(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
a(b(b(a(x)))) → a(c(a(b(x))))
a(c(x)) → c(c(a(x)))
c(c(c(x))) → b(c(b(x)))
Q is empty.
(1) QTRS Reverse (EQUIVALENT transformation)
We applied the QTRS Reverse Processor [REVERSE].
(2) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
a(b(b(a(x)))) → b(a(c(a(x))))
c(a(x)) → a(c(c(x)))
c(c(c(x))) → b(c(b(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:
A(b(b(a(x)))) → A(c(a(x)))
A(b(b(a(x)))) → C(a(x))
C(a(x)) → A(c(c(x)))
C(a(x)) → C(c(x))
C(a(x)) → C(x)
C(c(c(x))) → C(b(x))
The TRS R consists of the following rules:
a(b(b(a(x)))) → b(a(c(a(x))))
c(a(x)) → a(c(c(x)))
c(c(c(x))) → b(c(b(x)))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(5) DependencyGraphProof (EQUIVALENT transformation)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.
(6) Obligation:
Q DP problem:
The TRS P consists of the following rules:
A(b(b(a(x)))) → C(a(x))
C(a(x)) → A(c(c(x)))
A(b(b(a(x)))) → A(c(a(x)))
C(a(x)) → C(c(x))
C(a(x)) → C(x)
The TRS R consists of the following rules:
a(b(b(a(x)))) → b(a(c(a(x))))
c(a(x)) → a(c(c(x)))
c(c(c(x))) → b(c(b(x)))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(7) MRRProof (EQUIVALENT transformation)
By using the rule removal processor [LPAR04] with the following ordering, at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented.
Strictly oriented dependency pairs:
A(b(b(a(x)))) → C(a(x))
C(a(x)) → C(c(x))
C(a(x)) → C(x)
Used ordering: Polynomial interpretation [POLO]:
POL(A(x1)) = 3 + 3·x1
POL(C(x1)) = 3·x1
POL(a(x1)) = 1 + x1
POL(b(x1)) = x1
POL(c(x1)) = x1
(8) Obligation:
Q DP problem:
The TRS P consists of the following rules:
C(a(x)) → A(c(c(x)))
A(b(b(a(x)))) → A(c(a(x)))
The TRS R consists of the following rules:
a(b(b(a(x)))) → b(a(c(a(x))))
c(a(x)) → a(c(c(x)))
c(c(c(x))) → b(c(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 1 SCC with 1 less node.
(10) Obligation:
Q DP problem:
The TRS P consists of the following rules:
A(b(b(a(x)))) → A(c(a(x)))
The TRS R consists of the following rules:
a(b(b(a(x)))) → b(a(c(a(x))))
c(a(x)) → a(c(c(x)))
c(c(c(x))) → b(c(b(x)))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(11) QDPOrderProof (EQUIVALENT transformation)
We use the reduction pair processor [LPAR04,JAR06].
The following pairs can be oriented strictly and are deleted.
A(b(b(a(x)))) → A(c(a(x)))
The remaining pairs can at least be oriented weakly.
Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
| POL(b(x1)) = | | + | | / | -I | 0A | 0A | \ |
| | | -I | -I | -I | | |
| \ | -I | 0A | 1A | / |
| · | x1 |
| POL(a(x1)) = | | + | | / | 0A | 0A | 0A | \ |
| | | 0A | 0A | 0A | | |
| \ | 0A | 0A | 0A | / |
| · | x1 |
| POL(c(x1)) = | | + | | / | 0A | 0A | 0A | \ |
| | | 0A | 0A | -I | | |
| \ | -I | 0A | -I | / |
| · | x1 |
The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
a(b(b(a(x)))) → b(a(c(a(x))))
c(a(x)) → a(c(c(x)))
c(c(c(x))) → b(c(b(x)))
(12) Obligation:
Q DP problem:
P is empty.
The TRS R consists of the following rules:
a(b(b(a(x)))) → b(a(c(a(x))))
c(a(x)) → a(c(c(x)))
c(c(c(x))) → b(c(b(x)))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(13) PisEmptyProof (EQUIVALENT transformation)
The TRS P is empty. Hence, there is no (P,Q,R) chain.
(14) YES