(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
f(x, f(a, y)) → f(a, f(f(a, h(f(a, x))), y))
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(x, f(a, y)) → F(a, f(f(a, h(f(a, x))), y))
F(x, f(a, y)) → F(f(a, h(f(a, x))), y)
F(x, f(a, y)) → F(a, h(f(a, x)))
F(x, f(a, y)) → F(a, x)
The TRS R consists of the following rules:
f(x, f(a, y)) → f(a, f(f(a, h(f(a, x))), y))
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:
F(x, f(a, y)) → F(f(a, h(f(a, x))), y)
F(x, f(a, y)) → F(a, f(f(a, h(f(a, x))), y))
F(x, f(a, y)) → F(a, x)
The TRS R consists of the following rules:
f(x, f(a, y)) → f(a, f(f(a, h(f(a, x))), y))
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(x, f(a, y)) → F(a, x)
The remaining pairs can at least be oriented weakly.
Used ordering: Polynomial Order [NEGPOLO,POLO] with Interpretation:
| POL( F(x1, x2) ) = x1 + x2 + 1 |
| POL( f(x1, x2) ) = 2x2 + 1 |
| POL( h(x1) ) = max{0, -2} |
The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
f(x, f(a, y)) → f(a, f(f(a, h(f(a, x))), y))
(6) Obligation:
Q DP problem:
The TRS P consists of the following rules:
F(x, f(a, y)) → F(f(a, h(f(a, x))), y)
F(x, f(a, y)) → F(a, f(f(a, h(f(a, x))), y))
The TRS R consists of the following rules:
f(x, f(a, y)) → f(a, f(f(a, h(f(a, x))), y))
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.
F(x, f(a, y)) → F(f(a, h(f(a, x))), y)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
F(
x1,
x2) =
x2
f(
x1,
x2) =
f(
x2)
Knuth-Bendix order [KBO] with precedence:
trivial
and weight map:
dummyConstant=1
f_1=1
The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
f(x, f(a, y)) → f(a, f(f(a, h(f(a, x))), y))
(8) Obligation:
Q DP problem:
The TRS P consists of the following rules:
F(x, f(a, y)) → F(a, f(f(a, h(f(a, x))), y))
The TRS R consists of the following rules:
f(x, f(a, y)) → f(a, f(f(a, h(f(a, x))), y))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(9) TransformationProof (EQUIVALENT transformation)
By instantiating [LPAR04] the rule
F(
x,
f(
a,
y)) →
F(
a,
f(
f(
a,
h(
f(
a,
x))),
y)) we obtained the following new rules [LPAR04]:
F(a, f(a, x1)) → F(a, f(f(a, h(f(a, a))), x1)) → F(a, f(a, x1)) → F(a, f(f(a, h(f(a, a))), x1))
(10) Obligation:
Q DP problem:
The TRS P consists of the following rules:
F(a, f(a, x1)) → F(a, f(f(a, h(f(a, a))), x1))
The TRS R consists of the following rules:
f(x, f(a, y)) → f(a, f(f(a, h(f(a, x))), y))
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.
F(a, f(a, x1)) → F(a, f(f(a, h(f(a, a))), x1))
The remaining pairs can at least be oriented weakly.
Used ordering: Polynomial Order [NEGPOLO,POLO] with Interpretation:
| POL( F(x1, x2) ) = x2 + 2 |
| POL( f(x1, x2) ) = max{0, x1 + 2x2 - 1} |
The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
f(x, f(a, y)) → f(a, f(f(a, h(f(a, x))), y))
(12) Obligation:
Q DP problem:
P is empty.
The TRS R consists of the following rules:
f(x, f(a, y)) → f(a, f(f(a, h(f(a, x))), y))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(13) PisEmptyProof (EQUIVALENT transformation)
The TRS P is empty. Hence, there is no (P,Q,R) chain.
(14) YES