(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
b(b(b(a(b(x))))) → b(a(b(b(a(b(x))))))
b(a(b(b(x)))) → b(b(a(b(a(b(x))))))
b(a(b(a(a(b(b(x))))))) → b(a(a(b(a(a(b(b(b(a(b(x)))))))))))
Q is empty.
(1) DependencyPairsProof (EQUIVALENT transformation)
Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
(2) Obligation:
Q DP problem:
The TRS P consists of the following rules:
B(b(b(a(b(x))))) → B(a(b(b(a(b(x))))))
B(a(b(b(x)))) → B(b(a(b(a(b(x))))))
B(a(b(b(x)))) → B(a(b(a(b(x)))))
B(a(b(b(x)))) → B(a(b(x)))
B(a(b(a(a(b(b(x))))))) → B(a(a(b(a(a(b(b(b(a(b(x)))))))))))
B(a(b(a(a(b(b(x))))))) → B(a(a(b(b(b(a(b(x))))))))
B(a(b(a(a(b(b(x))))))) → B(b(b(a(b(x)))))
B(a(b(a(a(b(b(x))))))) → B(b(a(b(x))))
B(a(b(a(a(b(b(x))))))) → B(a(b(x)))
The TRS R consists of the following rules:
b(b(b(a(b(x))))) → b(a(b(b(a(b(x))))))
b(a(b(b(x)))) → b(b(a(b(a(b(x))))))
b(a(b(a(a(b(b(x))))))) → b(a(a(b(a(a(b(b(b(a(b(x)))))))))))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(3) DependencyGraphProof (EQUIVALENT transformation)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 2 less nodes.
(4) Obligation:
Q DP problem:
The TRS P consists of the following rules:
B(a(b(b(x)))) → B(b(a(b(a(b(x))))))
B(b(b(a(b(x))))) → B(a(b(b(a(b(x))))))
B(a(b(b(x)))) → B(a(b(a(b(x)))))
B(a(b(b(x)))) → B(a(b(x)))
B(a(b(a(a(b(b(x))))))) → B(b(b(a(b(x)))))
B(a(b(a(a(b(b(x))))))) → B(b(a(b(x))))
B(a(b(a(a(b(b(x))))))) → B(a(b(x)))
The TRS R consists of the following rules:
b(b(b(a(b(x))))) → b(a(b(b(a(b(x))))))
b(a(b(b(x)))) → b(b(a(b(a(b(x))))))
b(a(b(a(a(b(b(x))))))) → b(a(a(b(a(a(b(b(b(a(b(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(a(b(a(a(b(b(x))))))) → B(b(b(a(b(x)))))
B(a(b(a(a(b(b(x))))))) → B(b(a(b(x))))
B(a(b(a(a(b(b(x))))))) → B(a(b(x)))
The remaining pairs can at least be oriented weakly.
Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
POL(a(x1)) = | | + | / | 0A | -I | 0A | \ |
| | 0A | 0A | 0A | | |
\ | 1A | 0A | 0A | / |
| · | x1 |
POL(b(x1)) = | | + | / | -I | -I | -I | \ |
| | 0A | 0A | 0A | | |
\ | 0A | 0A | -I | / |
| · | x1 |
The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
b(a(b(b(x)))) → b(b(a(b(a(b(x))))))
b(b(b(a(b(x))))) → b(a(b(b(a(b(x))))))
b(a(b(a(a(b(b(x))))))) → b(a(a(b(a(a(b(b(b(a(b(x)))))))))))
(6) Obligation:
Q DP problem:
The TRS P consists of the following rules:
B(a(b(b(x)))) → B(b(a(b(a(b(x))))))
B(b(b(a(b(x))))) → B(a(b(b(a(b(x))))))
B(a(b(b(x)))) → B(a(b(a(b(x)))))
B(a(b(b(x)))) → B(a(b(x)))
The TRS R consists of the following rules:
b(b(b(a(b(x))))) → b(a(b(b(a(b(x))))))
b(a(b(b(x)))) → b(b(a(b(a(b(x))))))
b(a(b(a(a(b(b(x))))))) → b(a(a(b(a(a(b(b(b(a(b(x)))))))))))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(7) QDPOrderProof (EQUIVALENT transformation)
We use the reduction pair processor [LPAR04,JAR06].
The following pairs can be oriented strictly and are deleted.
B(b(b(a(b(x))))) → B(a(b(b(a(b(x))))))
B(a(b(b(x)))) → B(a(b(a(b(x)))))
B(a(b(b(x)))) → B(a(b(x)))
The remaining pairs can at least be oriented weakly.
Used ordering: Polynomial Order [NEGPOLO,POLO] with Interpretation:
POL( B(x1) ) = max{0, 2x1 - 2} |
POL( a(x1) ) = max{0, x1 - 2} |
The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
b(a(b(b(x)))) → b(b(a(b(a(b(x))))))
b(b(b(a(b(x))))) → b(a(b(b(a(b(x))))))
b(a(b(a(a(b(b(x))))))) → b(a(a(b(a(a(b(b(b(a(b(x)))))))))))
(8) Obligation:
Q DP problem:
The TRS P consists of the following rules:
B(a(b(b(x)))) → B(b(a(b(a(b(x))))))
The TRS R consists of the following rules:
b(b(b(a(b(x))))) → b(a(b(b(a(b(x))))))
b(a(b(b(x)))) → b(b(a(b(a(b(x))))))
b(a(b(a(a(b(b(x))))))) → b(a(a(b(a(a(b(b(b(a(b(x)))))))))))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(9) DependencyGraphProof (EQUIVALENT transformation)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node.
(10) TRUE