(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
b(b(x)) → b(a(b(x)))
b(b(a(b(x)))) → b(a(b(a(a(b(b(x)))))))
b(a(b(x))) → b(a(a(b(x))))
b(a(a(b(a(b(x)))))) → b(b(x))
Q is empty.
(1) RootLabelingProof (EQUIVALENT transformation)
We used plain root labeling [ROOTLAB] with the following heuristic:
LabelAll: All function symbols get labeled
As Q is empty the root labeling was sound AND complete.
(2) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
b_{b_1}(b_{b_1}(x)) → b_{a_1}(a_{b_1}(b_{b_1}(x)))
b_{b_1}(b_{a_1}(x)) → b_{a_1}(a_{b_1}(b_{a_1}(x)))
b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x)))) → b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x)))))))
b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x)))) → b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x)))))))
b_{a_1}(a_{b_1}(b_{b_1}(x))) → b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x))))
b_{a_1}(a_{b_1}(b_{a_1}(x))) → b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x))))
b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x)))))) → b_{b_1}(b_{b_1}(x))
b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x)))))) → b_{b_1}(b_{a_1}(x))
Q is empty.
(3) DependencyPairsProof (EQUIVALENT transformation)
Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
(4) Obligation:
Q DP problem:
The TRS P consists of the following rules:
B_{B_1}(b_{b_1}(x)) → B_{A_1}(a_{b_1}(b_{b_1}(x)))
B_{B_1}(b_{a_1}(x)) → B_{A_1}(a_{b_1}(b_{a_1}(x)))
B_{B_1}(b_{a_1}(a_{b_1}(b_{b_1}(x)))) → B_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x)))))))
B_{B_1}(b_{a_1}(a_{b_1}(b_{b_1}(x)))) → B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x)))))
B_{B_1}(b_{a_1}(a_{b_1}(b_{b_1}(x)))) → B_{B_1}(b_{b_1}(x))
B_{B_1}(b_{a_1}(a_{b_1}(b_{a_1}(x)))) → B_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x)))))))
B_{B_1}(b_{a_1}(a_{b_1}(b_{a_1}(x)))) → B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x)))))
B_{B_1}(b_{a_1}(a_{b_1}(b_{a_1}(x)))) → B_{B_1}(b_{a_1}(x))
B_{A_1}(a_{b_1}(b_{b_1}(x))) → B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(x))))
B_{A_1}(a_{b_1}(b_{a_1}(x))) → B_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(x))))
B_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x)))))) → B_{B_1}(b_{b_1}(x))
B_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x)))))) → B_{B_1}(b_{a_1}(x))
The TRS R consists of the following rules:
b_{b_1}(b_{b_1}(x)) → b_{a_1}(a_{b_1}(b_{b_1}(x)))
b_{b_1}(b_{a_1}(x)) → b_{a_1}(a_{b_1}(b_{a_1}(x)))
b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x)))) → b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x)))))))
b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x)))) → b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x)))))))
b_{a_1}(a_{b_1}(b_{b_1}(x))) → b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x))))
b_{a_1}(a_{b_1}(b_{a_1}(x))) → b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x))))
b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x)))))) → b_{b_1}(b_{b_1}(x))
b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x)))))) → b_{b_1}(b_{a_1}(x))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(5) QDPOrderProof (EQUIVALENT transformation)
We use the reduction pair processor [LPAR04,JAR06].
The following pairs can be oriented strictly and are deleted.
B_{B_1}(b_{a_1}(a_{b_1}(b_{b_1}(x)))) → B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x)))))
B_{B_1}(b_{a_1}(a_{b_1}(b_{b_1}(x)))) → B_{B_1}(b_{b_1}(x))
B_{B_1}(b_{a_1}(a_{b_1}(b_{a_1}(x)))) → B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x)))))
B_{B_1}(b_{a_1}(a_{b_1}(b_{a_1}(x)))) → B_{B_1}(b_{a_1}(x))
B_{A_1}(a_{b_1}(b_{b_1}(x))) → B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(x))))
B_{A_1}(a_{b_1}(b_{a_1}(x))) → B_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(x))))
The remaining pairs can at least be oriented weakly.
Used ordering: Polynomial Order [NEGPOLO,POLO] with Interpretation:
| POL( B_{B_1}(x1) ) = 2x1 + 2 |
| POL( B_{A_1}(x1) ) = max{0, 2x1 - 2} |
| POL( a_{b_1}(x1) ) = x1 + 2 |
| POL( b_{b_1}(x1) ) = x1 + 2 |
| POL( a_{a_1}(x1) ) = max{0, x1 - 2} |
The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
b_{a_1}(a_{b_1}(b_{b_1}(x))) → b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x))))
b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x)))))) → b_{b_1}(b_{b_1}(x))
b_{b_1}(b_{b_1}(x)) → b_{a_1}(a_{b_1}(b_{b_1}(x)))
b_{a_1}(a_{b_1}(b_{a_1}(x))) → b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x))))
b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x)))))) → b_{b_1}(b_{a_1}(x))
b_{b_1}(b_{a_1}(x)) → b_{a_1}(a_{b_1}(b_{a_1}(x)))
b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x)))) → b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x)))))))
b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x)))) → b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x)))))))
(6) Obligation:
Q DP problem:
The TRS P consists of the following rules:
B_{B_1}(b_{b_1}(x)) → B_{A_1}(a_{b_1}(b_{b_1}(x)))
B_{B_1}(b_{a_1}(x)) → B_{A_1}(a_{b_1}(b_{a_1}(x)))
B_{B_1}(b_{a_1}(a_{b_1}(b_{b_1}(x)))) → B_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x)))))))
B_{B_1}(b_{a_1}(a_{b_1}(b_{a_1}(x)))) → B_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x)))))))
B_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x)))))) → B_{B_1}(b_{b_1}(x))
B_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x)))))) → B_{B_1}(b_{a_1}(x))
The TRS R consists of the following rules:
b_{b_1}(b_{b_1}(x)) → b_{a_1}(a_{b_1}(b_{b_1}(x)))
b_{b_1}(b_{a_1}(x)) → b_{a_1}(a_{b_1}(b_{a_1}(x)))
b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x)))) → b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x)))))))
b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x)))) → b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x)))))))
b_{a_1}(a_{b_1}(b_{b_1}(x))) → b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x))))
b_{a_1}(a_{b_1}(b_{a_1}(x))) → b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x))))
b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x)))))) → b_{b_1}(b_{b_1}(x))
b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x)))))) → b_{b_1}(b_{a_1}(x))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(7) DependencyGraphProof (EQUIVALENT transformation)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 6 less nodes.
(8) TRUE