(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
b(b(y, z), c(a, a, a)) → f(c(z, y, z))
f(b(b(a, z), c(a, x, y))) → z
c(y, x, f(z)) → b(f(b(z, x)), 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(b(y, z), c(a, a, a)) → F(c(z, y, z))
B(b(y, z), c(a, a, a)) → C(z, y, z)
C(y, x, f(z)) → B(f(b(z, x)), z)
C(y, x, f(z)) → F(b(z, x))
C(y, x, f(z)) → B(z, x)
The TRS R consists of the following rules:
b(b(y, z), c(a, a, a)) → f(c(z, y, z))
f(b(b(a, z), c(a, x, y))) → z
c(y, x, f(z)) → b(f(b(z, x)), 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(b(y, z), c(a, a, a)) → C(z, y, z)
C(y, x, f(z)) → B(f(b(z, x)), z)
C(y, x, f(z)) → B(z, x)
The TRS R consists of the following rules:
b(b(y, z), c(a, a, a)) → f(c(z, y, z))
f(b(b(a, z), c(a, x, y))) → z
c(y, x, f(z)) → b(f(b(z, x)), 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.
C(y, x, f(z)) → B(z, x)
The remaining pairs can at least be oriented weakly.
Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
| POL(B(x1, x2)) = | 4A | + | 1A | · | x1 | + | 0A | · | x2 |
| POL(b(x1, x2)) = | -I | + | 2A | · | x1 | + | 2A | · | x2 |
| POL(c(x1, x2, x3)) = | -I | + | 0A | · | x1 | + | 4A | · | x2 | + | 4A | · | x3 |
| POL(C(x1, x2, x3)) = | 5A | + | -I | · | x1 | + | 3A | · | x2 | + | 3A | · | x3 |
The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
b(b(y, z), c(a, a, a)) → f(c(z, y, z))
f(b(b(a, z), c(a, x, y))) → z
c(y, x, f(z)) → b(f(b(z, x)), z)
(6) Obligation:
Q DP problem:
The TRS P consists of the following rules:
B(b(y, z), c(a, a, a)) → C(z, y, z)
C(y, x, f(z)) → B(f(b(z, x)), z)
The TRS R consists of the following rules:
b(b(y, z), c(a, a, a)) → f(c(z, y, z))
f(b(b(a, z), c(a, x, y))) → z
c(y, x, f(z)) → b(f(b(z, x)), z)
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.
B(b(y, z), c(a, a, a)) → C(z, y, z)
The remaining pairs can at least be oriented weakly.
Used ordering: Matrix interpretation [MATRO] to (N^2, +, *, >=, >) :
| POL(B(x1, x2)) = | 2 | + | | · | x1 | + | | · | x2 |
| POL(b(x1, x2)) = | | + | | · | x1 | + | | · | x2 |
| POL(c(x1, x2, x3)) = | | + | | · | x1 | + | | · | x2 | + | | · | x3 |
| POL(C(x1, x2, x3)) = | 2 | + | | · | x1 | + | | · | x2 | + | | · | x3 |
The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
b(b(y, z), c(a, a, a)) → f(c(z, y, z))
f(b(b(a, z), c(a, x, y))) → z
c(y, x, f(z)) → b(f(b(z, x)), z)
(8) Obligation:
Q DP problem:
The TRS P consists of the following rules:
C(y, x, f(z)) → B(f(b(z, x)), z)
The TRS R consists of the following rules:
b(b(y, z), c(a, a, a)) → f(c(z, y, z))
f(b(b(a, z), c(a, x, y))) → z
c(y, x, f(z)) → b(f(b(z, x)), z)
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 0 SCCs with 1 less node.
(10) TRUE