(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
b(f(b(x, z)), y) → f(f(f(b(z, b(y, z)))))
c(f(f(c(x, a, z))), a, y) → b(y, f(b(a, z)))
b(b(c(b(a, a), a, z), f(a)), y) → z
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(f(b(x, z)), y) → B(z, b(y, z))
B(f(b(x, z)), y) → B(y, z)
C(f(f(c(x, a, z))), a, y) → B(y, f(b(a, z)))
C(f(f(c(x, a, z))), a, y) → B(a, z)
The TRS R consists of the following rules:
b(f(b(x, z)), y) → f(f(f(b(z, b(y, z)))))
c(f(f(c(x, a, z))), a, y) → b(y, f(b(a, z)))
b(b(c(b(a, a), a, z), f(a)), y) → z
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 2 less nodes.
(4) Obligation:
Q DP problem:
The TRS P consists of the following rules:
B(f(b(x, z)), y) → B(y, z)
B(f(b(x, z)), y) → B(z, b(y, z))
The TRS R consists of the following rules:
b(f(b(x, z)), y) → f(f(f(b(z, b(y, z)))))
c(f(f(c(x, a, z))), a, y) → b(y, f(b(a, z)))
b(b(c(b(a, a), a, z), f(a)), y) → z
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.
B(f(b(x, z)), y) → B(y, z)
B(f(b(x, z)), y) → B(z, b(y, z))
The remaining pairs can at least be oriented weakly.
Used ordering: Matrix interpretation [MATRO] to (N^2, +, *, >=, >) :
POL(B(x1, x2)) = | 0 | + | | · | x1 | + | | · | x2 |
POL(b(x1, x2)) = | | + | | · | x1 | + | | · | x2 |
POL(c(x1, x2, x3)) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 |
The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
b(f(b(x, z)), y) → f(f(f(b(z, b(y, z)))))
b(b(c(b(a, a), a, z), f(a)), y) → z
(6) Obligation:
Q DP problem:
P is empty.
The TRS R consists of the following rules:
b(f(b(x, z)), y) → f(f(f(b(z, b(y, z)))))
c(f(f(c(x, a, z))), a, y) → b(y, f(b(a, z)))
b(b(c(b(a, a), a, z), f(a)), y) → z
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(7) PisEmptyProof (EQUIVALENT transformation)
The TRS P is empty. Hence, there is no (P,Q,R) chain.
(8) YES