(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,JAR06].
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
F(
x1) =
x1
f(
x1) =
x1
c(
x1,
x2,
x3) =
c(
x1,
x2)
a =
a
Knuth-Bendix order [KBO] with precedence:
trivial
and weight map:
a=1
c_2=1
The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] 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.
(9) TransformationProof (EQUIVALENT transformation)
By instantiating [LPAR04] the rule
B(
y,
b(
a,
z)) →
B(
f(
c(
y,
y,
a)),
b(
f(
z),
a)) we obtained the following new rules [LPAR04]:
B(f(c(y_0, y_1, a)), b(a, a)) → B(f(c(f(c(y_0, y_1, a)), f(c(y_0, y_1, a)), a)), b(f(a), a)) → B(f(c(y_0, y_1, a)), b(a, a)) → B(f(c(f(c(y_0, y_1, a)), f(c(y_0, y_1, a)), a)), b(f(a), a))
(10) Obligation:
Q DP problem:
The TRS P consists of the following rules:
B(f(c(y_0, y_1, a)), b(a, a)) → B(f(c(f(c(y_0, y_1, a)), f(c(y_0, y_1, a)), a)), b(f(a), 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.
(11) DependencyGraphProof (EQUIVALENT transformation)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node.
(12) TRUE