(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
b(b(x)) → b(a(b(a(a(b(x))))))
b(b(a(b(x)))) → b(a(b(b(x))))
b(a(a(a(b(a(a(b(x)))))))) → b(a(a(a(b(b(x))))))
b(b(a(a(b(x))))) → b(a(a(b(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(x)) → B(a(b(a(a(b(x))))))
B(b(x)) → B(a(a(b(x))))
B(b(a(b(x)))) → B(a(b(b(x))))
B(b(a(b(x)))) → B(b(x))
B(a(a(a(b(a(a(b(x)))))))) → B(a(a(a(b(b(x))))))
B(a(a(a(b(a(a(b(x)))))))) → B(b(x))
B(b(a(a(b(x))))) → B(a(a(b(b(x)))))
B(b(a(a(b(x))))) → B(b(x))
The TRS R consists of the following rules:
b(b(x)) → b(a(b(a(a(b(x))))))
b(b(a(b(x)))) → b(a(b(b(x))))
b(a(a(a(b(a(a(b(x)))))))) → b(a(a(a(b(b(x))))))
b(b(a(a(b(x))))) → b(a(a(b(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 2 SCCs with 5 less nodes.
(4) Complex Obligation (AND)
(5) Obligation:
Q DP problem:
The TRS P consists of the following rules:
B(b(a(a(b(x))))) → B(b(x))
B(b(a(b(x)))) → B(b(x))
The TRS R consists of the following rules:
b(b(x)) → b(a(b(a(a(b(x))))))
b(b(a(b(x)))) → b(a(b(b(x))))
b(a(a(a(b(a(a(b(x)))))))) → b(a(a(a(b(b(x))))))
b(b(a(a(b(x))))) → b(a(a(b(b(x)))))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(6) QDPSizeChangeProof (EQUIVALENT transformation)
By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem.
From the DPs we obtained the following set of size-change graphs:
- B(b(a(a(b(x))))) → B(b(x))
The graph contains the following edges 1 > 1
- B(b(a(b(x)))) → B(b(x))
The graph contains the following edges 1 > 1
(7) YES
(8) Obligation:
Q DP problem:
The TRS P consists of the following rules:
B(a(a(a(b(a(a(b(x)))))))) → B(a(a(a(b(b(x))))))
The TRS R consists of the following rules:
b(b(x)) → b(a(b(a(a(b(x))))))
b(b(a(b(x)))) → b(a(b(b(x))))
b(a(a(a(b(a(a(b(x)))))))) → b(a(a(a(b(b(x))))))
b(b(a(a(b(x))))) → b(a(a(b(b(x)))))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(9) QDPOrderProof (EQUIVALENT transformation)
We use the reduction pair processor [LPAR04,JAR06].
The following pairs can be oriented strictly and are deleted.
B(a(a(a(b(a(a(b(x)))))))) → B(a(a(a(b(b(x))))))
The remaining pairs can at least be oriented weakly.
Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
| POL(a(x1)) = | | + | | / | -I | -I | -I | \ |
| | | -I | 0A | 0A | | |
| \ | 0A | -I | -I | / |
| · | x1 |
| POL(b(x1)) = | | + | | / | 0A | 1A | -I | \ |
| | | -I | -I | -I | | |
| \ | -I | -I | -I | / |
| · | x1 |
The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
b(b(x)) → b(a(b(a(a(b(x))))))
b(b(a(b(x)))) → b(a(b(b(x))))
b(a(a(a(b(a(a(b(x)))))))) → b(a(a(a(b(b(x))))))
b(b(a(a(b(x))))) → b(a(a(b(b(x)))))
(10) Obligation:
Q DP problem:
P is empty.
The TRS R consists of the following rules:
b(b(x)) → b(a(b(a(a(b(x))))))
b(b(a(b(x)))) → b(a(b(b(x))))
b(a(a(a(b(a(a(b(x)))))))) → b(a(a(a(b(b(x))))))
b(b(a(a(b(x))))) → b(a(a(b(b(x)))))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(11) PisEmptyProof (EQUIVALENT transformation)
The TRS P is empty. Hence, there is no (P,Q,R) chain.
(12) YES