(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
c(c(b(x))) → a(c(b(x)))
a(c(b(a(x)))) → b(c(c(x)))
b(a(c(x))) → a(b(c(a(x))))
b(c(a(x))) → c(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:
b(c(c(x))) → b(c(a(x)))
a(b(c(a(x)))) → c(c(b(x)))
c(a(b(x))) → a(c(b(a(x))))
a(c(b(x))) → b(a(c(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(c(c(x))) → B(c(a(x)))
B(c(c(x))) → C(a(x))
B(c(c(x))) → A(x)
A(b(c(a(x)))) → C(c(b(x)))
A(b(c(a(x)))) → C(b(x))
A(b(c(a(x)))) → B(x)
C(a(b(x))) → A(c(b(a(x))))
C(a(b(x))) → C(b(a(x)))
C(a(b(x))) → B(a(x))
C(a(b(x))) → A(x)
A(c(b(x))) → B(a(c(x)))
A(c(b(x))) → A(c(x))
A(c(b(x))) → C(x)
The TRS R consists of the following rules:
b(c(c(x))) → b(c(a(x)))
a(b(c(a(x)))) → c(c(b(x)))
c(a(b(x))) → a(c(b(a(x))))
a(c(b(x))) → b(a(c(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(b(c(a(x)))) → B(x)
C(a(b(x))) → B(a(x))
C(a(b(x))) → A(x)
A(c(b(x))) → B(a(c(x)))
A(c(b(x))) → A(c(x))
A(c(b(x))) → C(x)
The remaining pairs can at least be oriented weakly.
Used ordering: Polynomial interpretation [POLO]:
POL(A(x1)) = x1
POL(B(x1)) = x1
POL(C(x1)) = x1
POL(a(x1)) = x1
POL(b(x1)) = 1 + x1
POL(c(x1)) = x1
The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
a(b(c(a(x)))) → c(c(b(x)))
c(a(b(x))) → a(c(b(a(x))))
a(c(b(x))) → b(a(c(x)))
b(c(c(x))) → b(c(a(x)))
(6) Obligation:
Q DP problem:
The TRS P consists of the following rules:
B(c(c(x))) → B(c(a(x)))
B(c(c(x))) → C(a(x))
B(c(c(x))) → A(x)
A(b(c(a(x)))) → C(c(b(x)))
A(b(c(a(x)))) → C(b(x))
C(a(b(x))) → A(c(b(a(x))))
C(a(b(x))) → C(b(a(x)))
The TRS R consists of the following rules:
b(c(c(x))) → b(c(a(x)))
a(b(c(a(x)))) → c(c(b(x)))
c(a(b(x))) → a(c(b(a(x))))
a(c(b(x))) → b(a(c(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 2 less nodes.
(8) Complex Obligation (AND)
(9) Obligation:
Q DP problem:
The TRS P consists of the following rules:
C(a(b(x))) → A(c(b(a(x))))
A(b(c(a(x)))) → C(c(b(x)))
C(a(b(x))) → C(b(a(x)))
A(b(c(a(x)))) → C(b(x))
The TRS R consists of the following rules:
b(c(c(x))) → b(c(a(x)))
a(b(c(a(x)))) → c(c(b(x)))
c(a(b(x))) → a(c(b(a(x))))
a(c(b(x))) → b(a(c(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.
C(a(b(x))) → A(c(b(a(x))))
A(b(c(a(x)))) → C(c(b(x)))
C(a(b(x))) → C(b(a(x)))
A(b(c(a(x)))) → C(b(x))
The remaining pairs can at least be oriented weakly.
Used ordering: Polynomial interpretation [POLO]:
POL(A(x1)) = 4 + 4·x1
POL(C(x1)) = 2 + 4·x1
POL(a(x1)) = 5 + 4·x1
POL(b(x1)) = 5
POL(c(x1)) = x1
The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
a(b(c(a(x)))) → c(c(b(x)))
c(a(b(x))) → a(c(b(a(x))))
a(c(b(x))) → b(a(c(x)))
b(c(c(x))) → b(c(a(x)))
(11) Obligation:
Q DP problem:
P is empty.
The TRS R consists of the following rules:
b(c(c(x))) → b(c(a(x)))
a(b(c(a(x)))) → c(c(b(x)))
c(a(b(x))) → a(c(b(a(x))))
a(c(b(x))) → b(a(c(x)))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(12) PisEmptyProof (EQUIVALENT transformation)
The TRS P is empty. Hence, there is no (P,Q,R) chain.
(13) YES
(14) Obligation:
Q DP problem:
The TRS P consists of the following rules:
B(c(c(x))) → B(c(a(x)))
The TRS R consists of the following rules:
b(c(c(x))) → b(c(a(x)))
a(b(c(a(x)))) → c(c(b(x)))
c(a(b(x))) → a(c(b(a(x))))
a(c(b(x))) → b(a(c(x)))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(15) QDPOrderProof (EQUIVALENT transformation)
We use the reduction pair processor [LPAR04,JAR06].
The following pairs can be oriented strictly and are deleted.
B(c(c(x))) → B(c(a(x)))
The remaining pairs can at least be oriented weakly.
Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
| POL(c(x1)) = | | + | | / | 0A | 1A | 0A | \ |
| | | 0A | 0A | 1A | | |
| \ | 0A | 0A | 0A | / |
| · | x1 |
| POL(a(x1)) = | | + | | / | 0A | 0A | 1A | \ |
| | | -I | -I | 0A | | |
| \ | -I | 0A | 0A | / |
| · | x1 |
| POL(b(x1)) = | | + | | / | 0A | 0A | 0A | \ |
| | | -I | 0A | 0A | | |
| \ | 0A | 1A | 1A | / |
| · | x1 |
The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
a(b(c(a(x)))) → c(c(b(x)))
c(a(b(x))) → a(c(b(a(x))))
a(c(b(x))) → b(a(c(x)))
b(c(c(x))) → b(c(a(x)))
(16) Obligation:
Q DP problem:
P is empty.
The TRS R consists of the following rules:
b(c(c(x))) → b(c(a(x)))
a(b(c(a(x)))) → c(c(b(x)))
c(a(b(x))) → a(c(b(a(x))))
a(c(b(x))) → b(a(c(x)))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(17) PisEmptyProof (EQUIVALENT transformation)
The TRS P is empty. Hence, there is no (P,Q,R) chain.
(18) YES