(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
c(c(c(a(x, y)))) → b(c(c(c(c(y)))), x)
c(c(b(c(y), 0))) → a(0, c(c(a(y, 0))))
c(c(a(a(y, 0), x))) → c(y)
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:
C(c(c(a(x, y)))) → C(c(c(c(y))))
C(c(c(a(x, y)))) → C(c(c(y)))
C(c(c(a(x, y)))) → C(c(y))
C(c(c(a(x, y)))) → C(y)
C(c(b(c(y), 0))) → C(c(a(y, 0)))
C(c(b(c(y), 0))) → C(a(y, 0))
C(c(a(a(y, 0), x))) → C(y)
The TRS R consists of the following rules:
c(c(c(a(x, y)))) → b(c(c(c(c(y)))), x)
c(c(b(c(y), 0))) → a(0, c(c(a(y, 0))))
c(c(a(a(y, 0), x))) → c(y)
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 1 less node.
(4) Obligation:
Q DP problem:
The TRS P consists of the following rules:
C(c(c(a(x, y)))) → C(c(c(y)))
C(c(c(a(x, y)))) → C(c(c(c(y))))
C(c(c(a(x, y)))) → C(c(y))
C(c(c(a(x, y)))) → C(y)
C(c(b(c(y), 0))) → C(c(a(y, 0)))
C(c(a(a(y, 0), x))) → C(y)
The TRS R consists of the following rules:
c(c(c(a(x, y)))) → b(c(c(c(c(y)))), x)
c(c(b(c(y), 0))) → a(0, c(c(a(y, 0))))
c(c(a(a(y, 0), x))) → c(y)
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.
C(c(c(a(x, y)))) → C(c(c(y)))
C(c(c(a(x, y)))) → C(c(y))
C(c(c(a(x, y)))) → C(y)
C(c(b(c(y), 0))) → C(c(a(y, 0)))
C(c(a(a(y, 0), x))) → C(y)
The remaining pairs can at least be oriented weakly.
Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
| POL(a(x1, x2)) = | 1A | + | 0A | · | x1 | + | 1A | · | x2 |
| POL(b(x1, x2)) = | 2A | + | 0A | · | x1 | + | 0A | · | x2 |
The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
c(c(c(a(x, y)))) → b(c(c(c(c(y)))), x)
c(c(b(c(y), 0))) → a(0, c(c(a(y, 0))))
c(c(a(a(y, 0), x))) → c(y)
(6) Obligation:
Q DP problem:
The TRS P consists of the following rules:
C(c(c(a(x, y)))) → C(c(c(c(y))))
The TRS R consists of the following rules:
c(c(c(a(x, y)))) → b(c(c(c(c(y)))), x)
c(c(b(c(y), 0))) → a(0, c(c(a(y, 0))))
c(c(a(a(y, 0), x))) → c(y)
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.
C(c(c(a(x, y)))) → C(c(c(c(y))))
The remaining pairs can at least be oriented weakly.
Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
| POL(a(x1, x2)) = | 3A | + | 0A | · | x1 | + | 2A | · | x2 |
| POL(b(x1, x2)) = | 5A | + | 1A | · | x1 | + | 3A | · | x2 |
The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
c(c(c(a(x, y)))) → b(c(c(c(c(y)))), x)
c(c(b(c(y), 0))) → a(0, c(c(a(y, 0))))
c(c(a(a(y, 0), x))) → c(y)
(8) Obligation:
Q DP problem:
P is empty.
The TRS R consists of the following rules:
c(c(c(a(x, y)))) → b(c(c(c(c(y)))), x)
c(c(b(c(y), 0))) → a(0, c(c(a(y, 0))))
c(c(a(a(y, 0), x))) → c(y)
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(9) PisEmptyProof (EQUIVALENT transformation)
The TRS P is empty. Hence, there is no (P,Q,R) chain.
(10) YES