(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
a(a(a(x))) → b(b(a(x)))
a(b(a(x))) → b(b(a(x)))
b(a(b(x))) → a(a(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(a(a(x))) → a(b(b(x)))
a(b(a(x))) → a(b(b(x)))
b(a(b(x))) → b(a(a(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(a(a(x))) → A(b(b(x)))
A(a(a(x))) → B(b(x))
A(a(a(x))) → B(x)
A(b(a(x))) → A(b(b(x)))
A(b(a(x))) → B(b(x))
A(b(a(x))) → B(x)
B(a(b(x))) → B(a(a(x)))
B(a(b(x))) → A(a(x))
B(a(b(x))) → A(x)
The TRS R consists of the following rules:
a(a(a(x))) → a(b(b(x)))
a(b(a(x))) → a(b(b(x)))
b(a(b(x))) → b(a(a(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.
A(a(a(x))) → B(b(x))
A(a(a(x))) → B(x)
A(b(a(x))) → B(b(x))
A(b(a(x))) → B(x)
B(a(b(x))) → A(x)
The remaining pairs can at least be oriented weakly.
Used ordering: Polynomial interpretation [POLO]:
POL(A(x1)) = 1 + x1
POL(B(x1)) = x1
POL(a(x1)) = 1 + x1
POL(b(x1)) = 1 + x1
The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
b(a(b(x))) → b(a(a(x)))
a(b(a(x))) → a(b(b(x)))
a(a(a(x))) → a(b(b(x)))
(6) Obligation:
Q DP problem:
The TRS P consists of the following rules:
A(a(a(x))) → A(b(b(x)))
A(b(a(x))) → A(b(b(x)))
B(a(b(x))) → B(a(a(x)))
B(a(b(x))) → A(a(x))
The TRS R consists of the following rules:
a(a(a(x))) → a(b(b(x)))
a(b(a(x))) → a(b(b(x)))
b(a(b(x))) → b(a(a(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 2 SCCs with 1 less node.
(8) Complex Obligation (AND)
(9) Obligation:
Q DP problem:
The TRS P consists of the following rules:
A(b(a(x))) → A(b(b(x)))
A(a(a(x))) → A(b(b(x)))
The TRS R consists of the following rules:
a(a(a(x))) → a(b(b(x)))
a(b(a(x))) → a(b(b(x)))
b(a(b(x))) → b(a(a(x)))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(10) QDPOrderProof (EQUIVALENT transformation)
We use the reduction pair processor [LPAR04,JAR06].
The following pairs can be oriented strictly and are deleted.
A(a(a(x))) → A(b(b(x)))
The remaining pairs can at least be oriented weakly.
Used ordering: Polynomial interpretation [POLO]:
POL(A(x1)) = x1
POL(a(x1)) = 1 + x1
POL(b(x1)) = 1
The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
b(a(b(x))) → b(a(a(x)))
(11) Obligation:
Q DP problem:
The TRS P consists of the following rules:
A(b(a(x))) → A(b(b(x)))
The TRS R consists of the following rules:
a(a(a(x))) → a(b(b(x)))
a(b(a(x))) → a(b(b(x)))
b(a(b(x))) → b(a(a(x)))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(12) QDPOrderProof (EQUIVALENT transformation)
We use the reduction pair processor [LPAR04,JAR06].
The following pairs can be oriented strictly and are deleted.
A(b(a(x))) → A(b(b(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 | -I | \ |
| | | -I | -I | -I | | |
| \ | 0A | 0A | 0A | / |
| · | x1 |
| POL(a(x1)) = | | + | | / | 0A | -I | -I | \ |
| | | 0A | 0A | 0A | | |
| \ | 0A | -I | -I | / |
| · | x1 |
The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
b(a(b(x))) → b(a(a(x)))
a(b(a(x))) → a(b(b(x)))
a(a(a(x))) → a(b(b(x)))
(13) Obligation:
Q DP problem:
P is empty.
The TRS R consists of the following rules:
a(a(a(x))) → a(b(b(x)))
a(b(a(x))) → a(b(b(x)))
b(a(b(x))) → b(a(a(x)))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(14) PisEmptyProof (EQUIVALENT transformation)
The TRS P is empty. Hence, there is no (P,Q,R) chain.
(15) YES
(16) Obligation:
Q DP problem:
The TRS P consists of the following rules:
B(a(b(x))) → B(a(a(x)))
The TRS R consists of the following rules:
a(a(a(x))) → a(b(b(x)))
a(b(a(x))) → a(b(b(x)))
b(a(b(x))) → b(a(a(x)))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(17) QDPOrderProof (EQUIVALENT transformation)
We use the reduction pair processor [LPAR04,JAR06].
The following pairs can be oriented strictly and are deleted.
B(a(b(x))) → B(a(a(x)))
The remaining pairs can at least be oriented weakly.
Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
| POL(a(x1)) = | | + | | / | 0A | 0A | 0A | \ |
| | | -I | -I | 0A | | |
| \ | -I | 1A | -I | / |
| · | x1 |
| POL(b(x1)) = | | + | | / | -I | -I | 0A | \ |
| | | -I | -I | 1A | | |
| \ | -I | -I | -I | / |
| · | x1 |
The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
a(b(a(x))) → a(b(b(x)))
a(a(a(x))) → a(b(b(x)))
b(a(b(x))) → b(a(a(x)))
(18) Obligation:
Q DP problem:
P is empty.
The TRS R consists of the following rules:
a(a(a(x))) → a(b(b(x)))
a(b(a(x))) → a(b(b(x)))
b(a(b(x))) → b(a(a(x)))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(19) PisEmptyProof (EQUIVALENT transformation)
The TRS P is empty. Hence, there is no (P,Q,R) chain.
(20) YES