a.1(a.0(b.0(B.1(x)))))))) → B1.0(a.1(a.0(b.0(b.1(a.0(b.1(a.0(B.1(x)))))))))
B1.0(a.1(a.0(b.0(x)))) → B1.1(a.0(a.1(a.0(b.0(x)))))
B1.0(b.1(a.0(a.1(a.0(b.0(B.0(x0))))))) → B1.1(a.0(b.0(a.1(a.0(b.0(b.1(a.0(B.0(x0)))))))))
B1.0(b.1(a.0(a.1(a.0(b.0(b.1(x0))))))) → B1.1(a.0(b.1(a.0(b.1(a.0(b.1(a.0(a.1(a.0(b.1(x0)))))))))))
B1.0(b.1(a.0(a.1(a.0(b.1(a.0(b.0(x0)))))))) → B1.1(a.0(b.0(a.1(a.0(b.0(b.0(x0)))))))
B1.1(a.0(a.1(a.0(b.0(b.0(x)))))) → B1.1(a.0(a.1(a.0(b.0(x)))))
B1.0(b.1(a.0(b.1(a.0(a.1(a.0(b.1(a.0(b.0(B.0(x0))))))))))) → B1.1(a.0(b.1(a.0(b.0(a.1(a.0(b.0(b.1(a.0(b.1(a.0(b.1(a.0(B.0(x0)))))))))))))))
B1.0(b.1(a.0(b.1(a.0(a.1(a.0(b.1(a.0(b.0(B.1(x0))))))))))) → B1.1(a.0(b.1(a.0(b.0(a.1(a.0(b.0(b.1(a.0(b.1(a.0(B.1(x0)))))))))))))
B1.1(a.0(b.1(a.0(a.1(a.0(b.0(B.0(x)))))))) → B1.0(b.1(a.0(B.0(x))))
B1.0(b.1(a.0(b.1(a.0(a.1(a.0(b.1(a.0(b.0(x0)))))))))) → B1.1(a.0(b.1(a.0(b.1(a.0(b.0(a.1(a.0(b.0(b.0(x0)))))))))))
B1.0(b.1(a.0(a.1(a.0(b.0(b.0(B.0(x0)))))))) → B1.1(a.0(b.1(a.0(b.1(a.0(b.1(a.0(b.1(a.0(a.1(a.0(b.1(a.0(b.1(a.0(B.0(x0)))))))))))))))))
B1.0(b.0(a.1(a.0(b.0(B.1(x0)))))) → B1.1(a.0(b.1(a.0(b.1(a.0(b.1(a.0(a.1(a.0(b.1(a.0(b.1(a.0(b.1(a.0(B.1(x0)))))))))))))))))
B1.0(b.1(a.0(a.1(a.0(b.1(a.0(b.0(B.1(x0))))))))) → B1.1(a.0(b.0(a.1(a.0(b.0(b.1(a.0(b.1(a.0(b.1(a.0(B.1(x0)))))))))))))
B1.1(a.0(b.1(a.0(a.1(a.0(b.1(a.0(B.0(x))))))))) → B1.0(a.1(a.0(b.0(B.0(x)))))
B1.0(b.1(a.0(b.1(a.0(a.1(a.0(b.1(a.0(b.0(B.0(x0))))))))))) → B1.1(a.0(b.1(a.0(b.1(a.0(b.0(a.1(a.0(b.0(b.1(a.0(b.1(a.0(B.0(x0)))))))))))))))
B1.0(b.1(a.0(a.1(a.0(b.0(b.0(x0))))))) → B1.1(a.0(b.1(a.0(b.1(a.0(b.1(a.0(b.1(a.0(a.1(a.0(b.0(x0)))))))))))))
B1.0(b.1(a.0(b.1(a.0(a.1(a.0(b.0(B.0(x0))))))))) → B1.1(a.0(b.1(a.0(b.1(a.0(b.0(a.1(a.0(b.0(b.1(a.0(b.1(a.0(B.0(x0)))))))))))))))
B1.0(b.1(a.0(a.1(a.0(b.1(a.0(b.0(B.0(x0))))))))) → B1.1(a.0(b.1(a.0(b.0(a.1(a.0(b.0(b.1(a.0(b.1(a.0(b.1(a.0(B.0(x0)))))))))))))))
B1.0(b.1(a.0(b.1(a.0(a.1(a.0(b.0(B.1(x0))))))))) → B1.1(a.0(b.1(a.0(b.0(a.1(a.0(b.0(b.1(a.0(B.1(x0)))))))))))
B1.0(b.1(a.0(a.1(a.0(b.0(b.1(x0))))))) → B1.1(a.0(b.1(a.0(b.1(a.0(b.1(a.0(b.1(a.0(a.1(a.0(b.1(x0)))))))))))))
B1.0(b.1(a.0(b.1(a.0(a.1(a.0(b.1(a.0(b.0(B.0(x0))))))))))) → B1.1(a.0(b.1(a.0(b.0(a.1(a.0(b.0(b.1(a.0(b.1(a.0(B.0(x0)))))))))))))
B1.0(b.1(a.0(b.1(a.0(a.1(a.0(b.1(a.0(b.0(B.1(x0))))))))))) → B1.1(a.0(b.1(a.0(b.1(a.0(b.0(a.1(a.0(b.0(b.1(a.0(b.1(a.0(b.1(a.0(B.1(x0)))))))))))))))))
B1.0(b.1(a.0(a.1(a.0(b.1(a.0(b.0(B.0(x0))))))))) → B1.1(a.0(b.0(a.1(a.0(b.0(b.1(a.0(b.1(a.0(B.0(x0)))))))))))
B1.0(b.0(a.1(a.0(b.1(x0))))) → B1.1(a.0(b.1(a.0(b.1(a.0(b.1(a.0(a.1(a.0(b.1(x0)))))))))))
B1.0(b.1(a.0(a.1(a.0(b.0(b.0(x0))))))) → B1.1(a.0(b.1(a.0(b.1(a.0(b.1(a.0(a.1(a.0(b.0(x0)))))))))))
B1.0(b.1(a.0(a.1(a.0(b.1(a.0(B.1(x0)))))))) → B1.1(a.0(b.1(a.0(b.0(a.1(a.0(b.0(B.1(x0)))))))))
B1.0(b.1(a.0(a.1(a.0(b.1(a.0(b.1(x0)))))))) → B1.1(a.0(b.0(a.1(a.0(b.0(b.1(x0)))))))
B1.0(b.1(a.0(b.1(a.0(a.1(a.0(b.1(a.0(B.1(x0)))))))))) → B1.1(a.0(b.1(a.0(b.0(a.1(a.0(b.0(B.1(x0)))))))))
B1.0(b.1(a.0(a.1(a.0(b.0(b.0(B.1(x0)))))))) → B1.1(a.0(b.1(a.0(b.1(a.0(b.1(a.0(b.1(a.0(a.1(a.0(b.1(a.0(b.1(a.0(B.1(x0)))))))))))))))))
B1.0(b.1(a.0(a.1(a.0(b.0(B.0(x0))))))) → B1.1(a.0(b.0(a.1(a.0(b.0(b.1(a.0(b.1(a.0(B.0(x0)))))))))))
B1.0(b.0(b.0(x0))) → B1.1(a.0(b.1(a.0(b.1(a.0(b.1(a.0(b.1(a.0(b.1(a.0(b.0(x0)))))))))))))
B1.0(b.1(a.0(b.1(a.0(a.1(a.0(b.0(B.0(x0))))))))) → B1.1(a.0(b.1(a.0(b.1(a.0(b.0(a.1(a.0(b.0(b.1(a.0(B.0(x0)))))))))))))
B1.0(b.0(a.1(a.0(b.0(x0))))) → B1.1(a.0(b.1(a.0(b.1(a.0(b.1(a.0(a.1(a.0(b.0(x0)))))))))))
B1.0(b.1(a.0(a.1(a.0(b.1(a.0(B.1(x0)))))))) → B1.1(a.0(b.0(a.1(a.0(b.0(B.1(x0)))))))
B1.0(b.1(a.0(a.1(a.0(b.1(a.0(B.0(x0)))))))) → B1.1(a.0(b.1(a.0(b.0(a.1(a.0(b.0(B.0(x0)))))))))
B1.0(b.1(a.0(b.1(a.0(a.1(a.0(b.1(a.0(B.1(x0)))))))))) → B1.1(a.0(b.1(a.0(b.1(a.0(b.0(a.1(a.0(b.0(B.1(x0)))))))))))
B1.1(a.0(b.1(a.0(a.1(a.0(b.1(a.0(b.1(x))))))))) → B1.0(a.1(a.0(b.0(b.1(x)))))
B1.0(b.1(a.0(a.1(a.0(b.0(b.0(B.1(x0)))))))) → B1.1(a.0(b.1(a.0(b.1(a.0(b.1(a.0(a.1(a.0(b.1(a.0(b.1(a.0(b.1(a.0(B.1(x0)))))))))))))))))
B1.0(b.1(a.0(b.1(a.0(a.1(a.0(b.0(B.0(x0))))))))) → B1.1(a.0(b.1(a.0(b.0(a.1(a.0(b.0(b.1(a.0(B.0(x0)))))))))))
B1.1(a.0(b.1(a.0(a.1(a.0(b.1(a.0(b.0(x))))))))) → B1.0(b.0(x))
B1.1(a.0(a.1(a.0(b.0(b.0(B.0(x))))))) → B1.1(a.0(b.1(a.0(a.1(a.0(b.1(a.0(b.1(a.0(b.1(a.0(B.0(x)))))))))))))
B1.1(a.0(b.1(a.0(a.1(a.0(b.1(a.0(b.0(B.0(x)))))))))) → B1.0(b.1(a.0(b.1(a.0(b.1(a.0(B.0(x))))))))
B1.0(b.1(a.0(a.1(a.0(b.0(B.1(x0))))))) → B1.1(a.0(b.1(a.0(b.0(a.1(a.0(b.0(b.1(a.0(b.1(a.0(B.1(x0)))))))))))))
B1.0(b.1(a.0(a.1(a.0(b.0(B.1(x0))))))) → B1.1(a.0(b.0(a.1(a.0(b.0(b.1(a.0(b.1(a.0(B.1(x0)))))))))))
B1.1(a.0(a.1(a.0(b.0(b.0(B.1(x))))))) → B1.1(a.0(b.1(a.0(a.1(a.0(b.1(a.0(b.1(a.0(B.1(x)))))))))))
B1.1(a.0(b.1(a.0(a.1(a.0(b.1(a.0(b.0(B.1(x)))))))))) → B1.0(b.1(a.0(b.1(a.0(b.1(a.0(B.1(x))))))))
B1.0(b.1(a.0(b.1(a.0(a.1(a.0(b.1(a.0(b.1(x0)))))))))) → B1.1(a.0(b.1(a.0(b.0(a.1(a.0(b.0(b.1(x0)))))))))
B1.0(b.1(a.0(a.1(a.0(b.1(a.0(b.0(B.0(x0))))))))) → B1.1(a.0(b.1(a.0(b.0(a.1(a.0(b.0(b.1(a.0(b.1(a.0(B.0(x0)))))))))))))
B1.1(a.0(b.1(a.0(a.1(a.0(b.0(B.0(x)))))))) → B1.0(a.1(a.0(b.0(b.1(a.0(B.0(x)))))))
B1.0(b.0(a.1(a.0(b.0(B.1(x0)))))) → B1.1(a.0(b.1(a.0(b.1(a.0(b.1(a.0(a.1(a.0(b.1(a.0(b.1(a.0(B.1(x0)))))))))))))))
B1.0(b.1(a.0(b.1(a.0(a.1(a.0(b.1(a.0(b.0(B.1(x0))))))))))) → B1.1(a.0(b.1(a.0(b.0(a.1(a.0(b.0(b.1(a.0(b.1(a.0(b.1(a.0(B.1(x0)))))))))))))))
B1.0(b.0(x)) → B1.1(a.0(b.0(x)))
B1.0(b.1(a.0(a.1(a.0(b.0(B.1(x0))))))) → B1.1(a.0(b.0(a.1(a.0(b.0(b.1(a.0(B.1(x0)))))))))
B1.0(b.1(a.0(b.1(a.0(a.1(a.0(b.0(B.1(x0))))))))) → B1.1(a.0(b.1(a.0(b.0(a.1(a.0(b.0(b.1(a.0(b.1(a.0(B.1(x0)))))))))))))
B1.0(a.1(a.0(b.1(x)))) → B1.1(a.0(a.1(a.0(b.1(x)))))
B1.0(b.1(a.0(a.1(a.0(b.0(B.1(x0))))))) → B1.1(a.0(b.1(a.0(b.0(a.1(a.0(b.0(b.1(a.0(B.1(x0)))))))))))
B1.1(a.0(b.1(a.0(a.1(a.0(b.0(B.0(x)))))))) → B1.0(a.1(a.0(b.0(b.1(a.0(b.1(a.0(B.0(x)))))))))
The TRS R consists of the following rules:
b.1(a.0(a.1(a.0(b.0(b.0(B.0(x))))))) → b.1(a.0(b.1(a.0(a.1(a.0(b.1(a.0(b.1(a.0(b.1(a.0(B.0(x)))))))))))))
b.0(b.1(x)) → b.1(a.0(b.1(a.0(b.1(a.0(b.1(x)))))))
b.0(b.0(B.0(x))) → b.1(a.0(b.1(a.0(b.1(a.0(b.1(a.0(b.1(a.0(b.1(a.0(B.0(x)))))))))))))
b.1(a.0(a.1(a.0(b.0(b.0(B.1(x))))))) → b.1(a.0(b.1(a.0(a.1(a.0(b.1(a.0(b.1(a.0(B.1(x)))))))))))
b.1(a.0(a.1(a.0(b.0(b.1(x)))))) → b.1(a.0(b.1(a.0(a.1(a.0(b.1(x)))))))
b.0(a.1(a.0(b.0(B.0(x))))) → b.1(a.0(a.1(a.0(b.1(a.0(b.1(a.0(b.1(a.0(B.0(x)))))))))))
b.1(a.0(b.1(a.0(a.1(a.0(b.0(B.0(x)))))))) → b.0(a.1(a.0(b.0(b.1(a.0(b.1(a.0(B.0(x)))))))))
b.1(a.0(a.1(a.0(b.0(b.0(B.1(x))))))) → b.1(a.0(b.1(a.0(a.1(a.0(b.1(a.0(b.1(a.0(b.1(a.0(B.1(x)))))))))))))
b.0(B.1(x)) → b.1(a.0(B.1(x)))
b.0(a.1(a.0(b.0(B.1(x))))) → b.1(a.0(a.1(a.0(b.1(a.0(b.1(a.0(B.1(x)))))))))
b.1(a.0(b.1(a.0(a.1(a.0(b.1(a.0(b.0(B.1(x)))))))))) → b.0(a.1(a.0(b.0(b.1(a.0(b.1(a.0(B.1(x)))))))))
b.0(B.0(x)) → b.1(a.0(B.0(x)))
b.1(a.0(b.1(a.0(a.1(a.0(b.0(B.0(x)))))))) → b.0(a.1(a.0(b.0(b.1(a.0(B.0(x)))))))
b.0(b.0(x)) → b.1(a.0(b.1(a.0(b.1(a.0(b.0(x)))))))
b.1(a.0(b.1(a.0(a.1(a.0(b.0(B.1(x)))))))) → b.0(a.1(a.0(b.0(b.1(a.0(B.1(x)))))))
b.1(a.0(a.1(a.0(b.0(b.0(B.0(x))))))) → b.1(a.0(b.1(a.0(a.1(a.0(b.1(a.0(b.1(a.0(B.0(x)))))))))))
b.1(a.0(b.1(a.0(a.1(a.0(b.0(B.1(x)))))))) → b.0(a.1(a.0(b.0(b.1(a.0(b.1(a.0(B.1(x)))))))))
b.1(a.0(b.1(a.0(a.1(a.0(b.1(a.0(b.0(B.1(x)))))))))) → b.0(a.1(a.0(b.0(b.1(a.0(b.1(a.0(b.1(a.0(B.1(x)))))))))))
b.1(a.0(b.1(a.0(a.1(a.0(b.1(a.0(b.0(B.0(x)))))))))) → b.0(a.1(a.0(b.0(b.1(a.0(b.1(a.0(B.0(x)))))))))
b.1(a.0(b.1(a.0(a.1(a.0(b.1(a.0(B.0(x))))))))) → b.0(a.1(a.0(b.0(B.0(x)))))
b.0(b.0(B.0(x))) → b.1(a.0(b.1(a.0(b.1(a.0(b.1(a.0(b.1(a.0(B.0(x)))))))))))
b.0(b.0(B.1(x))) → b.1(a.0(b.1(a.0(b.1(a.0(b.1(a.0(b.1(a.0(B.1(x)))))))))))
b.0(a.1(a.0(b.0(B.1(x))))) → b.1(a.0(a.1(a.0(b.1(a.0(b.1(a.0(b.1(a.0(B.1(x)))))))))))
b.1(a.0(a.1(a.0(b.0(b.0(x)))))) → b.1(a.0(b.1(a.0(a.1(a.0(b.0(x)))))))
b.0(a.1(a.0(b.1(x)))) → b.1(a.0(a.1(a.0(b.1(x)))))
b.0(a.1(a.0(b.0(B.0(x))))) → b.1(a.0(a.1(a.0(b.1(a.0(b.1(a.0(B.0(x)))))))))
b.1(a.0(b.1(a.0(a.1(a.0(b.1(a.0(B.1(x))))))))) → b.0(a.1(a.0(b.0(B.1(x)))))
b.1(a.0(b.1(a.0(a.1(a.0(b.1(a.0(b.0(x))))))))) → b.0(a.1(a.0(b.0(b.0(x)))))
b.0(a.1(a.0(b.0(x)))) → b.1(a.0(a.1(a.0(b.0(x)))))
b.0(b.0(B.1(x))) → b.1(a.0(b.1(a.0(b.1(a.0(b.1(a.0(b.1(a.0(b.1(a.0(B.1(x)))))))))))))
b.1(a.0(b.1(a.0(a.1(a.0(b.1(a.0(b.1(x))))))))) → b.0(a.1(a.0(b.0(b.1(x)))))
b.1(a.0(b.1(a.0(a.1(a.0(b.1(a.0(b.0(B.0(x)))))))))) → b.0(a.1(a.0(b.0(b.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
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ SemLabProof
↳ SemLabProof2
↳ QDP
↳ QDPToSRSProof
↳ QTRS
↳ QTRS Reverse
↳ QTRS
↳ QTRS Reverse
↳ QTRS Reverse
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ SemLabProof
↳ QDP
↳ RuleRemovalProof
↳ SemLabProof2
↳ QTRS Reverse
↳ QTRS Reverse
Q DP problem:
The TRS P consists of the following rules:
B1.1(a.0(a.1(a.0(b.0(b.0(x)))))) → B1.1(a.0(b.1(a.0(a.1(a.0(b.0(x)))))))
B1.1(a.0(b.1(a.0(a.1(a.0(b.0(B.1(x)))))))) → B1.0(b.1(a.0(B.1(x))))
B1.0(b.1(a.0(a.1(a.0(b.0(B.0(x0))))))) → B1.1(a.0(b.1(a.0(b.0(a.1(a.0(b.0(b.1(a.0(b.1(a.0(B.0(x0)))))))))))))
B1.0(b.1(a.0(a.1(a.0(b.1(a.0(b.0(B.0(x0))))))))) → B1.1(a.0(b.0(a.1(a.0(b.0(b.1(a.0(b.1(a.0(b.1(a.0(B.0(x0)))))))))))))
B1.1(a.0(b.1(a.0(a.1(a.0(b.1(a.0(b.0(B.0(x)))))))))) → B1.0(a.1(a.0(b.0(b.1(a.0(b.1(a.0(B.0(x)))))))))
B1.1(a.0(b.1(a.0(a.1(a.0(b.1(a.0(b.0(B.1(x)))))))))) → B1.0(a.1(a.0(b.0(b.1(a.0(b.1(a.0(b.1(a.0(B.1(x)))))))))))
B1.0(b.1(a.0(b.1(a.0(a.1(a.0(b.0(B.1(x0))))))))) → B1.1(a.0(b.1(a.0(b.1(a.0(b.0(a.1(a.0(b.0(b.1(a.0(B.1(x0)))))))))))))
B1.0(b.1(a.0(b.1(a.0(a.1(a.0(b.1(a.0(b.0(B.0(x0))))))))))) → B1.1(a.0(b.1(a.0(b.1(a.0(b.0(a.1(a.0(b.0(b.1(a.0(b.1(a.0(b.1(a.0(B.0(x0)))))))))))))))))
B1.0(b.0(a.1(a.0(b.0(B.0(x0)))))) → B1.1(a.0(b.1(a.0(b.1(a.0(a.1(a.0(b.1(a.0(b.1(a.0(b.1(a.0(B.0(x0)))))))))))))))
B1.1(a.0(a.1(a.0(b.0(b.0(B.1(x))))))) → B1.1(a.0(b.1(a.0(a.1(a.0(b.1(a.0(b.1(a.0(b.1(a.0(B.1(x)))))))))))))
B1.0(b.0(a.1(a.0(b.0(B.0(x0)))))) → B1.1(a.0(b.1(a.0(b.1(a.0(a.1(a.0(b.1(a.0(b.1(a.0(B.0(x0)))))))))))))
B1.0(b.1(a.0(a.1(a.0(b.1(a.0(B.0(x0)))))))) → B1.1(a.0(b.0(a.1(a.0(b.0(B.0(x0)))))))
B1.0(b.1(a.0(a.1(a.0(b.0(b.0(B.1(x0)))))))) → B1.1(a.0(b.1(a.0(b.1(a.0(b.1(a.0(a.1(a.0(b.1(a.0(b.1(a.0(B.1(x0)))))))))))))))
B1.0(b.1(x)) → B1.1(a.0(b.1(x)))
B1.0(b.1(a.0(b.1(a.0(a.1(a.0(b.0(B.0(x0))))))))) → B1.1(a.0(b.1(a.0(b.0(a.1(a.0(b.0(b.1(a.0(b.1(a.0(B.0(x0)))))))))))))
B1.0(b.0(b.1(x0))) → B1.1(a.0(b.1(a.0(b.1(a.0(b.1(a.0(b.1(a.0(b.1(x0)))))))))))
B1.1(a.0(a.1(a.0(b.0(b.1(x)))))) → B1.1(a.0(a.1(a.0(b.1(x)))))
B1.0(b.1(a.0(a.1(a.0(b.1(a.0(b.1(x0)))))))) → B1.1(a.0(b.1(a.0(b.0(a.1(a.0(b.0(b.1(x0)))))))))
B1.0(b.1(a.0(a.1(a.0(b.1(a.0(b.0(x0)))))))) → B1.1(a.0(b.1(a.0(b.0(a.1(a.0(b.0(b.0(x0)))))))))
B1.0(b.1(a.0(a.1(a.0(b.0(b.0(B.0(x0)))))))) → B1.1(a.0(b.1(a.0(b.1(a.0(b.1(a.0(a.1(a.0(b.1(a.0(b.1(a.0(b.1(a.0(B.0(x0)))))))))))))))))
B1.1(a.0(a.1(a.0(b.0(b.1(x)))))) → B1.1(a.0(b.1(a.0(a.1(a.0(b.1(x)))))))
B1.0(b.0(b.0(x0))) → B1.1(a.0(b.1(a.0(b.1(a.0(b.1(a.0(b.1(a.0(b.0(x0)))))))))))
B1.1(a.0(b.1(a.0(a.1(a.0(b.1(a.0(b.0(B.0(x)))))))))) → B1.0(a.1(a.0(b.0(b.1(a.0(b.1(a.0(b.1(a.0(B.0(x)))))))))))
B1.0(b.0(a.1(a.0(b.0(B.1(x0)))))) → B1.1(a.0(b.1(a.0(b.1(a.0(a.1(a.0(b.1(a.0(b.1(a.0(b.1(a.0(B.1(x0)))))))))))))))
B1.1(a.0(b.1(a.0(a.1(a.0(b.1(a.0(b.0(B.0(x)))))))))) → B1.0(b.1(a.0(b.1(a.0(B.0(x))))))
B1.1(a.0(b.1(a.0(a.1(a.0(b.0(B.1(x)))))))) → B1.0(a.1(a.0(b.0(b.1(a.0(B.1(x)))))))
B1.0(b.0(a.1(a.0(b.0(B.0(x0)))))) → B1.1(a.0(b.1(a.0(b.1(a.0(b.1(a.0(a.1(a.0(b.1(a.0(b.1(a.0(B.0(x0)))))))))))))))
B1.0(b.1(a.0(b.1(a.0(a.1(a.0(b.1(a.0(b.0(B.1(x0))))))))))) → B1.1(a.0(b.1(a.0(b.1(a.0(b.0(a.1(a.0(b.0(b.1(a.0(b.1(a.0(B.1(x0)))))))))))))))
B1.1(a.0(b.1(a.0(a.1(a.0(b.0(B.1(x)))))))) → B1.0(b.1(a.0(b.1(a.0(B.1(x))))))
B1.0(b.0(b.1(x0))) → B1.1(a.0(b.1(a.0(b.1(a.0(b.1(a.0(b.1(a.0(b.1(a.0(b.1(x0)))))))))))))
B1.1(a.0(a.1(a.0(b.0(b.0(B.0(x))))))) → B1.1(a.0(b.1(a.0(a.1(a.0(b.1(a.0(b.1(a.0(B.0(x)))))))))))
B1.1(a.0(b.1(a.0(a.1(a.0(b.1(a.0(b.0(B.1(x)))))))))) → B1.0(b.1(a.0(b.1(a.0(B.1(x))))))
B1.0(b.0(a.1(a.0(b.0(B.1(x0)))))) → B1.1(a.0(b.1(a.0(b.1(a.0(a.1(a.0(b.1(a.0(b.1(a.0(B.1(x0)))))))))))))
B1.1(a.0(b.1(a.0(a.1(a.0(b.0(B.0(x)))))))) → B1.0(b.1(a.0(b.1(a.0(B.0(x))))))
B1.1(a.0(b.1(a.0(a.1(a.0(b.1(a.0(b.0(x))))))))) → B1.0(a.1(a.0(b.0(b.0(x)))))
B1.0(b.1(a.0(b.1(a.0(a.1(a.0(b.1(a.0(B.0(x0)))))))))) → B1.1(a.0(b.1(a.0(b.1(a.0(b.0(a.1(a.0(b.0(B.0(x0)))))))))))
B1.0(b.1(a.0(b.1(a.0(a.1(a.0(b.1(a.0(b.0(x0)))))))))) → B1.1(a.0(b.1(a.0(b.0(a.1(a.0(b.0(b.0(x0)))))))))
B1.1(a.0(b.1(a.0(a.1(a.0(b.1(a.0(b.1(x))))))))) → B1.0(b.1(x))
B1.0(b.1(a.0(a.1(a.0(b.0(b.0(B.0(x0)))))))) → B1.1(a.0(b.1(a.0(b.1(a.0(b.1(a.0(a.1(a.0(b.1(a.0(b.1(a.0(B.0(x0)))))))))))))))
B1.0(b.1(a.0(a.1(a.0(b.1(a.0(b.0(B.1(x0))))))))) → B1.1(a.0(b.1(a.0(b.0(a.1(a.0(b.0(b.1(a.0(b.1(a.0(B.1(x0)))))))))))))
B1.0(b.1(a.0(a.1(a.0(b.1(a.0(b.0(B.1(x0))))))))) → B1.1(a.0(b.0(a.1(a.0(b.0(b.1(a.0(b.1(a.0(B.1(x0)))))))))))
B1.1(a.0(b.1(a.0(a.1(a.0(b.1(a.0(B.1(x))))))))) → B1.0(a.1(a.0(b.0(B.1(x)))))
B1.0(b.0(a.1(a.0(b.1(x0))))) → B1.1(a.0(b.1(a.0(b.1(a.0(a.1(a.0(b.1(x0)))))))))
B1.0(b.1(a.0(a.1(a.0(b.1(a.0(b.0(B.1(x0))))))))) → B1.1(a.0(b.1(a.0(b.0(a.1(a.0(b.0(b.1(a.0(b.1(a.0(b.1(a.0(B.1(x0)))))))))))))))
B1.0(b.0(a.1(a.0(b.0(x0))))) → B1.1(a.0(b.1(a.0(b.1(a.0(a.1(a.0(b.0(x0)))))))))
B1.0(b.1(a.0(b.1(a.0(a.1(a.0(b.0(B.1(x0))))))))) → B1.1(a.0(b.1(a.0(b.1(a.0(b.0(a.1(a.0(b.0(b.1(a.0(b.1(a.0(B.1(x0)))))))))))))))
B1.0(b.1(a.0(a.1(a.0(b.0(b.0(B.1(x0)))))))) → B1.1(a.0(b.1(a.0(b.1(a.0(b.1(a.0(b.1(a.0(a.1(a.0(b.1(a.0(b.1(a.0(b.1(a.0(B.1(x0)))))))))))))))))))
B1.0(b.1(a.0(b.1(a.0(a.1(a.0(b.1(a.0(b.1(x0)))))))))) → B1.1(a.0(b.1(a.0(b.1(a.0(b.0(a.1(a.0(b.0(b.1(x0)))))))))))
B1.0(b.1(a.0(a.1(a.0(b.0(b.0(B.0(x0)))))))) → B1.1(a.0(b.1(a.0(b.1(a.0(b.1(a.0(b.1(a.0(a.1(a.0(b.1(a.0(b.1(a.0(b.1(a.0(B.0(x0)))))))))))))))))))
B1.0(b.0(a.1(a.0(b.0(B.0(x0)))))) → B1.1(a.0(b.1(a.0(b.1(a.0(b.1(a.0(a.1(a.0(b.1(a.0(b.1(a.0(b.1(a.0(B.0(x0)))))))))))))))))
B1.0(b.1(a.0(b.1(a.0(a.1(a.0(b.1(a.0(B.0(x0)))))))))) → B1.1(a.0(b.1(a.0(b.0(a.1(a.0(b.0(B.0(x0)))))))))
B1.1(a.0(b.1(a.0(a.1(a.0(b.1(a.0(b.0(B.1(x)))))))))) → B1.0(a.1(a.0(b.0(b.1(a.0(b.1(a.0(B.1(x)))))))))
B1.0(b.1(a.0(a.1(a.0(b.0(B.0(x0))))))) → B1.1(a.0(b.1(a.0(b.0(a.1(a.0(b.0(b.1(a.0(B.0(x0)))))))))))
B1.1(a.0(b.1(a.0(a.1(a.0(b.0(B.1(x)))))))) → B1.0(a.1(a.0(b.0(b.1(a.0(b.1(a.0(B.1(x)))))))))
B1.0(a.1(a.0(b.0(x)))) → B1.1(a.0(a.1(a.0(b.0(x)))))
B1.0(b.1(a.0(a.1(a.0(b.0(B.0(x0))))))) → B1.1(a.0(b.0(a.1(a.0(b.0(b.1(a.0(B.0(x0)))))))))
B1.0(b.1(a.0(a.1(a.0(b.0(b.1(x0))))))) → B1.1(a.0(b.1(a.0(b.1(a.0(b.1(a.0(a.1(a.0(b.1(x0)))))))))))
B1.0(b.1(a.0(a.1(a.0(b.1(a.0(b.0(x0)))))))) → B1.1(a.0(b.0(a.1(a.0(b.0(b.0(x0)))))))
B1.1(a.0(a.1(a.0(b.0(b.0(x)))))) → B1.1(a.0(a.1(a.0(b.0(x)))))
B1.0(b.1(a.0(b.1(a.0(a.1(a.0(b.1(a.0(b.0(B.0(x0))))))))))) → B1.1(a.0(b.1(a.0(b.0(a.1(a.0(b.0(b.1(a.0(b.1(a.0(b.1(a.0(B.0(x0)))))))))))))))
B1.0(b.1(a.0(b.1(a.0(a.1(a.0(b.1(a.0(b.0(B.1(x0))))))))))) → B1.1(a.0(b.1(a.0(b.0(a.1(a.0(b.0(b.1(a.0(b.1(a.0(B.1(x0)))))))))))))
B1.1(a.0(b.1(a.0(a.1(a.0(b.0(B.0(x)))))))) → B1.0(b.1(a.0(B.0(x))))
B1.0(b.1(a.0(b.1(a.0(a.1(a.0(b.1(a.0(b.0(x0)))))))))) → B1.1(a.0(b.1(a.0(b.1(a.0(b.0(a.1(a.0(b.0(b.0(x0)))))))))))
B1.0(b.1(a.0(a.1(a.0(b.0(b.0(B.0(x0)))))))) → B1.1(a.0(b.1(a.0(b.1(a.0(b.1(a.0(b.1(a.0(a.1(a.0(b.1(a.0(b.1(a.0(B.0(x0)))))))))))))))))
B1.0(b.0(a.1(a.0(b.0(B.1(x0)))))) → B1.1(a.0(b.1(a.0(b.1(a.0(b.1(a.0(a.1(a.0(b.1(a.0(b.1(a.0(b.1(a.0(B.1(x0)))))))))))))))))
B1.0(b.1(a.0(a.1(a.0(b.1(a.0(b.0(B.1(x0))))))))) → B1.1(a.0(b.0(a.1(a.0(b.0(b.1(a.0(b.1(a.0(b.1(a.0(B.1(x0)))))))))))))
B1.1(a.0(b.1(a.0(a.1(a.0(b.1(a.0(B.0(x))))))))) → B1.0(a.1(a.0(b.0(B.0(x)))))
B1.0(b.1(a.0(b.1(a.0(a.1(a.0(b.1(a.0(b.0(B.0(x0))))))))))) → B1.1(a.0(b.1(a.0(b.1(a.0(b.0(a.1(a.0(b.0(b.1(a.0(b.1(a.0(B.0(x0)))))))))))))))
B1.0(b.1(a.0(a.1(a.0(b.0(b.0(x0))))))) → B1.1(a.0(b.1(a.0(b.1(a.0(b.1(a.0(b.1(a.0(a.1(a.0(b.0(x0)))))))))))))
B1.0(b.1(a.0(b.1(a.0(a.1(a.0(b.0(B.0(x0))))))))) → B1.1(a.0(b.1(a.0(b.1(a.0(b.0(a.1(a.0(b.0(b.1(a.0(b.1(a.0(B.0(x0)))))))))))))))
B1.0(b.1(a.0(a.1(a.0(b.1(a.0(b.0(B.0(x0))))))))) → B1.1(a.0(b.1(a.0(b.0(a.1(a.0(b.0(b.1(a.0(b.1(a.0(b.1(a.0(B.0(x0)))))))))))))))
B1.0(b.1(a.0(b.1(a.0(a.1(a.0(b.0(B.1(x0))))))))) → B1.1(a.0(b.1(a.0(b.0(a.1(a.0(b.0(b.1(a.0(B.1(x0)))))))))))
B1.0(b.1(a.0(a.1(a.0(b.0(b.1(x0))))))) → B1.1(a.0(b.1(a.0(b.1(a.0(b.1(a.0(b.1(a.0(a.1(a.0(b.1(x0)))))))))))))
B1.0(b.1(a.0(b.1(a.0(a.1(a.0(b.1(a.0(b.0(B.0(x0))))))))))) → B1.1(a.0(b.1(a.0(b.0(a.1(a.0(b.0(b.1(a.0(b.1(a.0(B.0(x0)))))))))))))
B1.0(b.1(a.0(b.1(a.0(a.1(a.0(b.1(a.0(b.0(B.1(x0))))))))))) → B1.1(a.0(b.1(a.0(b.1(a.0(b.0(a.1(a.0(b.0(b.1(a.0(b.1(a.0(b.1(a.0(B.1(x0)))))))))))))))))
B1.0(b.1(a.0(a.1(a.0(b.1(a.0(b.0(B.0(x0))))))))) → B1.1(a.0(b.0(a.1(a.0(b.0(b.1(a.0(b.1(a.0(B.0(x0)))))))))))
B1.0(b.0(a.1(a.0(b.1(x0))))) → B1.1(a.0(b.1(a.0(b.1(a.0(b.1(a.0(a.1(a.0(b.1(x0)))))))))))
B1.0(b.1(a.0(a.1(a.0(b.0(b.0(x0))))))) → B1.1(a.0(b.1(a.0(b.1(a.0(b.1(a.0(a.1(a.0(b.0(x0)))))))))))
B1.0(b.1(a.0(a.1(a.0(b.1(a.0(B.1(x0)))))))) → B1.1(a.0(b.1(a.0(b.0(a.1(a.0(b.0(B.1(x0)))))))))
B1.0(b.1(a.0(a.1(a.0(b.1(a.0(b.1(x0)))))))) → B1.1(a.0(b.0(a.1(a.0(b.0(b.1(x0)))))))
B1.0(b.1(a.0(b.1(a.0(a.1(a.0(b.1(a.0(B.1(x0)))))))))) → B1.1(a.0(b.1(a.0(b.0(a.1(a.0(b.0(B.1(x0)))))))))
B1.0(b.1(a.0(a.1(a.0(b.0(b.0(B.1(x0)))))))) → B1.1(a.0(b.1(a.0(b.1(a.0(b.1(a.0(b.1(a.0(a.1(a.0(b.1(a.0(b.1(a.0(B.1(x0)))))))))))))))))
B1.0(b.1(a.0(a.1(a.0(b.0(B.0(x0))))))) → B1.1(a.0(b.0(a.1(a.0(b.0(b.1(a.0(b.1(a.0(B.0(x0)))))))))))
B1.0(b.0(b.0(x0))) → B1.1(a.0(b.1(a.0(b.1(a.0(b.1(a.0(b.1(a.0(b.1(a.0(b.0(x0)))))))))))))
B1.0(b.1(a.0(b.1(a.0(a.1(a.0(b.0(B.0(x0))))))))) → B1.1(a.0(b.1(a.0(b.1(a.0(b.0(a.1(a.0(b.0(b.1(a.0(B.0(x0)))))))))))))
B1.0(b.0(a.1(a.0(b.0(x0))))) → B1.1(a.0(b.1(a.0(b.1(a.0(b.1(a.0(a.1(a.0(b.0(x0)))))))))))
B1.0(b.1(a.0(a.1(a.0(b.1(a.0(B.1(x0)))))))) → B1.1(a.0(b.0(a.1(a.0(b.0(B.1(x0)))))))
B1.0(b.1(a.0(a.1(a.0(b.1(a.0(B.0(x0)))))))) → B1.1(a.0(b.1(a.0(b.0(a.1(a.0(b.0(B.0(x0)))))))))
B1.0(b.1(a.0(b.1(a.0(a.1(a.0(b.1(a.0(B.1(x0)))))))))) → B1.1(a.0(b.1(a.0(b.1(a.0(b.0(a.1(a.0(b.0(B.1(x0)))))))))))
B1.1(a.0(b.1(a.0(a.1(a.0(b.1(a.0(b.1(x))))))))) → B1.0(a.1(a.0(b.0(b.1(x)))))
B1.0(b.1(a.0(a.1(a.0(b.0(b.0(B.1(x0)))))))) → B1.1(a.0(b.1(a.0(b.1(a.0(b.1(a.0(a.1(a.0(b.1(a.0(b.1(a.0(b.1(a.0(B.1(x0)))))))))))))))))
B1.0(b.1(a.0(b.1(a.0(a.1(a.0(b.0(B.0(x0))))))))) → B1.1(a.0(b.1(a.0(b.0(a.1(a.0(b.0(b.1(a.0(B.0(x0)))))))))))
B1.1(a.0(b.1(a.0(a.1(a.0(b.1(a.0(b.0(x))))))))) → B1.0(b.0(x))
B1.1(a.0(a.1(a.0(b.0(b.0(B.0(x))))))) → B1.1(a.0(b.1(a.0(a.1(a.0(b.1(a.0(b.1(a.0(b.1(a.0(B.0(x)))))))))))))
B1.1(a.0(b.1(a.0(a.1(a.0(b.1(a.0(b.0(B.0(x)))))))))) → B1.0(b.1(a.0(b.1(a.0(b.1(a.0(B.0(x))))))))
B1.0(b.1(a.0(a.1(a.0(b.0(B.1(x0))))))) → B1.1(a.0(b.1(a.0(b.0(a.1(a.0(b.0(b.1(a.0(b.1(a.0(B.1(x0)))))))))))))
B1.0(b.1(a.0(a.1(a.0(b.0(B.1(x0))))))) → B1.1(a.0(b.0(a.1(a.0(b.0(b.1(a.0(b.1(a.0(B.1(x0)))))))))))
B1.1(a.0(a.1(a.0(b.0(b.0(B.1(x))))))) → B1.1(a.0(b.1(a.0(a.1(a.0(b.1(a.0(b.1(a.0(B.1(x)))))))))))
B1.1(a.0(b.1(a.0(a.1(a.0(b.1(a.0(b.0(B.1(x)))))))))) → B1.0(b.1(a.0(b.1(a.0(b.1(a.0(B.1(x))))))))
B1.0(b.1(a.0(b.1(a.0(a.1(a.0(b.1(a.0(b.1(x0)))))))))) → B1.1(a.0(b.1(a.0(b.0(a.1(a.0(b.0(b.1(x0)))))))))
B1.0(b.1(a.0(a.1(a.0(b.1(a.0(b.0(B.0(x0))))))))) → B1.1(a.0(b.1(a.0(b.0(a.1(a.0(b.0(b.1(a.0(b.1(a.0(B.0(x0)))))))))))))
B1.1(a.0(b.1(a.0(a.1(a.0(b.0(B.0(x)))))))) → B1.0(a.1(a.0(b.0(b.1(a.0(B.0(x)))))))
B1.0(b.0(a.1(a.0(b.0(B.1(x0)))))) → B1.1(a.0(b.1(a.0(b.1(a.0(b.1(a.0(a.1(a.0(b.1(a.0(b.1(a.0(B.1(x0)))))))))))))))
B1.0(b.1(a.0(b.1(a.0(a.1(a.0(b.1(a.0(b.0(B.1(x0))))))))))) → B1.1(a.0(b.1(a.0(b.0(a.1(a.0(b.0(b.1(a.0(b.1(a.0(b.1(a.0(B.1(x0)))))))))))))))
B1.0(b.0(x)) → B1.1(a.0(b.0(x)))
B1.0(b.1(a.0(a.1(a.0(b.0(B.1(x0))))))) → B1.1(a.0(b.0(a.1(a.0(b.0(b.1(a.0(B.1(x0)))))))))
B1.0(b.1(a.0(b.1(a.0(a.1(a.0(b.0(B.1(x0))))))))) → B1.1(a.0(b.1(a.0(b.0(a.1(a.0(b.0(b.1(a.0(b.1(a.0(B.1(x0)))))))))))))
B1.0(a.1(a.0(b.1(x)))) → B1.1(a.0(a.1(a.0(b.1(x)))))
B1.0(b.1(a.0(a.1(a.0(b.0(B.1(x0))))))) → B1.1(a.0(b.1(a.0(b.0(a.1(a.0(b.0(b.1(a.0(B.1(x0)))))))))))
B1.1(a.0(b.1(a.0(a.1(a.0(b.0(B.0(x)))))))) → B1.0(a.1(a.0(b.0(b.1(a.0(b.1(a.0(B.0(x)))))))))
The TRS R consists of the following rules:
b.1(a.0(a.1(a.0(b.0(b.0(B.0(x))))))) → b.1(a.0(b.1(a.0(a.1(a.0(b.1(a.0(b.1(a.0(b.1(a.0(B.0(x)))))))))))))
b.0(b.1(x)) → b.1(a.0(b.1(a.0(b.1(a.0(b.1(x)))))))
b.0(b.0(B.0(x))) → b.1(a.0(b.1(a.0(b.1(a.0(b.1(a.0(b.1(a.0(b.1(a.0(B.0(x)))))))))))))
b.1(a.0(a.1(a.0(b.0(b.0(B.1(x))))))) → b.1(a.0(b.1(a.0(a.1(a.0(b.1(a.0(b.1(a.0(B.1(x)))))))))))
b.1(a.0(a.1(a.0(b.0(b.1(x)))))) → b.1(a.0(b.1(a.0(a.1(a.0(b.1(x)))))))
b.0(a.1(a.0(b.0(B.0(x))))) → b.1(a.0(a.1(a.0(b.1(a.0(b.1(a.0(b.1(a.0(B.0(x)))))))))))
b.1(a.0(b.1(a.0(a.1(a.0(b.0(B.0(x)))))))) → b.0(a.1(a.0(b.0(b.1(a.0(b.1(a.0(B.0(x)))))))))
b.1(a.0(a.1(a.0(b.0(b.0(B.1(x))))))) → b.1(a.0(b.1(a.0(a.1(a.0(b.1(a.0(b.1(a.0(b.1(a.0(B.1(x)))))))))))))
b.0(B.1(x)) → b.1(a.0(B.1(x)))
b.0(a.1(a.0(b.0(B.1(x))))) → b.1(a.0(a.1(a.0(b.1(a.0(b.1(a.0(B.1(x)))))))))
b.1(a.0(b.1(a.0(a.1(a.0(b.1(a.0(b.0(B.1(x)))))))))) → b.0(a.1(a.0(b.0(b.1(a.0(b.1(a.0(B.1(x)))))))))
b.0(B.0(x)) → b.1(a.0(B.0(x)))
b.1(a.0(b.1(a.0(a.1(a.0(b.0(B.0(x)))))))) → b.0(a.1(a.0(b.0(b.1(a.0(B.0(x)))))))
b.0(b.0(x)) → b.1(a.0(b.1(a.0(b.1(a.0(b.0(x)))))))
b.1(a.0(b.1(a.0(a.1(a.0(b.0(B.1(x)))))))) → b.0(a.1(a.0(b.0(b.1(a.0(B.1(x)))))))
b.1(a.0(a.1(a.0(b.0(b.0(B.0(x))))))) → b.1(a.0(b.1(a.0(a.1(a.0(b.1(a.0(b.1(a.0(B.0(x)))))))))))
b.1(a.0(b.1(a.0(a.1(a.0(b.0(B.1(x)))))))) → b.0(a.1(a.0(b.0(b.1(a.0(b.1(a.0(B.1(x)))))))))
b.1(a.0(b.1(a.0(a.1(a.0(b.1(a.0(b.0(B.1(x)))))))))) → b.0(a.1(a.0(b.0(b.1(a.0(b.1(a.0(b.1(a.0(B.1(x)))))))))))
b.1(a.0(b.1(a.0(a.1(a.0(b.1(a.0(b.0(B.0(x)))))))))) → b.0(a.1(a.0(b.0(b.1(a.0(b.1(a.0(B.0(x)))))))))
b.1(a.0(b.1(a.0(a.1(a.0(b.1(a.0(B.0(x))))))))) → b.0(a.1(a.0(b.0(B.0(x)))))
b.0(b.0(B.0(x))) → b.1(a.0(b.1(a.0(b.1(a.0(b.1(a.0(b.1(a.0(B.0(x)))))))))))
b.0(b.0(B.1(x))) → b.1(a.0(b.1(a.0(b.1(a.0(b.1(a.0(b.1(a.0(B.1(x)))))))))))
b.0(a.1(a.0(b.0(B.1(x))))) → b.1(a.0(a.1(a.0(b.1(a.0(b.1(a.0(b.1(a.0(B.1(x)))))))))))
b.1(a.0(a.1(a.0(b.0(b.0(x)))))) → b.1(a.0(b.1(a.0(a.1(a.0(b.0(x)))))))
b.0(a.1(a.0(b.1(x)))) → b.1(a.0(a.1(a.0(b.1(x)))))
b.0(a.1(a.0(b.0(B.0(x))))) → b.1(a.0(a.1(a.0(b.1(a.0(b.1(a.0(B.0(x)))))))))
b.1(a.0(b.1(a.0(a.1(a.0(b.1(a.0(B.1(x))))))))) → b.0(a.1(a.0(b.0(B.1(x)))))
b.1(a.0(b.1(a.0(a.1(a.0(b.1(a.0(b.0(x))))))))) → b.0(a.1(a.0(b.0(b.0(x)))))
b.0(a.1(a.0(b.0(x)))) → b.1(a.0(a.1(a.0(b.0(x)))))
b.0(b.0(B.1(x))) → b.1(a.0(b.1(a.0(b.1(a.0(b.1(a.0(b.1(a.0(b.1(a.0(B.1(x)))))))))))))
b.1(a.0(b.1(a.0(a.1(a.0(b.1(a.0(b.1(x))))))))) → b.0(a.1(a.0(b.0(b.1(x)))))
b.1(a.0(b.1(a.0(a.1(a.0(b.1(a.0(b.0(B.0(x)))))))))) → b.0(a.1(a.0(b.0(b.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.
By using the rule removal processor [15] with the following polynomial ordering [25], at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented.
Strictly oriented dependency pairs:
B1.1(a.0(b.1(a.0(a.1(a.0(b.0(B.1(x)))))))) → B1.0(b.1(a.0(B.1(x))))
B1.1(a.0(b.1(a.0(a.1(a.0(b.1(a.0(b.0(B.0(x)))))))))) → B1.0(b.1(a.0(b.1(a.0(B.0(x))))))
B1.1(a.0(b.1(a.0(a.1(a.0(b.0(B.1(x)))))))) → B1.0(b.1(a.0(b.1(a.0(B.1(x))))))
B1.1(a.0(b.1(a.0(a.1(a.0(b.1(a.0(b.0(B.1(x)))))))))) → B1.0(b.1(a.0(b.1(a.0(B.1(x))))))
B1.1(a.0(b.1(a.0(a.1(a.0(b.0(B.0(x)))))))) → B1.0(b.1(a.0(b.1(a.0(B.0(x))))))
B1.1(a.0(b.1(a.0(a.1(a.0(b.1(a.0(b.1(x))))))))) → B1.0(b.1(x))
B1.1(a.0(b.1(a.0(a.1(a.0(b.0(B.0(x)))))))) → B1.0(b.1(a.0(B.0(x))))
B1.1(a.0(b.1(a.0(a.1(a.0(b.1(a.0(b.0(x))))))))) → B1.0(b.0(x))
B1.1(a.0(b.1(a.0(a.1(a.0(b.1(a.0(b.0(B.0(x)))))))))) → B1.0(b.1(a.0(b.1(a.0(b.1(a.0(B.0(x))))))))
B1.1(a.0(b.1(a.0(a.1(a.0(b.1(a.0(b.0(B.1(x)))))))))) → B1.0(b.1(a.0(b.1(a.0(b.1(a.0(B.1(x))))))))
Used ordering: POLO with Polynomial interpretation [25]:
POL(B.0(x1)) = x1
POL(B.1(x1)) = x1
POL(B1.0(x1)) = x1
POL(B1.1(x1)) = x1
POL(a.0(x1)) = x1
POL(a.1(x1)) = 1 + x1
POL(b.0(x1)) = x1
POL(b.1(x1)) = x1
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ SemLabProof
↳ SemLabProof2
↳ QDP
↳ QDPToSRSProof
↳ QTRS
↳ QTRS Reverse
↳ QTRS
↳ QTRS Reverse
↳ QTRS Reverse
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ SemLabProof
↳ QDP
↳ RuleRemovalProof
↳ QDP
↳ DependencyGraphProof
↳ SemLabProof2
↳ QTRS Reverse
↳ QTRS Reverse
Q DP problem:
The TRS P consists of the following rules:
B1.1(a.0(a.1(a.0(b.0(b.0(x)))))) → B1.1(a.0(b.1(a.0(a.1(a.0(b.0(x)))))))
B1.0(b.1(a.0(a.1(a.0(b.0(B.0(x0))))))) → B1.1(a.0(b.1(a.0(b.0(a.1(a.0(b.0(b.1(a.0(b.1(a.0(B.0(x0)))))))))))))
B1.0(b.1(a.0(a.1(a.0(b.1(a.0(b.0(B.0(x0))))))))) → B1.1(a.0(b.0(a.1(a.0(b.0(b.1(a.0(b.1(a.0(b.1(a.0(B.0(x0)))))))))))))
B1.1(a.0(b.1(a.0(a.1(a.0(b.1(a.0(b.0(B.0(x)))))))))) → B1.0(a.1(a.0(b.0(b.1(a.0(b.1(a.0(B.0(x)))))))))
B1.1(a.0(b.1(a.0(a.1(a.0(b.1(a.0(b.0(B.1(x)))))))))) → B1.0(a.1(a.0(b.0(b.1(a.0(b.1(a.0(b.1(a.0(B.1(x)))))))))))
B1.0(b.1(a.0(b.1(a.0(a.1(a.0(b.0(B.1(x0))))))))) → B1.1(a.0(b.1(a.0(b.1(a.0(b.0(a.1(a.0(b.0(b.1(a.0(B.1(x0)))))))))))))
B1.0(b.0(a.1(a.0(b.0(B.0(x0)))))) → B1.1(a.0(b.1(a.0(b.1(a.0(a.1(a.0(b.1(a.0(b.1(a.0(b.1(a.0(B.0(x0)))))))))))))))
B1.0(b.1(a.0(b.1(a.0(a.1(a.0(b.1(a.0(b.0(B.0(x0))))))))))) → B1.1(a.0(b.1(a.0(b.1(a.0(b.0(a.1(a.0(b.0(b.1(a.0(b.1(a.0(b.1(a.0(B.0(x0)))))))))))))))))
B1.1(a.0(a.1(a.0(b.0(b.0(B.1(x))))))) → B1.1(a.0(b.1(a.0(a.1(a.0(b.1(a.0(b.1(a.0(b.1(a.0(B.1(x)))))))))))))
B1.0(b.1(a.0(a.1(a.0(b.1(a.0(B.0(x0)))))))) → B1.1(a.0(b.0(a.1(a.0(b.0(B.0(x0)))))))
B1.0(b.0(a.1(a.0(b.0(B.0(x0)))))) → B1.1(a.0(b.1(a.0(b.1(a.0(a.1(a.0(b.1(a.0(b.1(a.0(B.0(x0)))))))))))))
B1.0(b.1(x)) → B1.1(a.0(b.1(x)))
B1.0(b.1(a.0(a.1(a.0(b.0(b.0(B.1(x0)))))))) → B1.1(a.0(b.1(a.0(b.1(a.0(b.1(a.0(a.1(a.0(b.1(a.0(b.1(a.0(B.1(x0)))))))))))))))
B1.0(b.1(a.0(b.1(a.0(a.1(a.0(b.0(B.0(x0))))))))) → B1.1(a.0(b.1(a.0(b.0(a.1(a.0(b.0(b.1(a.0(b.1(a.0(B.0(x0)))))))))))))
B1.0(b.0(b.1(x0))) → B1.1(a.0(b.1(a.0(b.1(a.0(b.1(a.0(b.1(a.0(b.1(x0)))))))))))
B1.1(a.0(a.1(a.0(b.0(b.1(x)))))) → B1.1(a.0(a.1(a.0(b.1(x)))))
B1.0(b.1(a.0(a.1(a.0(b.1(a.0(b.1(x0)))))))) → B1.1(a.0(b.1(a.0(b.0(a.1(a.0(b.0(b.1(x0)))))))))
B1.0(b.1(a.0(a.1(a.0(b.1(a.0(b.0(x0)))))))) → B1.1(a.0(b.1(a.0(b.0(a.1(a.0(b.0(b.0(x0)))))))))
B1.0(b.1(a.0(a.1(a.0(b.0(b.0(B.0(x0)))))))) → B1.1(a.0(b.1(a.0(b.1(a.0(b.1(a.0(a.1(a.0(b.1(a.0(b.1(a.0(b.1(a.0(B.0(x0)))))))))))))))))
B1.1(a.0(a.1(a.0(b.0(b.1(x)))))) → B1.1(a.0(b.1(a.0(a.1(a.0(b.1(x)))))))
B1.0(b.0(b.0(x0))) → B1.1(a.0(b.1(a.0(b.1(a.0(b.1(a.0(b.1(a.0(b.0(x0)))))))))))
B1.1(a.0(b.1(a.0(a.1(a.0(b.1(a.0(b.0(B.0(x)))))))))) → B1.0(a.1(a.0(b.0(b.1(a.0(b.1(a.0(b.1(a.0(B.0(x)))))))))))
B1.0(b.0(a.1(a.0(b.0(B.1(x0)))))) → B1.1(a.0(b.1(a.0(b.1(a.0(a.1(a.0(b.1(a.0(b.1(a.0(b.1(a.0(B.1(x0)))))))))))))))
B1.1(a.0(b.1(a.0(a.1(a.0(b.0(B.1(x)))))))) → B1.0(a.1(a.0(b.0(b.1(a.0(B.1(x)))))))
B1.0(b.0(a.1(a.0(b.0(B.0(x0)))))) → B1.1(a.0(b.1(a.0(b.1(a.0(b.1(a.0(a.1(a.0(b.1(a.0(b.1(a.0(B.0(x0)))))))))))))))
B1.0(b.1(a.0(b.1(a.0(a.1(a.0(b.1(a.0(b.0(B.1(x0))))))))))) → B1.1(a.0(b.1(a.0(b.1(a.0(b.0(a.1(a.0(b.0(b.1(a.0(b.1(a.0(B.1(x0)))))))))))))))
B1.0(b.0(b.1(x0))) → B1.1(a.0(b.1(a.0(b.1(a.0(b.1(a.0(b.1(a.0(b.1(a.0(b.1(x0)))))))))))))
B1.1(a.0(a.1(a.0(b.0(b.0(B.0(x))))))) → B1.1(a.0(b.1(a.0(a.1(a.0(b.1(a.0(b.1(a.0(B.0(x)))))))))))
B1.0(b.0(a.1(a.0(b.0(B.1(x0)))))) → B1.1(a.0(b.1(a.0(b.1(a.0(a.1(a.0(b.1(a.0(b.1(a.0(B.1(x0)))))))))))))
B1.0(b.1(a.0(b.1(a.0(a.1(a.0(b.1(a.0(B.0(x0)))))))))) → B1.1(a.0(b.1(a.0(b.1(a.0(b.0(a.1(a.0(b.0(B.0(x0)))))))))))
B1.1(a.0(b.1(a.0(a.1(a.0(b.1(a.0(b.0(x))))))))) → B1.0(a.1(a.0(b.0(b.0(x)))))
B1.0(b.1(a.0(b.1(a.0(a.1(a.0(b.1(a.0(b.0(x0)))))))))) → B1.1(a.0(b.1(a.0(b.0(a.1(a.0(b.0(b.0(x0)))))))))
B1.0(b.1(a.0(a.1(a.0(b.1(a.0(b.0(B.1(x0))))))))) → B1.1(a.0(b.1(a.0(b.0(a.1(a.0(b.0(b.1(a.0(b.1(a.0(B.1(x0)))))))))))))
B1.0(b.1(a.0(a.1(a.0(b.0(b.0(B.0(x0)))))))) → B1.1(a.0(b.1(a.0(b.1(a.0(b.1(a.0(a.1(a.0(b.1(a.0(b.1(a.0(B.0(x0)))))))))))))))
B1.0(b.1(a.0(a.1(a.0(b.1(a.0(b.0(B.1(x0))))))))) → B1.1(a.0(b.0(a.1(a.0(b.0(b.1(a.0(b.1(a.0(B.1(x0)))))))))))
B1.0(b.0(a.1(a.0(b.1(x0))))) → B1.1(a.0(b.1(a.0(b.1(a.0(a.1(a.0(b.1(x0)))))))))
B1.1(a.0(b.1(a.0(a.1(a.0(b.1(a.0(B.1(x))))))))) → B1.0(a.1(a.0(b.0(B.1(x)))))
B1.0(b.1(a.0(a.1(a.0(b.1(a.0(b.0(B.1(x0))))))))) → B1.1(a.0(b.1(a.0(b.0(a.1(a.0(b.0(b.1(a.0(b.1(a.0(b.1(a.0(B.1(x0)))))))))))))))
B1.0(b.0(a.1(a.0(b.0(x0))))) → B1.1(a.0(b.1(a.0(b.1(a.0(a.1(a.0(b.0(x0)))))))))
B1.0(b.1(a.0(b.1(a.0(a.1(a.0(b.0(B.1(x0))))))))) → B1.1(a.0(b.1(a.0(b.1(a.0(b.0(a.1(a.0(b.0(b.1(a.0(b.1(a.0(B.1(x0)))))))))))))))
B1.0(b.1(a.0(a.1(a.0(b.0(b.0(B.1(x0)))))))) → B1.1(a.0(b.1(a.0(b.1(a.0(b.1(a.0(b.1(a.0(a.1(a.0(b.1(a.0(b.1(a.0(b.1(a.0(B.1(x0)))))))))))))))))))
B1.0(b.1(a.0(b.1(a.0(a.1(a.0(b.1(a.0(b.1(x0)))))))))) → B1.1(a.0(b.1(a.0(b.1(a.0(b.0(a.1(a.0(b.0(b.1(x0)))))))))))
B1.0(b.0(a.1(a.0(b.0(B.0(x0)))))) → B1.1(a.0(b.1(a.0(b.1(a.0(b.1(a.0(a.1(a.0(b.1(a.0(b.1(a.0(b.1(a.0(B.0(x0)))))))))))))))))
B1.0(b.1(a.0(a.1(a.0(b.0(b.0(B.0(x0)))))))) → B1.1(a.0(b.1(a.0(b.1(a.0(b.1(a.0(b.1(a.0(a.1(a.0(b.1(a.0(b.1(a.0(b.1(a.0(B.0(x0)))))))))))))))))))
B1.0(b.1(a.0(b.1(a.0(a.1(a.0(b.1(a.0(B.0(x0)))))))))) → B1.1(a.0(b.1(a.0(b.0(a.1(a.0(b.0(B.0(x0)))))))))
B1.1(a.0(b.1(a.0(a.1(a.0(b.1(a.0(b.0(B.1(x)))))))))) → B1.0(a.1(a.0(b.0(b.1(a.0(b.1(a.0(B.1(x)))))))))
B1.0(b.1(a.0(a.1(a.0(b.0(B.0(x0))))))) → B1.1(a.0(b.1(a.0(b.0(a.1(a.0(b.0(b.1(a.0(B.0(x0)))))))))))
B1.1(a.0(b.1(a.0(a.1(a.0(b.0(B.1(x)))))))) → B1.0(a.1(a.0(b.0(b.1(a.0(b.1(a.0(B.1(x)))))))))
B1.0(a.1(a.0(b.0(x)))) → B1.1(a.0(a.1(a.0(b.0(x)))))
B1.0(b.1(a.0(a.1(a.0(b.0(B.0(x0))))))) → B1.1(a.0(b.0(a.1(a.0(b.0(b.1(a.0(B.0(x0)))))))))
B1.0(b.1(a.0(a.1(a.0(b.0(b.1(x0))))))) → B1.1(a.0(b.1(a.0(b.1(a.0(b.1(a.0(a.1(a.0(b.1(x0)))))))))))
B1.0(b.1(a.0(a.1(a.0(b.1(a.0(b.0(x0)))))))) → B1.1(a.0(b.0(a.1(a.0(b.0(b.0(x0)))))))
B1.1(a.0(a.1(a.0(b.0(b.0(x)))))) → B1.1(a.0(a.1(a.0(b.0(x)))))
B1.0(b.1(a.0(b.1(a.0(a.1(a.0(b.1(a.0(b.0(B.0(x0))))))))))) → B1.1(a.0(b.1(a.0(b.0(a.1(a.0(b.0(b.1(a.0(b.1(a.0(b.1(a.0(B.0(x0)))))))))))))))
B1.0(b.1(a.0(b.1(a.0(a.1(a.0(b.1(a.0(b.0(B.1(x0))))))))))) → B1.1(a.0(b.1(a.0(b.0(a.1(a.0(b.0(b.1(a.0(b.1(a.0(B.1(x0)))))))))))))
B1.0(b.1(a.0(b.1(a.0(a.1(a.0(b.1(a.0(b.0(x0)))))))))) → B1.1(a.0(b.1(a.0(b.1(a.0(b.0(a.1(a.0(b.0(b.0(x0)))))))))))
B1.0(b.1(a.0(a.1(a.0(b.0(b.0(B.0(x0)))))))) → B1.1(a.0(b.1(a.0(b.1(a.0(b.1(a.0(b.1(a.0(a.1(a.0(b.1(a.0(b.1(a.0(B.0(x0)))))))))))))))))
B1.0(b.0(a.1(a.0(b.0(B.1(x0)))))) → B1.1(a.0(b.1(a.0(b.1(a.0(b.1(a.0(a.1(a.0(b.1(a.0(b.1(a.0(b.1(a.0(B.1(x0)))))))))))))))))
B1.0(b.1(a.0(a.1(a.0(b.1(a.0(b.0(B.1(x0))))))))) → B1.1(a.0(b.0(a.1(a.0(b.0(b.1(a.0(b.1(a.0(b.1(a.0(B.1(x0)))))))))))))
B1.1(a.0(b.1(a.0(a.1(a.0(b.1(a.0(B.0(x))))))))) → B1.0(a.1(a.0(b.0(B.0(x)))))
B1.0(b.1(a.0(b.1(a.0(a.1(a.0(b.1(a.0(b.0(B.0(x0))))))))))) → B1.1(a.0(b.1(a.0(b.1(a.0(b.0(a.1(a.0(b.0(b.1(a.0(b.1(a.0(B.0(x0)))))))))))))))
B1.0(b.1(a.0(a.1(a.0(b.0(b.0(x0))))))) → B1.1(a.0(b.1(a.0(b.1(a.0(b.1(a.0(b.1(a.0(a.1(a.0(b.0(x0)))))))))))))
B1.0(b.1(a.0(b.1(a.0(a.1(a.0(b.0(B.0(x0))))))))) → B1.1(a.0(b.1(a.0(b.1(a.0(b.0(a.1(a.0(b.0(b.1(a.0(b.1(a.0(B.0(x0)))))))))))))))
B1.0(b.1(a.0(a.1(a.0(b.1(a.0(b.0(B.0(x0))))))))) → B1.1(a.0(b.1(a.0(b.0(a.1(a.0(b.0(b.1(a.0(b.1(a.0(b.1(a.0(B.0(x0)))))))))))))))
B1.0(b.1(a.0(b.1(a.0(a.1(a.0(b.0(B.1(x0))))))))) → B1.1(a.0(b.1(a.0(b.0(a.1(a.0(b.0(b.1(a.0(B.1(x0)))))))))))
B1.0(b.1(a.0(a.1(a.0(b.0(b.1(x0))))))) → B1.1(a.0(b.1(a.0(b.1(a.0(b.1(a.0(b.1(a.0(a.1(a.0(b.1(x0)))))))))))))
B1.0(b.1(a.0(b.1(a.0(a.1(a.0(b.1(a.0(b.0(B.0(x0))))))))))) → B1.1(a.0(b.1(a.0(b.0(a.1(a.0(b.0(b.1(a.0(b.1(a.0(B.0(x0)))))))))))))
B1.0(b.1(a.0(b.1(a.0(a.1(a.0(b.1(a.0(b.0(B.1(x0))))))))))) → B1.1(a.0(b.1(a.0(b.1(a.0(b.0(a.1(a.0(b.0(b.1(a.0(b.1(a.0(b.1(a.0(B.1(x0)))))))))))))))))
B1.0(b.1(a.0(a.1(a.0(b.1(a.0(b.0(B.0(x0))))))))) → B1.1(a.0(b.0(a.1(a.0(b.0(b.1(a.0(b.1(a.0(B.0(x0)))))))))))
B1.0(b.0(a.1(a.0(b.1(x0))))) → B1.1(a.0(b.1(a.0(b.1(a.0(b.1(a.0(a.1(a.0(b.1(x0)))))))))))
B1.0(b.1(a.0(a.1(a.0(b.0(b.0(x0))))))) → B1.1(a.0(b.1(a.0(b.1(a.0(b.1(a.0(a.1(a.0(b.0(x0)))))))))))
B1.0(b.1(a.0(a.1(a.0(b.1(a.0(B.1(x0)))))))) → B1.1(a.0(b.1(a.0(b.0(a.1(a.0(b.0(B.1(x0)))))))))
B1.0(b.1(a.0(a.1(a.0(b.1(a.0(b.1(x0)))))))) → B1.1(a.0(b.0(a.1(a.0(b.0(b.1(x0)))))))
B1.0(b.1(a.0(b.1(a.0(a.1(a.0(b.1(a.0(B.1(x0)))))))))) → B1.1(a.0(b.1(a.0(b.0(a.1(a.0(b.0(B.1(x0)))))))))
B1.0(b.1(a.0(a.1(a.0(b.0(b.0(B.1(x0)))))))) → B1.1(a.0(b.1(a.0(b.1(a.0(b.1(a.0(b.1(a.0(a.1(a.0(b.1(a.0(b.1(a.0(B.1(x0)))))))))))))))))
B1.0(b.1(a.0(a.1(a.0(b.0(B.0(x0))))))) → B1.1(a.0(b.0(a.1(a.0(b.0(b.1(a.0(b.1(a.0(B.0(x0)))))))))))
B1.0(b.0(b.0(x0))) → B1.1(a.0(b.1(a.0(b.1(a.0(b.1(a.0(b.1(a.0(b.1(a.0(b.0(x0)))))))))))))
B1.0(b.1(a.0(b.1(a.0(a.1(a.0(b.0(B.0(x0))))))))) → B1.1(a.0(b.1(a.0(b.1(a.0(b.0(a.1(a.0(b.0(b.1(a.0(B.0(x0)))))))))))))
B1.0(b.0(a.1(a.0(b.0(x0))))) → B1.1(a.0(b.1(a.0(b.1(a.0(b.1(a.0(a.1(a.0(b.0(x0)))))))))))
B1.0(b.1(a.0(a.1(a.0(b.1(a.0(B.1(x0)))))))) → B1.1(a.0(b.0(a.1(a.0(b.0(B.1(x0)))))))
B1.0(b.1(a.0(a.1(a.0(b.1(a.0(B.0(x0)))))))) → B1.1(a.0(b.1(a.0(b.0(a.1(a.0(b.0(B.0(x0)))))))))
B1.0(b.1(a.0(b.1(a.0(a.1(a.0(b.1(a.0(B.1(x0)))))))))) → B1.1(a.0(b.1(a.0(b.1(a.0(b.0(a.1(a.0(b.0(B.1(x0)))))))))))
B1.1(a.0(b.1(a.0(a.1(a.0(b.1(a.0(b.1(x))))))))) → B1.0(a.1(a.0(b.0(b.1(x)))))
B1.0(b.1(a.0(a.1(a.0(b.0(b.0(B.1(x0)))))))) → B1.1(a.0(b.1(a.0(b.1(a.0(b.1(a.0(a.1(a.0(b.1(a.0(b.1(a.0(b.1(a.0(B.1(x0)))))))))))))))))
B1.0(b.1(a.0(b.1(a.0(a.1(a.0(b.0(B.0(x0))))))))) → B1.1(a.0(b.1(a.0(b.0(a.1(a.0(b.0(b.1(a.0(B.0(x0)))))))))))
B1.1(a.0(a.1(a.0(b.0(b.0(B.0(x))))))) → B1.1(a.0(b.1(a.0(a.1(a.0(b.1(a.0(b.1(a.0(b.1(a.0(B.0(x)))))))))))))
B1.0(b.1(a.0(a.1(a.0(b.0(B.1(x0))))))) → B1.1(a.0(b.1(a.0(b.0(a.1(a.0(b.0(b.1(a.0(b.1(a.0(B.1(x0)))))))))))))
B1.0(b.1(a.0(a.1(a.0(b.0(B.1(x0))))))) → B1.1(a.0(b.0(a.1(a.0(b.0(b.1(a.0(b.1(a.0(B.1(x0)))))))))))
B1.1(a.0(a.1(a.0(b.0(b.0(B.1(x))))))) → B1.1(a.0(b.1(a.0(a.1(a.0(b.1(a.0(b.1(a.0(B.1(x)))))))))))
B1.0(b.1(a.0(b.1(a.0(a.1(a.0(b.1(a.0(b.1(x0)))))))))) → B1.1(a.0(b.1(a.0(b.0(a.1(a.0(b.0(b.1(x0)))))))))
B1.0(b.1(a.0(a.1(a.0(b.1(a.0(b.0(B.0(x0))))))))) → B1.1(a.0(b.1(a.0(b.0(a.1(a.0(b.0(b.1(a.0(b.1(a.0(B.0(x0)))))))))))))
B1.1(a.0(b.1(a.0(a.1(a.0(b.0(B.0(x)))))))) → B1.0(a.1(a.0(b.0(b.1(a.0(B.0(x)))))))
B1.0(b.0(a.1(a.0(b.0(B.1(x0)))))) → B1.1(a.0(b.1(a.0(b.1(a.0(b.1(a.0(a.1(a.0(b.1(a.0(b.1(a.0(B.1(x0)))))))))))))))
B1.0(b.1(a.0(b.1(a.0(a.1(a.0(b.1(a.0(b.0(B.1(x0))))))))))) → B1.1(a.0(b.1(a.0(b.0(a.1(a.0(b.0(b.1(a.0(b.1(a.0(b.1(a.0(B.1(x0)))))))))))))))
B1.0(b.0(x)) → B1.1(a.0(b.0(x)))
B1.0(b.1(a.0(a.1(a.0(b.0(B.1(x0))))))) → B1.1(a.0(b.0(a.1(a.0(b.0(b.1(a.0(B.1(x0)))))))))
B1.0(b.1(a.0(b.1(a.0(a.1(a.0(b.0(B.1(x0))))))))) → B1.1(a.0(b.1(a.0(b.0(a.1(a.0(b.0(b.1(a.0(b.1(a.0(B.1(x0)))))))))))))
B1.0(a.1(a.0(b.1(x)))) → B1.1(a.0(a.1(a.0(b.1(x)))))
B1.0(b.1(a.0(a.1(a.0(b.0(B.1(x0))))))) → B1.1(a.0(b.1(a.0(b.0(a.1(a.0(b.0(b.1(a.0(B.1(x0)))))))))))
B1.1(a.0(b.1(a.0(a.1(a.0(b.0(B.0(x)))))))) → B1.0(a.1(a.0(b.0(b.1(a.0(b.1(a.0(B.0(x)))))))))
The TRS R consists of the following rules:
b.1(a.0(a.1(a.0(b.0(b.0(B.0(x))))))) → b.1(a.0(b.1(a.0(a.1(a.0(b.1(a.0(b.1(a.0(b.1(a.0(B.0(x)))))))))))))
b.0(b.1(x)) → b.1(a.0(b.1(a.0(b.1(a.0(b.1(x)))))))
b.0(b.0(B.0(x))) → b.1(a.0(b.1(a.0(b.1(a.0(b.1(a.0(b.1(a.0(b.1(a.0(B.0(x)))))))))))))
b.1(a.0(a.1(a.0(b.0(b.0(B.1(x))))))) → b.1(a.0(b.1(a.0(a.1(a.0(b.1(a.0(b.1(a.0(B.1(x)))))))))))
b.1(a.0(a.1(a.0(b.0(b.1(x)))))) → b.1(a.0(b.1(a.0(a.1(a.0(b.1(x)))))))
b.0(a.1(a.0(b.0(B.0(x))))) → b.1(a.0(a.1(a.0(b.1(a.0(b.1(a.0(b.1(a.0(B.0(x)))))))))))
b.1(a.0(b.1(a.0(a.1(a.0(b.0(B.0(x)))))))) → b.0(a.1(a.0(b.0(b.1(a.0(b.1(a.0(B.0(x)))))))))
b.1(a.0(a.1(a.0(b.0(b.0(B.1(x))))))) → b.1(a.0(b.1(a.0(a.1(a.0(b.1(a.0(b.1(a.0(b.1(a.0(B.1(x)))))))))))))
b.0(B.1(x)) → b.1(a.0(B.1(x)))
b.0(a.1(a.0(b.0(B.1(x))))) → b.1(a.0(a.1(a.0(b.1(a.0(b.1(a.0(B.1(x)))))))))
b.1(a.0(b.1(a.0(a.1(a.0(b.1(a.0(b.0(B.1(x)))))))))) → b.0(a.1(a.0(b.0(b.1(a.0(b.1(a.0(B.1(x)))))))))
b.0(B.0(x)) → b.1(a.0(B.0(x)))
b.1(a.0(b.1(a.0(a.1(a.0(b.0(B.0(x)))))))) → b.0(a.1(a.0(b.0(b.1(a.0(B.0(x)))))))
b.0(b.0(x)) → b.1(a.0(b.1(a.0(b.1(a.0(b.0(x)))))))
b.1(a.0(b.1(a.0(a.1(a.0(b.0(B.1(x)))))))) → b.0(a.1(a.0(b.0(b.1(a.0(B.1(x)))))))
b.1(a.0(a.1(a.0(b.0(b.0(B.0(x))))))) → b.1(a.0(b.1(a.0(a.1(a.0(b.1(a.0(b.1(a.0(B.0(x)))))))))))
b.1(a.0(b.1(a.0(a.1(a.0(b.0(B.1(x)))))))) → b.0(a.1(a.0(b.0(b.1(a.0(b.1(a.0(B.1(x)))))))))
b.1(a.0(b.1(a.0(a.1(a.0(b.1(a.0(b.0(B.1(x)))))))))) → b.0(a.1(a.0(b.0(b.1(a.0(b.1(a.0(b.1(a.0(B.1(x)))))))))))
b.1(a.0(b.1(a.0(a.1(a.0(b.1(a.0(b.0(B.0(x)))))))))) → b.0(a.1(a.0(b.0(b.1(a.0(b.1(a.0(B.0(x)))))))))
b.1(a.0(b.1(a.0(a.1(a.0(b.1(a.0(B.0(x))))))))) → b.0(a.1(a.0(b.0(B.0(x)))))
b.0(b.0(B.0(x))) → b.1(a.0(b.1(a.0(b.1(a.0(b.1(a.0(b.1(a.0(B.0(x)))))))))))
b.0(b.0(B.1(x))) → b.1(a.0(b.1(a.0(b.1(a.0(b.1(a.0(b.1(a.0(B.1(x)))))))))))
b.0(a.1(a.0(b.0(B.1(x))))) → b.1(a.0(a.1(a.0(b.1(a.0(b.1(a.0(b.1(a.0(B.1(x)))))))))))
b.1(a.0(a.1(a.0(b.0(b.0(x)))))) → b.1(a.0(b.1(a.0(a.1(a.0(b.0(x)))))))
b.0(a.1(a.0(b.1(x)))) → b.1(a.0(a.1(a.0(b.1(x)))))
b.0(a.1(a.0(b.0(B.0(x))))) → b.1(a.0(a.1(a.0(b.1(a.0(b.1(a.0(B.0(x)))))))))
b.1(a.0(b.1(a.0(a.1(a.0(b.1(a.0(B.1(x))))))))) → b.0(a.1(a.0(b.0(B.1(x)))))
b.1(a.0(b.1(a.0(a.1(a.0(b.1(a.0(b.0(x))))))))) → b.0(a.1(a.0(b.0(b.0(x)))))
b.0(a.1(a.0(b.0(x)))) → b.1(a.0(a.1(a.0(b.0(x)))))
b.0(b.0(B.1(x))) → b.1(a.0(b.1(a.0(b.1(a.0(b.1(a.0(b.1(a.0(b.1(a.0(B.1(x)))))))))))))
b.1(a.0(b.1(a.0(a.1(a.0(b.1(a.0(b.1(x))))))))) → b.0(a.1(a.0(b.0(b.1(x)))))
b.1(a.0(b.1(a.0(a.1(a.0(b.1(a.0(b.0(B.0(x)))))))))) → b.0(a.1(a.0(b.0(b.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 78 less nodes.
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ SemLabProof
↳ SemLabProof2
↳ QDP
↳ QDPToSRSProof
↳ QTRS
↳ QTRS Reverse
↳ QTRS
↳ QTRS Reverse
↳ QTRS Reverse
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ SemLabProof
↳ QDP
↳ RuleRemovalProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ SemLabProof2
↳ QTRS Reverse
↳ QTRS Reverse
Q DP problem:
The TRS P consists of the following rules:
B1.1(a.0(a.1(a.0(b.0(b.0(x)))))) → B1.1(a.0(b.1(a.0(a.1(a.0(b.0(x)))))))
B1.1(a.0(b.1(a.0(a.1(a.0(b.1(a.0(B.1(x))))))))) → B1.0(a.1(a.0(b.0(B.1(x)))))
B1.1(a.0(b.1(a.0(a.1(a.0(b.0(B.1(x)))))))) → B1.0(a.1(a.0(b.0(b.1(a.0(b.1(a.0(B.1(x)))))))))
B1.1(a.0(b.1(a.0(a.1(a.0(b.1(a.0(b.0(x))))))))) → B1.0(a.1(a.0(b.0(b.0(x)))))
B1.0(a.1(a.0(b.0(x)))) → B1.1(a.0(a.1(a.0(b.0(x)))))
B1.1(a.0(a.1(a.0(b.0(b.1(x)))))) → B1.1(a.0(a.1(a.0(b.1(x)))))
B1.1(a.0(a.1(a.0(b.0(b.0(x)))))) → B1.1(a.0(a.1(a.0(b.0(x)))))
B1.1(a.0(b.1(a.0(a.1(a.0(b.1(a.0(b.1(x))))))))) → B1.0(a.1(a.0(b.0(b.1(x)))))
B1.1(a.0(b.1(a.0(a.1(a.0(b.1(a.0(b.0(B.0(x)))))))))) → B1.0(a.1(a.0(b.0(b.1(a.0(b.1(a.0(B.0(x)))))))))
B1.1(a.0(a.1(a.0(b.0(b.1(x)))))) → B1.1(a.0(b.1(a.0(a.1(a.0(b.1(x)))))))
B1.1(a.0(b.1(a.0(a.1(a.0(b.1(a.0(b.0(B.1(x)))))))))) → B1.0(a.1(a.0(b.0(b.1(a.0(b.1(a.0(b.1(a.0(B.1(x)))))))))))
B1.0(a.1(a.0(b.1(x)))) → B1.1(a.0(a.1(a.0(b.1(x)))))
B1.1(a.0(a.1(a.0(b.0(b.0(B.0(x))))))) → B1.1(a.0(b.1(a.0(a.1(a.0(b.1(a.0(b.1(a.0(b.1(a.0(B.0(x)))))))))))))
B1.1(a.0(b.1(a.0(a.1(a.0(b.1(a.0(b.0(B.0(x)))))))))) → B1.0(a.1(a.0(b.0(b.1(a.0(b.1(a.0(b.1(a.0(B.0(x)))))))))))
B1.1(a.0(a.1(a.0(b.0(b.0(B.1(x))))))) → B1.1(a.0(b.1(a.0(a.1(a.0(b.1(a.0(b.1(a.0(B.1(x)))))))))))
B1.1(a.0(b.1(a.0(a.1(a.0(b.0(B.1(x)))))))) → B1.0(a.1(a.0(b.0(b.1(a.0(B.1(x)))))))
B1.1(a.0(a.1(a.0(b.0(b.0(B.1(x))))))) → B1.1(a.0(b.1(a.0(a.1(a.0(b.1(a.0(b.1(a.0(b.1(a.0(B.1(x)))))))))))))
B1.1(a.0(b.1(a.0(a.1(a.0(b.1(a.0(B.0(x))))))))) → B1.0(a.1(a.0(b.0(B.0(x)))))
B1.1(a.0(b.1(a.0(a.1(a.0(b.1(a.0(b.0(B.1(x)))))))))) → B1.0(a.1(a.0(b.0(b.1(a.0(b.1(a.0(B.1(x)))))))))
B1.1(a.0(b.1(a.0(a.1(a.0(b.0(B.0(x)))))))) → B1.0(a.1(a.0(b.0(b.1(a.0(b.1(a.0(B.0(x)))))))))
B1.1(a.0(a.1(a.0(b.0(b.0(B.0(x))))))) → B1.1(a.0(b.1(a.0(a.1(a.0(b.1(a.0(b.1(a.0(B.0(x)))))))))))
B1.1(a.0(b.1(a.0(a.1(a.0(b.0(B.0(x)))))))) → B1.0(a.1(a.0(b.0(b.1(a.0(B.0(x)))))))
The TRS R consists of the following rules:
b.1(a.0(a.1(a.0(b.0(b.0(B.0(x))))))) → b.1(a.0(b.1(a.0(a.1(a.0(b.1(a.0(b.1(a.0(b.1(a.0(B.0(x)))))))))))))
b.0(b.1(x)) → b.1(a.0(b.1(a.0(b.1(a.0(b.1(x)))))))
b.0(b.0(B.0(x))) → b.1(a.0(b.1(a.0(b.1(a.0(b.1(a.0(b.1(a.0(b.1(a.0(B.0(x)))))))))))))
b.1(a.0(a.1(a.0(b.0(b.0(B.1(x))))))) → b.1(a.0(b.1(a.0(a.1(a.0(b.1(a.0(b.1(a.0(B.1(x)))))))))))
b.1(a.0(a.1(a.0(b.0(b.1(x)))))) → b.1(a.0(b.1(a.0(a.1(a.0(b.1(x)))))))
b.0(a.1(a.0(b.0(B.0(x))))) → b.1(a.0(a.1(a.0(b.1(a.0(b.1(a.0(b.1(a.0(B.0(x)))))))))))
b.1(a.0(b.1(a.0(a.1(a.0(b.0(B.0(x)))))))) → b.0(a.1(a.0(b.0(b.1(a.0(b.1(a.0(B.0(x)))))))))
b.1(a.0(a.1(a.0(b.0(b.0(B.1(x))))))) → b.1(a.0(b.1(a.0(a.1(a.0(b.1(a.0(b.1(a.0(b.1(a.0(B.1(x)))))))))))))
b.0(B.1(x)) → b.1(a.0(B.1(x)))
b.0(a.1(a.0(b.0(B.1(x))))) → b.1(a.0(a.1(a.0(b.1(a.0(b.1(a.0(B.1(x)))))))))
b.1(a.0(b.1(a.0(a.1(a.0(b.1(a.0(b.0(B.1(x)))))))))) → b.0(a.1(a.0(b.0(b.1(a.0(b.1(a.0(B.1(x)))))))))
b.0(B.0(x)) → b.1(a.0(B.0(x)))
b.1(a.0(b.1(a.0(a.1(a.0(b.0(B.0(x)))))))) → b.0(a.1(a.0(b.0(b.1(a.0(B.0(x)))))))
b.0(b.0(x)) → b.1(a.0(b.1(a.0(b.1(a.0(b.0(x)))))))
b.1(a.0(b.1(a.0(a.1(a.0(b.0(B.1(x)))))))) → b.0(a.1(a.0(b.0(b.1(a.0(B.1(x)))))))
b.1(a.0(a.1(a.0(b.0(b.0(B.0(x))))))) → b.1(a.0(b.1(a.0(a.1(a.0(b.1(a.0(b.1(a.0(B.0(x)))))))))))
b.1(a.0(b.1(a.0(a.1(a.0(b.0(B.1(x)))))))) → b.0(a.1(a.0(b.0(b.1(a.0(b.1(a.0(B.1(x)))))))))
b.1(a.0(b.1(a.0(a.1(a.0(b.1(a.0(b.0(B.1(x)))))))))) → b.0(a.1(a.0(b.0(b.1(a.0(b.1(a.0(b.1(a.0(B.1(x)))))))))))
b.1(a.0(b.1(a.0(a.1(a.0(b.1(a.0(b.0(B.0(x)))))))))) → b.0(a.1(a.0(b.0(b.1(a.0(b.1(a.0(B.0(x)))))))))
b.1(a.0(b.1(a.0(a.1(a.0(b.1(a.0(B.0(x))))))))) → b.0(a.1(a.0(b.0(B.0(x)))))
b.0(b.0(B.0(x))) → b.1(a.0(b.1(a.0(b.1(a.0(b.1(a.0(b.1(a.0(B.0(x)))))))))))
b.0(b.0(B.1(x))) → b.1(a.0(b.1(a.0(b.1(a.0(b.1(a.0(b.1(a.0(B.1(x)))))))))))
b.0(a.1(a.0(b.0(B.1(x))))) → b.1(a.0(a.1(a.0(b.1(a.0(b.1(a.0(b.1(a.0(B.1(x)))))))))))
b.1(a.0(a.1(a.0(b.0(b.0(x)))))) → b.1(a.0(b.1(a.0(a.1(a.0(b.0(x)))))))
b.0(a.1(a.0(b.1(x)))) → b.1(a.0(a.1(a.0(b.1(x)))))
b.0(a.1(a.0(b.0(B.0(x))))) → b.1(a.0(a.1(a.0(b.1(a.0(b.1(a.0(B.0(x)))))))))
b.1(a.0(b.1(a.0(a.1(a.0(b.1(a.0(B.1(x))))))))) → b.0(a.1(a.0(b.0(B.1(x)))))
b.1(a.0(b.1(a.0(a.1(a.0(b.1(a.0(b.0(x))))))))) → b.0(a.1(a.0(b.0(b.0(x)))))
b.0(a.1(a.0(b.0(x)))) → b.1(a.0(a.1(a.0(b.0(x)))))
b.0(b.0(B.1(x))) → b.1(a.0(b.1(a.0(b.1(a.0(b.1(a.0(b.1(a.0(b.1(a.0(B.1(x)))))))))))))
b.1(a.0(b.1(a.0(a.1(a.0(b.1(a.0(b.1(x))))))))) → b.0(a.1(a.0(b.0(b.1(x)))))
b.1(a.0(b.1(a.0(a.1(a.0(b.1(a.0(b.0(B.0(x)))))))))) → b.0(a.1(a.0(b.0(b.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
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ SemLabProof
↳ SemLabProof2
↳ QDP
↳ QDPToSRSProof
↳ QTRS
↳ QTRS Reverse
↳ QTRS
↳ QTRS Reverse
↳ QTRS Reverse
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ SemLabProof
↳ SemLabProof2
↳ QDP
↳ SemLabProof
↳ SemLabProof2
↳ QTRS Reverse
↳ QTRS Reverse
Q DP problem:
The TRS P consists of the following rules:
B1(a(a(a(b(b(x)))))) → B1(a(b(a(a(a(b(x)))))))
B1(a(b(a(a(a(b(B(x)))))))) → B1(a(a(b(b(a(b(a(B(x)))))))))
B1(a(b(a(a(a(b(B(x)))))))) → B1(a(a(b(b(a(B(x)))))))
B1(a(a(a(b(b(B(x))))))) → B1(a(b(a(a(a(b(a(b(a(B(x)))))))))))
B1(a(a(a(b(b(x)))))) → B1(a(a(a(b(x)))))
B1(a(a(a(b(b(B(x))))))) → B1(a(b(a(a(a(b(a(b(a(b(a(B(x)))))))))))))
B1(a(b(a(a(a(b(a(b(B(x)))))))))) → B1(a(a(b(b(a(b(a(B(x)))))))))
B1(a(b(a(a(a(b(a(B(x))))))))) → B1(a(a(b(B(x)))))
B1(a(b(a(a(a(b(a(b(x))))))))) → B1(a(a(b(b(x)))))
B1(a(a(b(x)))) → B1(a(a(a(b(x)))))
B1(a(b(a(a(a(b(a(b(B(x)))))))))) → B1(a(a(b(b(a(b(a(b(a(B(x)))))))))))
The TRS R consists of the following rules:
b(a(a(a(b(b(x)))))) → b(a(b(a(a(a(b(x)))))))
b(b(x)) → b(a(b(a(b(a(b(x)))))))
b(a(b(a(a(a(b(a(b(x))))))))) → b(a(a(b(b(x)))))
b(a(a(b(x)))) → b(a(a(a(b(x)))))
b(a(a(b(B(x))))) → b(a(a(a(b(a(b(a(B(x)))))))))
b(a(a(b(B(x))))) → b(a(a(a(b(a(b(a(b(a(B(x)))))))))))
b(b(B(x))) → b(a(b(a(b(a(b(a(b(a(b(a(B(x)))))))))))))
b(b(B(x))) → b(a(b(a(b(a(b(a(b(a(B(x)))))))))))
b(a(b(a(a(a(b(a(b(B(x)))))))))) → b(a(a(b(b(a(b(a(B(x)))))))))
b(B(x)) → b(a(B(x)))
b(a(b(a(a(a(b(a(B(x))))))))) → b(a(a(b(B(x)))))
b(a(a(a(b(b(B(x))))))) → b(a(b(a(a(a(b(a(b(a(b(a(B(x)))))))))))))
b(a(a(a(b(b(B(x))))))) → b(a(b(a(a(a(b(a(b(a(B(x)))))))))))
b(a(b(a(a(a(b(B(x)))))))) → b(a(a(b(b(a(B(x)))))))
b(a(b(a(a(a(b(a(b(B(x)))))))))) → b(a(a(b(b(a(b(a(b(a(B(x)))))))))))
b(a(b(a(a(a(b(B(x)))))))) → b(a(a(b(b(a(b(a(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: 1
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:
B1.1(a.1(a.0(b.1(x)))) → B1.1(a.1(a.1(a.0(b.1(x)))))
B1.1(a.1(a.0(b.0(x)))) → B1.1(a.1(a.1(a.0(b.0(x)))))
B1.1(a.1(a.1(a.0(b.0(b.0(x)))))) → B1.1(a.1(a.1(a.0(b.0(x)))))
B1.1(a.0(b.1(a.1(a.1(a.0(b.1(a.0(b.1(x))))))))) → B1.1(a.1(a.0(b.0(b.1(x)))))
B1.1(a.1(a.1(a.0(b.0(b.1(x)))))) → B1.1(a.1(a.1(a.0(b.1(x)))))
B1.1(a.0(b.1(a.1(a.1(a.0(b.1(a.0(b.0(x))))))))) → B1.1(a.1(a.0(b.0(b.0(x)))))
B1.1(a.0(b.1(a.1(a.1(a.0(b.1(a.1(B.0(x))))))))) → B1.1(a.1(a.0(b.1(B.0(x)))))
B1.1(a.0(b.1(a.1(a.1(a.0(b.1(B.1(x)))))))) → B1.1(a.1(a.0(b.0(b.1(a.0(b.1(a.1(B.1(x)))))))))
B1.1(a.0(b.1(a.1(a.1(a.0(b.1(a.1(B.1(x))))))))) → B1.1(a.1(a.0(b.1(B.1(x)))))
B1.1(a.0(b.1(a.1(a.1(a.0(b.1(a.0(b.1(B.1(x)))))))))) → B1.1(a.1(a.0(b.0(b.1(a.0(b.1(a.0(b.1(a.1(B.1(x)))))))))))
B1.1(a.1(a.1(a.0(b.0(b.1(B.1(x))))))) → B1.1(a.0(b.1(a.1(a.1(a.0(b.1(a.0(b.1(a.1(B.1(x)))))))))))
B1.1(a.0(b.1(a.1(a.1(a.0(b.1(a.0(b.1(B.1(x)))))))))) → B1.1(a.1(a.0(b.0(b.1(a.0(b.1(a.1(B.1(x)))))))))
B1.1(a.0(b.1(a.1(a.1(a.0(b.1(a.0(b.1(B.0(x)))))))))) → B1.1(a.1(a.0(b.0(b.1(a.0(b.1(a.1(B.0(x)))))))))
B1.1(a.0(b.1(a.1(a.1(a.0(b.1(B.0(x)))))))) → B1.1(a.1(a.0(b.0(b.1(a.1(B.0(x)))))))
B1.1(a.0(b.1(a.1(a.1(a.0(b.1(B.1(x)))))))) → B1.1(a.1(a.0(b.0(b.1(a.1(B.1(x)))))))
B1.1(a.0(b.1(a.1(a.1(a.0(b.1(a.0(b.1(B.0(x)))))))))) → B1.1(a.1(a.0(b.0(b.1(a.0(b.1(a.0(b.1(a.1(B.0(x)))))))))))
B1.1(a.0(b.1(a.1(a.1(a.0(b.1(B.0(x)))))))) → B1.1(a.1(a.0(b.0(b.1(a.0(b.1(a.1(B.0(x)))))))))
B1.1(a.1(a.1(a.0(b.0(b.1(B.0(x))))))) → B1.1(a.0(b.1(a.1(a.1(a.0(b.1(a.0(b.1(a.1(B.0(x)))))))))))
B1.1(a.1(a.1(a.0(b.0(b.1(B.1(x))))))) → B1.1(a.0(b.1(a.1(a.1(a.0(b.1(a.0(b.1(a.0(b.1(a.1(B.1(x)))))))))))))
B1.1(a.1(a.1(a.0(b.0(b.0(x)))))) → B1.1(a.0(b.1(a.1(a.1(a.0(b.0(x)))))))
B1.1(a.1(a.1(a.0(b.0(b.1(B.0(x))))))) → B1.1(a.0(b.1(a.1(a.1(a.0(b.1(a.0(b.1(a.0(b.1(a.1(B.0(x)))))))))))))
B1.1(a.1(a.1(a.0(b.0(b.1(x)))))) → B1.1(a.0(b.1(a.1(a.1(a.0(b.1(x)))))))
The TRS R consists of the following rules:
b.1(a.0(b.1(a.1(a.1(a.0(b.1(a.0(b.1(B.1(x)))))))))) → b.1(a.1(a.0(b.0(b.1(a.0(b.1(a.1(B.1(x)))))))))
b.1(a.0(b.1(a.1(a.1(a.0(b.1(a.0(b.1(x))))))))) → b.1(a.1(a.0(b.0(b.1(x)))))
b.0(b.1(x)) → b.1(a.0(b.1(a.0(b.1(a.0(b.1(x)))))))
b.1(a.0(b.1(a.1(a.1(a.0(b.1(B.0(x)))))))) → b.1(a.1(a.0(b.0(b.1(a.0(b.1(a.1(B.0(x)))))))))
b.1(a.1(a.0(b.1(B.0(x))))) → b.1(a.1(a.1(a.0(b.1(a.0(b.1(a.1(B.0(x)))))))))
b.1(a.0(b.1(a.1(a.1(a.0(b.1(B.1(x)))))))) → b.1(a.1(a.0(b.0(b.1(a.0(b.1(a.1(B.1(x)))))))))
b.1(a.0(b.1(a.1(a.1(a.0(b.1(a.0(b.1(B.0(x)))))))))) → b.1(a.1(a.0(b.0(b.1(a.0(b.1(a.1(B.0(x)))))))))
b.0(b.1(B.0(x))) → b.1(a.0(b.1(a.0(b.1(a.0(b.1(a.0(b.1(a.0(b.1(a.1(B.0(x)))))))))))))
b.1(a.1(a.0(b.1(B.1(x))))) → b.1(a.1(a.1(a.0(b.1(a.0(b.1(a.1(B.1(x)))))))))
b.1(a.0(b.1(a.1(a.1(a.0(b.1(a.1(B.1(x))))))))) → b.1(a.1(a.0(b.1(B.1(x)))))
b.1(a.0(b.1(a.1(a.1(a.0(b.1(a.1(B.0(x))))))))) → b.1(a.1(a.0(b.1(B.0(x)))))
b.0(b.0(x)) → b.1(a.0(b.1(a.0(b.1(a.0(b.0(x)))))))
b.1(B.1(x)) → b.1(a.1(B.1(x)))
b.1(a.1(a.1(a.0(b.0(b.1(B.0(x))))))) → b.1(a.0(b.1(a.1(a.1(a.0(b.1(a.0(b.1(a.0(b.1(a.1(B.0(x)))))))))))))
b.1(a.0(b.1(a.1(a.1(a.0(b.1(B.0(x)))))))) → b.1(a.1(a.0(b.0(b.1(a.1(B.0(x)))))))
b.1(a.1(a.1(a.0(b.0(b.1(x)))))) → b.1(a.0(b.1(a.1(a.1(a.0(b.1(x)))))))
b.1(a.0(b.1(a.1(a.1(a.0(b.1(a.0(b.0(x))))))))) → b.1(a.1(a.0(b.0(b.0(x)))))
b.0(b.1(B.0(x))) → b.1(a.0(b.1(a.0(b.1(a.0(b.1(a.0(b.1(a.1(B.0(x)))))))))))
b.0(b.1(B.1(x))) → b.1(a.0(b.1(a.0(b.1(a.0(b.1(a.0(b.1(a.1(B.1(x)))))))))))
b.1(a.1(a.1(a.0(b.0(b.1(B.1(x))))))) → b.1(a.0(b.1(a.1(a.1(a.0(b.1(a.0(b.1(a.0(b.1(a.1(B.1(x)))))))))))))
b.1(a.0(b.1(a.1(a.1(a.0(b.1(a.0(b.1(B.0(x)))))))))) → b.1(a.1(a.0(b.0(b.1(a.0(b.1(a.0(b.1(a.1(B.0(x)))))))))))
b.1(a.0(b.1(a.1(a.1(a.0(b.1(B.1(x)))))))) → b.1(a.1(a.0(b.0(b.1(a.1(B.1(x)))))))
b.1(a.1(a.0(b.1(B.1(x))))) → b.1(a.1(a.1(a.0(b.1(a.0(b.1(a.0(b.1(a.1(B.1(x)))))))))))
b.1(a.0(b.1(a.1(a.1(a.0(b.1(a.0(b.1(B.1(x)))))))))) → b.1(a.1(a.0(b.0(b.1(a.0(b.1(a.0(b.1(a.1(B.1(x)))))))))))
b.1(a.1(a.0(b.0(x)))) → b.1(a.1(a.1(a.0(b.0(x)))))
b.1(a.1(a.0(b.1(x)))) → b.1(a.1(a.1(a.0(b.1(x)))))
b.1(a.1(a.1(a.0(b.0(b.0(x)))))) → b.1(a.0(b.1(a.1(a.1(a.0(b.0(x)))))))
b.1(a.1(a.0(b.1(B.0(x))))) → b.1(a.1(a.1(a.0(b.1(a.0(b.1(a.0(b.1(a.1(B.0(x)))))))))))
b.0(b.1(B.1(x))) → b.1(a.0(b.1(a.0(b.1(a.0(b.1(a.0(b.1(a.0(b.1(a.1(B.1(x)))))))))))))
b.1(a.1(a.1(a.0(b.0(b.1(B.0(x))))))) → b.1(a.0(b.1(a.1(a.1(a.0(b.1(a.0(b.1(a.1(B.0(x)))))))))))
b.1(a.1(a.1(a.0(b.0(b.1(B.1(x))))))) → b.1(a.0(b.1(a.1(a.1(a.0(b.1(a.0(b.1(a.1(B.1(x)))))))))))
b.1(B.0(x)) → b.1(a.1(B.0(x)))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ SemLabProof
↳ SemLabProof2
↳ QDP
↳ QDPToSRSProof
↳ QTRS
↳ QTRS Reverse
↳ QTRS
↳ QTRS Reverse
↳ QTRS Reverse
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ SemLabProof
↳ SemLabProof2
↳ QDP
↳ SemLabProof
↳ QDP
↳ DependencyGraphProof
↳ SemLabProof2
↳ QTRS Reverse
↳ QTRS Reverse
Q DP problem:
The TRS P consists of the following rules:
B1.1(a.1(a.0(b.1(x)))) → B1.1(a.1(a.1(a.0(b.1(x)))))
B1.1(a.1(a.0(b.0(x)))) → B1.1(a.1(a.1(a.0(b.0(x)))))
B1.1(a.1(a.1(a.0(b.0(b.0(x)))))) → B1.1(a.1(a.1(a.0(b.0(x)))))
B1.1(a.0(b.1(a.1(a.1(a.0(b.1(a.0(b.1(x))))))))) → B1.1(a.1(a.0(b.0(b.1(x)))))
B1.1(a.1(a.1(a.0(b.0(b.1(x)))))) → B1.1(a.1(a.1(a.0(b.1(x)))))
B1.1(a.0(b.1(a.1(a.1(a.0(b.1(a.0(b.0(x))))))))) → B1.1(a.1(a.0(b.0(b.0(x)))))
B1.1(a.0(b.1(a.1(a.1(a.0(b.1(a.1(B.0(x))))))))) → B1.1(a.1(a.0(b.1(B.0(x)))))
B1.1(a.0(b.1(a.1(a.1(a.0(b.1(B.1(x)))))))) → B1.1(a.1(a.0(b.0(b.1(a.0(b.1(a.1(B.1(x)))))))))
B1.1(a.0(b.1(a.1(a.1(a.0(b.1(a.1(B.1(x))))))))) → B1.1(a.1(a.0(b.1(B.1(x)))))
B1.1(a.0(b.1(a.1(a.1(a.0(b.1(a.0(b.1(B.1(x)))))))))) → B1.1(a.1(a.0(b.0(b.1(a.0(b.1(a.0(b.1(a.1(B.1(x)))))))))))
B1.1(a.1(a.1(a.0(b.0(b.1(B.1(x))))))) → B1.1(a.0(b.1(a.1(a.1(a.0(b.1(a.0(b.1(a.1(B.1(x)))))))))))
B1.1(a.0(b.1(a.1(a.1(a.0(b.1(a.0(b.1(B.1(x)))))))))) → B1.1(a.1(a.0(b.0(b.1(a.0(b.1(a.1(B.1(x)))))))))
B1.1(a.0(b.1(a.1(a.1(a.0(b.1(a.0(b.1(B.0(x)))))))))) → B1.1(a.1(a.0(b.0(b.1(a.0(b.1(a.1(B.0(x)))))))))
B1.1(a.0(b.1(a.1(a.1(a.0(b.1(B.0(x)))))))) → B1.1(a.1(a.0(b.0(b.1(a.1(B.0(x)))))))
B1.1(a.0(b.1(a.1(a.1(a.0(b.1(B.1(x)))))))) → B1.1(a.1(a.0(b.0(b.1(a.1(B.1(x)))))))
B1.1(a.0(b.1(a.1(a.1(a.0(b.1(a.0(b.1(B.0(x)))))))))) → B1.1(a.1(a.0(b.0(b.1(a.0(b.1(a.0(b.1(a.1(B.0(x)))))))))))
B1.1(a.0(b.1(a.1(a.1(a.0(b.1(B.0(x)))))))) → B1.1(a.1(a.0(b.0(b.1(a.0(b.1(a.1(B.0(x)))))))))
B1.1(a.1(a.1(a.0(b.0(b.1(B.0(x))))))) → B1.1(a.0(b.1(a.1(a.1(a.0(b.1(a.0(b.1(a.1(B.0(x)))))))))))
B1.1(a.1(a.1(a.0(b.0(b.1(B.1(x))))))) → B1.1(a.0(b.1(a.1(a.1(a.0(b.1(a.0(b.1(a.0(b.1(a.1(B.1(x)))))))))))))
B1.1(a.1(a.1(a.0(b.0(b.0(x)))))) → B1.1(a.0(b.1(a.1(a.1(a.0(b.0(x)))))))
B1.1(a.1(a.1(a.0(b.0(b.1(B.0(x))))))) → B1.1(a.0(b.1(a.1(a.1(a.0(b.1(a.0(b.1(a.0(b.1(a.1(B.0(x)))))))))))))
B1.1(a.1(a.1(a.0(b.0(b.1(x)))))) → B1.1(a.0(b.1(a.1(a.1(a.0(b.1(x)))))))
The TRS R consists of the following rules:
b.1(a.0(b.1(a.1(a.1(a.0(b.1(a.0(b.1(B.1(x)))))))))) → b.1(a.1(a.0(b.0(b.1(a.0(b.1(a.1(B.1(x)))))))))
b.1(a.0(b.1(a.1(a.1(a.0(b.1(a.0(b.1(x))))))))) → b.1(a.1(a.0(b.0(b.1(x)))))
b.0(b.1(x)) → b.1(a.0(b.1(a.0(b.1(a.0(b.1(x)))))))
b.1(a.0(b.1(a.1(a.1(a.0(b.1(B.0(x)))))))) → b.1(a.1(a.0(b.0(b.1(a.0(b.1(a.1(B.0(x)))))))))
b.1(a.1(a.0(b.1(B.0(x))))) → b.1(a.1(a.1(a.0(b.1(a.0(b.1(a.1(B.0(x)))))))))
b.1(a.0(b.1(a.1(a.1(a.0(b.1(B.1(x)))))))) → b.1(a.1(a.0(b.0(b.1(a.0(b.1(a.1(B.1(x)))))))))
b.1(a.0(b.1(a.1(a.1(a.0(b.1(a.0(b.1(B.0(x)))))))))) → b.1(a.1(a.0(b.0(b.1(a.0(b.1(a.1(B.0(x)))))))))
b.0(b.1(B.0(x))) → b.1(a.0(b.1(a.0(b.1(a.0(b.1(a.0(b.1(a.0(b.1(a.1(B.0(x)))))))))))))
b.1(a.1(a.0(b.1(B.1(x))))) → b.1(a.1(a.1(a.0(b.1(a.0(b.1(a.1(B.1(x)))))))))
b.1(a.0(b.1(a.1(a.1(a.0(b.1(a.1(B.1(x))))))))) → b.1(a.1(a.0(b.1(B.1(x)))))
b.1(a.0(b.1(a.1(a.1(a.0(b.1(a.1(B.0(x))))))))) → b.1(a.1(a.0(b.1(B.0(x)))))
b.0(b.0(x)) → b.1(a.0(b.1(a.0(b.1(a.0(b.0(x)))))))
b.1(B.1(x)) → b.1(a.1(B.1(x)))
b.1(a.1(a.1(a.0(b.0(b.1(B.0(x))))))) → b.1(a.0(b.1(a.1(a.1(a.0(b.1(a.0(b.1(a.0(b.1(a.1(B.0(x)))))))))))))
b.1(a.0(b.1(a.1(a.1(a.0(b.1(B.0(x)))))))) → b.1(a.1(a.0(b.0(b.1(a.1(B.0(x)))))))
b.1(a.1(a.1(a.0(b.0(b.1(x)))))) → b.1(a.0(b.1(a.1(a.1(a.0(b.1(x)))))))
b.1(a.0(b.1(a.1(a.1(a.0(b.1(a.0(b.0(x))))))))) → b.1(a.1(a.0(b.0(b.0(x)))))
b.0(b.1(B.0(x))) → b.1(a.0(b.1(a.0(b.1(a.0(b.1(a.0(b.1(a.1(B.0(x)))))))))))
b.0(b.1(B.1(x))) → b.1(a.0(b.1(a.0(b.1(a.0(b.1(a.0(b.1(a.1(B.1(x)))))))))))
b.1(a.1(a.1(a.0(b.0(b.1(B.1(x))))))) → b.1(a.0(b.1(a.1(a.1(a.0(b.1(a.0(b.1(a.0(b.1(a.1(B.1(x)))))))))))))
b.1(a.0(b.1(a.1(a.1(a.0(b.1(a.0(b.1(B.0(x)))))))))) → b.1(a.1(a.0(b.0(b.1(a.0(b.1(a.0(b.1(a.1(B.0(x)))))))))))
b.1(a.0(b.1(a.1(a.1(a.0(b.1(B.1(x)))))))) → b.1(a.1(a.0(b.0(b.1(a.1(B.1(x)))))))
b.1(a.1(a.0(b.1(B.1(x))))) → b.1(a.1(a.1(a.0(b.1(a.0(b.1(a.0(b.1(a.1(B.1(x)))))))))))
b.1(a.0(b.1(a.1(a.1(a.0(b.1(a.0(b.1(B.1(x)))))))))) → b.1(a.1(a.0(b.0(b.1(a.0(b.1(a.0(b.1(a.1(B.1(x)))))))))))
b.1(a.1(a.0(b.0(x)))) → b.1(a.1(a.1(a.0(b.0(x)))))
b.1(a.1(a.0(b.1(x)))) → b.1(a.1(a.1(a.0(b.1(x)))))
b.1(a.1(a.1(a.0(b.0(b.0(x)))))) → b.1(a.0(b.1(a.1(a.1(a.0(b.0(x)))))))
b.1(a.1(a.0(b.1(B.0(x))))) → b.1(a.1(a.1(a.0(b.1(a.0(b.1(a.0(b.1(a.1(B.0(x)))))))))))
b.0(b.1(B.1(x))) → b.1(a.0(b.1(a.0(b.1(a.0(b.1(a.0(b.1(a.0(b.1(a.1(B.1(x)))))))))))))
b.1(a.1(a.1(a.0(b.0(b.1(B.0(x))))))) → b.1(a.0(b.1(a.1(a.1(a.0(b.1(a.0(b.1(a.1(B.0(x)))))))))))
b.1(a.1(a.1(a.0(b.0(b.1(B.1(x))))))) → b.1(a.0(b.1(a.1(a.1(a.0(b.1(a.0(b.1(a.1(B.1(x)))))))))))
b.1(B.0(x)) → b.1(a.1(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 4 less nodes.
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ SemLabProof
↳ SemLabProof2
↳ QDP
↳ QDPToSRSProof
↳ QTRS
↳ QTRS Reverse
↳ QTRS
↳ QTRS Reverse
↳ QTRS Reverse
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ SemLabProof
↳ SemLabProof2
↳ QDP
↳ SemLabProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ SemLabProof2
↳ QTRS Reverse
↳ QTRS Reverse
Q DP problem:
The TRS P consists of the following rules:
B1.1(a.1(a.0(b.0(x)))) → B1.1(a.1(a.1(a.0(b.0(x)))))
B1.1(a.1(a.1(a.0(b.0(b.0(x)))))) → B1.1(a.1(a.1(a.0(b.0(x)))))
B1.1(a.0(b.1(a.1(a.1(a.0(b.1(a.0(b.1(x))))))))) → B1.1(a.1(a.0(b.0(b.1(x)))))
B1.1(a.0(b.1(a.1(a.1(a.0(b.1(a.0(b.0(x))))))))) → B1.1(a.1(a.0(b.0(b.0(x)))))
B1.1(a.0(b.1(a.1(a.1(a.0(b.1(B.1(x)))))))) → B1.1(a.1(a.0(b.0(b.1(a.0(b.1(a.1(B.1(x)))))))))
B1.1(a.0(b.1(a.1(a.1(a.0(b.1(a.0(b.1(B.1(x)))))))))) → B1.1(a.1(a.0(b.0(b.1(a.0(b.1(a.0(b.1(a.1(B.1(x)))))))))))
B1.1(a.1(a.1(a.0(b.0(b.1(B.1(x))))))) → B1.1(a.0(b.1(a.1(a.1(a.0(b.1(a.0(b.1(a.1(B.1(x)))))))))))
B1.1(a.0(b.1(a.1(a.1(a.0(b.1(a.0(b.1(B.1(x)))))))))) → B1.1(a.1(a.0(b.0(b.1(a.0(b.1(a.1(B.1(x)))))))))
B1.1(a.0(b.1(a.1(a.1(a.0(b.1(a.0(b.1(B.0(x)))))))))) → B1.1(a.1(a.0(b.0(b.1(a.0(b.1(a.1(B.0(x)))))))))
B1.1(a.0(b.1(a.1(a.1(a.0(b.1(B.0(x)))))))) → B1.1(a.1(a.0(b.0(b.1(a.1(B.0(x)))))))
B1.1(a.0(b.1(a.1(a.1(a.0(b.1(B.1(x)))))))) → B1.1(a.1(a.0(b.0(b.1(a.1(B.1(x)))))))
B1.1(a.0(b.1(a.1(a.1(a.0(b.1(a.0(b.1(B.0(x)))))))))) → B1.1(a.1(a.0(b.0(b.1(a.0(b.1(a.0(b.1(a.1(B.0(x)))))))))))
B1.1(a.0(b.1(a.1(a.1(a.0(b.1(B.0(x)))))))) → B1.1(a.1(a.0(b.0(b.1(a.0(b.1(a.1(B.0(x)))))))))
B1.1(a.1(a.1(a.0(b.0(b.1(B.0(x))))))) → B1.1(a.0(b.1(a.1(a.1(a.0(b.1(a.0(b.1(a.1(B.0(x)))))))))))
B1.1(a.1(a.1(a.0(b.0(b.1(B.1(x))))))) → B1.1(a.0(b.1(a.1(a.1(a.0(b.1(a.0(b.1(a.0(b.1(a.1(B.1(x)))))))))))))
B1.1(a.1(a.1(a.0(b.0(b.0(x)))))) → B1.1(a.0(b.1(a.1(a.1(a.0(b.0(x)))))))
B1.1(a.1(a.1(a.0(b.0(b.1(B.0(x))))))) → B1.1(a.0(b.1(a.1(a.1(a.0(b.1(a.0(b.1(a.0(b.1(a.1(B.0(x)))))))))))))
B1.1(a.1(a.1(a.0(b.0(b.1(x)))))) → B1.1(a.0(b.1(a.1(a.1(a.0(b.1(x)))))))
The TRS R consists of the following rules:
b.1(a.0(b.1(a.1(a.1(a.0(b.1(a.0(b.1(B.1(x)))))))))) → b.1(a.1(a.0(b.0(b.1(a.0(b.1(a.1(B.1(x)))))))))
b.1(a.0(b.1(a.1(a.1(a.0(b.1(a.0(b.1(x))))))))) → b.1(a.1(a.0(b.0(b.1(x)))))
b.0(b.1(x)) → b.1(a.0(b.1(a.0(b.1(a.0(b.1(x)))))))
b.1(a.0(b.1(a.1(a.1(a.0(b.1(B.0(x)))))))) → b.1(a.1(a.0(b.0(b.1(a.0(b.1(a.1(B.0(x)))))))))
b.1(a.1(a.0(b.1(B.0(x))))) → b.1(a.1(a.1(a.0(b.1(a.0(b.1(a.1(B.0(x)))))))))
b.1(a.0(b.1(a.1(a.1(a.0(b.1(B.1(x)))))))) → b.1(a.1(a.0(b.0(b.1(a.0(b.1(a.1(B.1(x)))))))))
b.1(a.0(b.1(a.1(a.1(a.0(b.1(a.0(b.1(B.0(x)))))))))) → b.1(a.1(a.0(b.0(b.1(a.0(b.1(a.1(B.0(x)))))))))
b.0(b.1(B.0(x))) → b.1(a.0(b.1(a.0(b.1(a.0(b.1(a.0(b.1(a.0(b.1(a.1(B.0(x)))))))))))))
b.1(a.1(a.0(b.1(B.1(x))))) → b.1(a.1(a.1(a.0(b.1(a.0(b.1(a.1(B.1(x)))))))))
b.1(a.0(b.1(a.1(a.1(a.0(b.1(a.1(B.1(x))))))))) → b.1(a.1(a.0(b.1(B.1(x)))))
b.1(a.0(b.1(a.1(a.1(a.0(b.1(a.1(B.0(x))))))))) → b.1(a.1(a.0(b.1(B.0(x)))))
b.0(b.0(x)) → b.1(a.0(b.1(a.0(b.1(a.0(b.0(x)))))))
b.1(B.1(x)) → b.1(a.1(B.1(x)))
b.1(a.1(a.1(a.0(b.0(b.1(B.0(x))))))) → b.1(a.0(b.1(a.1(a.1(a.0(b.1(a.0(b.1(a.0(b.1(a.1(B.0(x)))))))))))))
b.1(a.0(b.1(a.1(a.1(a.0(b.1(B.0(x)))))))) → b.1(a.1(a.0(b.0(b.1(a.1(B.0(x)))))))
b.1(a.1(a.1(a.0(b.0(b.1(x)))))) → b.1(a.0(b.1(a.1(a.1(a.0(b.1(x)))))))
b.1(a.0(b.1(a.1(a.1(a.0(b.1(a.0(b.0(x))))))))) → b.1(a.1(a.0(b.0(b.0(x)))))
b.0(b.1(B.0(x))) → b.1(a.0(b.1(a.0(b.1(a.0(b.1(a.0(b.1(a.1(B.0(x)))))))))))
b.0(b.1(B.1(x))) → b.1(a.0(b.1(a.0(b.1(a.0(b.1(a.0(b.1(a.1(B.1(x)))))))))))
b.1(a.1(a.1(a.0(b.0(b.1(B.1(x))))))) → b.1(a.0(b.1(a.1(a.1(a.0(b.1(a.0(b.1(a.0(b.1(a.1(B.1(x)))))))))))))
b.1(a.0(b.1(a.1(a.1(a.0(b.1(a.0(b.1(B.0(x)))))))))) → b.1(a.1(a.0(b.0(b.1(a.0(b.1(a.0(b.1(a.1(B.0(x)))))))))))
b.1(a.0(b.1(a.1(a.1(a.0(b.1(B.1(x)))))))) → b.1(a.1(a.0(b.0(b.1(a.1(B.1(x)))))))
b.1(a.1(a.0(b.1(B.1(x))))) → b.1(a.1(a.1(a.0(b.1(a.0(b.1(a.0(b.1(a.1(B.1(x)))))))))))
b.1(a.0(b.1(a.1(a.1(a.0(b.1(a.0(b.1(B.1(x)))))))))) → b.1(a.1(a.0(b.0(b.1(a.0(b.1(a.0(b.1(a.1(B.1(x)))))))))))
b.1(a.1(a.0(b.0(x)))) → b.1(a.1(a.1(a.0(b.0(x)))))
b.1(a.1(a.0(b.1(x)))) → b.1(a.1(a.1(a.0(b.1(x)))))
b.1(a.1(a.1(a.0(b.0(b.0(x)))))) → b.1(a.0(b.1(a.1(a.1(a.0(b.0(x)))))))
b.1(a.1(a.0(b.1(B.0(x))))) → b.1(a.1(a.1(a.0(b.1(a.0(b.1(a.0(b.1(a.1(B.0(x)))))))))))
b.0(b.1(B.1(x))) → b.1(a.0(b.1(a.0(b.1(a.0(b.1(a.0(b.1(a.0(b.1(a.1(B.1(x)))))))))))))
b.1(a.1(a.1(a.0(b.0(b.1(B.0(x))))))) → b.1(a.0(b.1(a.1(a.1(a.0(b.1(a.0(b.1(a.1(B.0(x)))))))))))
b.1(a.1(a.1(a.0(b.0(b.1(B.1(x))))))) → b.1(a.0(b.1(a.1(a.1(a.0(b.1(a.0(b.1(a.1(B.1(x)))))))))))
b.1(B.0(x)) → b.1(a.1(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
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ Narrowing
↳ QDP
↳ SemLabProof
↳ SemLabProof2
↳ QDP
↳ QDPToSRSProof
↳ QTRS
↳ QTRS Reverse
↳ QTRS
↳ QTRS Reverse
↳ QTRS Reverse
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ SemLabProof
↳ SemLabProof2
↳ QDP
↳ SemLabProof
↳ SemLabProof2
↳ QDP
↳ QTRS Reverse
↳ QTRS Reverse
Q DP problem:
The TRS P consists of the following rules:
B1(a(a(a(b(b(x)))))) → B1(a(b(a(a(a(b(x)))))))
B1(a(b(a(a(a(b(B(x)))))))) → B1(a(a(b(b(a(b(a(B(x)))))))))
B1(a(b(a(a(a(b(B(x)))))))) → B1(a(a(b(b(a(B(x)))))))
B1(a(a(a(b(b(B(x))))))) → B1(a(b(a(a(a(b(a(b(a(B(x)))))))))))
B1(a(a(a(b(b(x)))))) → B1(a(a(a(b(x)))))
B1(a(a(a(b(b(B(x))))))) → B1(a(b(a(a(a(b(a(b(a(b(a(B(x)))))))))))))
B1(a(b(a(a(a(b(a(b(B(x)))))))))) → B1(a(a(b(b(a(b(a(B(x)))))))))
B1(a(b(a(a(a(b(a(b(x))))))))) → B1(a(a(b(b(x)))))
B1(a(a(b(x)))) → B1(a(a(a(b(x)))))
B1(a(b(a(a(a(b(a(b(B(x)))))))))) → B1(a(a(b(b(a(b(a(b(a(B(x)))))))))))
The TRS R consists of the following rules:
b(a(a(a(b(b(x)))))) → b(a(b(a(a(a(b(x)))))))
b(b(x)) → b(a(b(a(b(a(b(x)))))))
b(a(b(a(a(a(b(a(b(x))))))))) → b(a(a(b(b(x)))))
b(a(a(b(x)))) → b(a(a(a(b(x)))))
b(a(a(b(B(x))))) → b(a(a(a(b(a(b(a(B(x)))))))))
b(a(a(b(B(x))))) → b(a(a(a(b(a(b(a(b(a(B(x)))))))))))
b(b(B(x))) → b(a(b(a(b(a(b(a(b(a(b(a(B(x)))))))))))))
b(b(B(x))) → b(a(b(a(b(a(b(a(b(a(B(x)))))))))))
b(a(b(a(a(a(b(a(b(B(x)))))))))) → b(a(a(b(b(a(b(a(B(x)))))))))
b(B(x)) → b(a(B(x)))
b(a(b(a(a(a(b(a(B(x))))))))) → b(a(a(b(B(x)))))
b(a(a(a(b(b(B(x))))))) → b(a(b(a(a(a(b(a(b(a(b(a(B(x)))))))))))))
b(a(a(a(b(b(B(x))))))) → b(a(b(a(a(a(b(a(b(a(B(x)))))))))))
b(a(b(a(a(a(b(B(x)))))))) → b(a(a(b(b(a(B(x)))))))
b(a(b(a(a(a(b(a(b(B(x)))))))))) → b(a(a(b(b(a(b(a(b(a(B(x)))))))))))
b(a(b(a(a(a(b(B(x)))))))) → b(a(a(b(b(a(b(a(B(x)))))))))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We have reversed the following QTRS:
The set of rules R is
b(b(a(a(a(b(x1)))))) → b(a(a(a(b(a(b(x1)))))))
b(b(x1)) → b(a(b(a(b(a(b(x1)))))))
b(a(b(a(a(a(b(a(b(x1))))))))) → b(b(a(a(b(x1)))))
b(a(a(b(x1)))) → b(a(a(a(b(x1)))))
The set Q is empty.
We have obtained the following QTRS:
b(a(a(a(b(b(x)))))) → b(a(b(a(a(a(b(x)))))))
b(b(x)) → b(a(b(a(b(a(b(x)))))))
b(a(b(a(a(a(b(a(b(x))))))))) → b(a(a(b(b(x)))))
b(a(a(b(x)))) → b(a(a(a(b(x)))))
The set Q is empty.
↳ QTRS
↳ DependencyPairsProof
↳ QTRS Reverse
↳ QTRS
↳ QTRS Reverse
Q restricted rewrite system:
The TRS R consists of the following rules:
b(a(a(a(b(b(x)))))) → b(a(b(a(a(a(b(x)))))))
b(b(x)) → b(a(b(a(b(a(b(x)))))))
b(a(b(a(a(a(b(a(b(x))))))))) → b(a(a(b(b(x)))))
b(a(a(b(x)))) → b(a(a(a(b(x)))))
Q is empty.
We have reversed the following QTRS:
The set of rules R is
b(b(a(a(a(b(x1)))))) → b(a(a(a(b(a(b(x1)))))))
b(b(x1)) → b(a(b(a(b(a(b(x1)))))))
b(a(b(a(a(a(b(a(b(x1))))))))) → b(b(a(a(b(x1)))))
b(a(a(b(x1)))) → b(a(a(a(b(x1)))))
The set Q is empty.
We have obtained the following QTRS:
b(a(a(a(b(b(x)))))) → b(a(b(a(a(a(b(x)))))))
b(b(x)) → b(a(b(a(b(a(b(x)))))))
b(a(b(a(a(a(b(a(b(x))))))))) → b(a(a(b(b(x)))))
b(a(a(b(x)))) → b(a(a(a(b(x)))))
The set Q is empty.
↳ QTRS
↳ DependencyPairsProof
↳ QTRS Reverse
↳ QTRS Reverse
↳ QTRS
Q restricted rewrite system:
The TRS R consists of the following rules:
b(a(a(a(b(b(x)))))) → b(a(b(a(a(a(b(x)))))))
b(b(x)) → b(a(b(a(b(a(b(x)))))))
b(a(b(a(a(a(b(a(b(x))))))))) → b(a(a(b(b(x)))))
b(a(a(b(x)))) → b(a(a(a(b(x)))))
Q is empty.