(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
c(z, x, a) → f(b(b(f(z), z), x))
b(y, b(z, a)) → f(b(c(f(a), y, z), z))
f(c(c(z, a, a), x, a)) → 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:
C(z, x, a) → F(b(b(f(z), z), x))
C(z, x, a) → B(b(f(z), z), x)
C(z, x, a) → B(f(z), z)
C(z, x, a) → F(z)
B(y, b(z, a)) → F(b(c(f(a), y, z), z))
B(y, b(z, a)) → B(c(f(a), y, z), z)
B(y, b(z, a)) → C(f(a), y, z)
B(y, b(z, a)) → F(a)
The TRS R consists of the following rules:
c(z, x, a) → f(b(b(f(z), z), x))
b(y, b(z, a)) → f(b(c(f(a), y, z), z))
f(c(c(z, a, a), x, a)) → 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 4 less nodes.
(4) Obligation:
Q DP problem:
The TRS P consists of the following rules:
C(z, x, a) → B(b(f(z), z), x)
B(y, b(z, a)) → B(c(f(a), y, z), z)
B(y, b(z, a)) → C(f(a), y, z)
C(z, x, a) → B(f(z), z)
The TRS R consists of the following rules:
c(z, x, a) → f(b(b(f(z), z), x))
b(y, b(z, a)) → f(b(c(f(a), y, z), z))
f(c(c(z, a, a), x, a)) → z
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(5) TransformationProof (EQUIVALENT transformation)
By instantiating [LPAR04] the rule
C(
z,
x,
a) →
B(
b(
f(
z),
z),
x) we obtained the following new rules [LPAR04]:
C(f(a), y_0, a) → B(b(f(f(a)), f(a)), y_0) → C(f(a), y_0, a) → B(b(f(f(a)), f(a)), y_0)
(6) Obligation:
Q DP problem:
The TRS P consists of the following rules:
B(y, b(z, a)) → B(c(f(a), y, z), z)
B(y, b(z, a)) → C(f(a), y, z)
C(z, x, a) → B(f(z), z)
C(f(a), y_0, a) → B(b(f(f(a)), f(a)), y_0)
The TRS R consists of the following rules:
c(z, x, a) → f(b(b(f(z), z), x))
b(y, b(z, a)) → f(b(c(f(a), y, z), z))
f(c(c(z, a, a), x, a)) → z
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(7) TransformationProof (EQUIVALENT transformation)
By instantiating [LPAR04] the rule
C(
z,
x,
a) →
B(
f(
z),
z) we obtained the following new rules [LPAR04]:
C(f(a), y_0, a) → B(f(f(a)), f(a)) → C(f(a), y_0, a) → B(f(f(a)), f(a))
(8) Obligation:
Q DP problem:
The TRS P consists of the following rules:
B(y, b(z, a)) → B(c(f(a), y, z), z)
B(y, b(z, a)) → C(f(a), y, z)
C(f(a), y_0, a) → B(b(f(f(a)), f(a)), y_0)
C(f(a), y_0, a) → B(f(f(a)), f(a))
The TRS R consists of the following rules:
c(z, x, a) → f(b(b(f(z), z), x))
b(y, b(z, a)) → f(b(c(f(a), y, z), z))
f(c(c(z, a, a), x, a)) → 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 1 SCC with 1 less node.
(10) Obligation:
Q DP problem:
The TRS P consists of the following rules:
B(y, b(z, a)) → C(f(a), y, z)
C(f(a), y_0, a) → B(b(f(f(a)), f(a)), y_0)
B(y, b(z, a)) → B(c(f(a), y, z), z)
The TRS R consists of the following rules:
c(z, x, a) → f(b(b(f(z), z), x))
b(y, b(z, a)) → f(b(c(f(a), y, z), z))
f(c(c(z, a, a), x, a)) → z
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(11) QDPOrderProof (EQUIVALENT transformation)
We use the reduction pair processor [LPAR04,JAR06].
The following pairs can be oriented strictly and are deleted.
C(f(a), y_0, a) → B(b(f(f(a)), f(a)), y_0)
B(y, b(z, a)) → B(c(f(a), y, z), z)
The remaining pairs can at least be oriented weakly.
Used ordering: Polynomial Order [NEGPOLO,POLO] with Interpretation:
| POL( B(x1, x2) ) = max{0, x1 + x2 - 1} |
| POL( c(x1, ..., x3) ) = 2x1 + x2 + 2 |
| POL( f(x1) ) = max{0, x1 - 2} |
| POL( b(x1, x2) ) = x1 + x2 + 2 |
| POL( C(x1, ..., x3) ) = x2 + x3 + 2 |
The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
c(z, x, a) → f(b(b(f(z), z), x))
b(y, b(z, a)) → f(b(c(f(a), y, z), z))
f(c(c(z, a, a), x, a)) → z
(12) Obligation:
Q DP problem:
The TRS P consists of the following rules:
B(y, b(z, a)) → C(f(a), y, z)
The TRS R consists of the following rules:
c(z, x, a) → f(b(b(f(z), z), x))
b(y, b(z, a)) → f(b(c(f(a), y, z), z))
f(c(c(z, a, a), x, a)) → z
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(13) DependencyGraphProof (EQUIVALENT transformation)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node.
(14) TRUE