(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
f(c(a, z, x)) → b(a, z)
b(x, b(z, y)) → f(b(f(f(z)), c(x, z, y)))
b(y, z) → 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:
F(c(a, z, x)) → B(a, z)
B(x, b(z, y)) → F(b(f(f(z)), c(x, z, y)))
B(x, b(z, y)) → B(f(f(z)), c(x, z, y))
B(x, b(z, y)) → F(f(z))
B(x, b(z, y)) → F(z)
The TRS R consists of the following rules:
f(c(a, z, x)) → b(a, z)
b(x, b(z, y)) → f(b(f(f(z)), c(x, z, y)))
b(y, z) → 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 1 less node.
(4) Obligation:
Q DP problem:
The TRS P consists of the following rules:
B(x, b(z, y)) → F(b(f(f(z)), c(x, z, y)))
F(c(a, z, x)) → B(a, z)
B(x, b(z, y)) → F(f(z))
B(x, b(z, y)) → F(z)
The TRS R consists of the following rules:
f(c(a, z, x)) → b(a, z)
b(x, b(z, y)) → f(b(f(f(z)), c(x, z, y)))
b(y, z) → z
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(5) TransformationProof (EQUIVALENT transformation)
By instantiating [LPAR04] the rule
B(
x,
b(
z,
y)) →
F(
b(
f(
f(
z)),
c(
x,
z,
y))) we obtained the following new rules [LPAR04]:
B(a, b(x1, x2)) → F(b(f(f(x1)), c(a, x1, x2))) → B(a, b(x1, x2)) → F(b(f(f(x1)), c(a, x1, x2)))
(6) Obligation:
Q DP problem:
The TRS P consists of the following rules:
F(c(a, z, x)) → B(a, z)
B(x, b(z, y)) → F(f(z))
B(x, b(z, y)) → F(z)
B(a, b(x1, x2)) → F(b(f(f(x1)), c(a, x1, x2)))
The TRS R consists of the following rules:
f(c(a, z, x)) → b(a, z)
b(x, b(z, y)) → f(b(f(f(z)), c(x, z, y)))
b(y, z) → z
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(7) TransformationProof (EQUIVALENT transformation)
By instantiating [LPAR04] the rule
B(
x,
b(
z,
y)) →
F(
f(
z)) we obtained the following new rules [LPAR04]:
B(a, b(x1, x2)) → F(f(x1)) → B(a, b(x1, x2)) → F(f(x1))
(8) Obligation:
Q DP problem:
The TRS P consists of the following rules:
F(c(a, z, x)) → B(a, z)
B(x, b(z, y)) → F(z)
B(a, b(x1, x2)) → F(b(f(f(x1)), c(a, x1, x2)))
B(a, b(x1, x2)) → F(f(x1))
The TRS R consists of the following rules:
f(c(a, z, x)) → b(a, z)
b(x, b(z, y)) → f(b(f(f(z)), c(x, z, y)))
b(y, z) → z
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(9) TransformationProof (EQUIVALENT transformation)
By instantiating [LPAR04] the rule
B(
x,
b(
z,
y)) →
F(
z) we obtained the following new rules [LPAR04]:
B(a, b(x1, x2)) → F(x1) → B(a, b(x1, x2)) → F(x1)
(10) Obligation:
Q DP problem:
The TRS P consists of the following rules:
F(c(a, z, x)) → B(a, z)
B(a, b(x1, x2)) → F(b(f(f(x1)), c(a, x1, x2)))
B(a, b(x1, x2)) → F(f(x1))
B(a, b(x1, x2)) → F(x1)
The TRS R consists of the following rules:
f(c(a, z, x)) → b(a, z)
b(x, b(z, y)) → f(b(f(f(z)), c(x, z, y)))
b(y, z) → 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.
B(a, b(x1, x2)) → F(f(x1))
B(a, b(x1, x2)) → F(x1)
The remaining pairs can at least be oriented weakly.
Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
| POL(c(x1, x2, x3)) = | -I | + | -I | · | x1 | + | 1A | · | x2 | + | 1A | · | x3 |
| POL(B(x1, x2)) = | 0A | + | -I | · | x1 | + | 1A | · | x2 |
| POL(b(x1, x2)) = | 4A | + | 1A | · | x1 | + | 1A | · | x2 |
The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
b(x, b(z, y)) → f(b(f(f(z)), c(x, z, y)))
f(c(a, z, x)) → b(a, z)
b(y, z) → z
(12) Obligation:
Q DP problem:
The TRS P consists of the following rules:
F(c(a, z, x)) → B(a, z)
B(a, b(x1, x2)) → F(b(f(f(x1)), c(a, x1, x2)))
The TRS R consists of the following rules:
f(c(a, z, x)) → b(a, z)
b(x, b(z, y)) → f(b(f(f(z)), c(x, z, y)))
b(y, z) → z
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(13) QDPOrderProof (EQUIVALENT transformation)
We use the reduction pair processor [LPAR04,JAR06].
The following pairs can be oriented strictly and are deleted.
F(c(a, z, x)) → B(a, z)
The remaining pairs can at least be oriented weakly.
Used ordering: Polynomial interpretation [POLO,RATPOLO]:
POL(B(x1, x2)) = [1] + [4]x1 + [4]x2
POL(F(x1)) = [1] + x1
POL(a) = 0
POL(b(x1, x2)) = [2] + [2]x1 + x2
POL(c(x1, x2, x3)) = [1] + [4]x2
POL(f(x1)) = [2] + [1/4]x1
The value of delta used in the strict ordering is 1.
The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
b(x, b(z, y)) → f(b(f(f(z)), c(x, z, y)))
f(c(a, z, x)) → b(a, z)
b(y, z) → z
(14) Obligation:
Q DP problem:
The TRS P consists of the following rules:
B(a, b(x1, x2)) → F(b(f(f(x1)), c(a, x1, x2)))
The TRS R consists of the following rules:
f(c(a, z, x)) → b(a, z)
b(x, b(z, y)) → f(b(f(f(z)), c(x, z, y)))
b(y, z) → z
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(15) DependencyGraphProof (EQUIVALENT transformation)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node.
(16) TRUE