(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
f(b(a, z)) → z
b(y, b(a, z)) → b(f(c(y, y, a)), b(f(z), a))
f(f(f(c(z, x, a)))) → b(f(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(y, b(a, z)) → B(f(c(y, y, a)), b(f(z), a))
B(y, b(a, z)) → F(c(y, y, a))
B(y, b(a, z)) → B(f(z), a)
B(y, b(a, z)) → F(z)
F(f(f(c(z, x, a)))) → B(f(x), z)
F(f(f(c(z, x, a)))) → F(x)
The TRS R consists of the following rules:
f(b(a, z)) → z
b(y, b(a, z)) → b(f(c(y, y, a)), b(f(z), a))
f(f(f(c(z, x, a)))) → b(f(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(y, b(a, z)) → F(z)
F(f(f(c(z, x, a)))) → B(f(x), z)
B(y, b(a, z)) → B(f(c(y, y, a)), b(f(z), a))
F(f(f(c(z, x, a)))) → F(x)
The TRS R consists of the following rules:
f(b(a, z)) → z
b(y, b(a, z)) → b(f(c(y, y, a)), b(f(z), a))
f(f(f(c(z, x, a)))) → b(f(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].
The following pairs can be oriented strictly and are deleted.
F(f(f(c(z, x, a)))) → B(f(x), z)
F(f(f(c(z, x, a)))) → F(x)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
B(
x1,
x2) =
x2
b(
x1,
x2) =
x2
a =
a
F(
x1) =
x1
f(
x1) =
f(
x1)
c(
x1,
x2,
x3) =
c(
x1,
x2)
Lexicographic Path Order [LPO].
Precedence:
[a, f1, c2]
The following usable rules [FROCOS05] were oriented:
none
(6) Obligation:
Q DP problem:
The TRS P consists of the following rules:
B(y, b(a, z)) → F(z)
B(y, b(a, z)) → B(f(c(y, y, a)), b(f(z), a))
The TRS R consists of the following rules:
f(b(a, z)) → z
b(y, b(a, z)) → b(f(c(y, y, a)), b(f(z), a))
f(f(f(c(z, x, a)))) → b(f(x), z)
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 1 SCC with 1 less node.
(8) Obligation:
Q DP problem:
The TRS P consists of the following rules:
B(y, b(a, z)) → B(f(c(y, y, a)), b(f(z), a))
The TRS R consists of the following rules:
f(b(a, z)) → z
b(y, b(a, z)) → b(f(c(y, y, a)), b(f(z), a))
f(f(f(c(z, x, a)))) → b(f(x), z)
Q is empty.
We have to consider all minimal (P,Q,R)-chains.