(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
g(f(x, y), z) → f(x, g(y, z))
g(h(x, y), z) → g(x, f(y, z))
g(x, h(y, z)) → h(g(x, y), 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:
G(f(x, y), z) → G(y, z)
G(h(x, y), z) → G(x, f(y, z))
G(x, h(y, z)) → G(x, y)
The TRS R consists of the following rules:
g(f(x, y), z) → f(x, g(y, z))
g(h(x, y), z) → g(x, f(y, z))
g(x, h(y, z)) → h(g(x, y), z)
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(3) QDPOrderProof (EQUIVALENT transformation)
We use the reduction pair processor [LPAR04].
The following pairs can be oriented strictly and are deleted.
G(h(x, y), z) → G(x, f(y, z))
G(x, h(y, z)) → G(x, y)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
G(
x1,
x2) =
G(
x1,
x2)
f(
x1,
x2) =
x2
h(
x1,
x2) =
h(
x1,
x2)
g(
x1,
x2) =
x2
Recursive Path Order [RPO].
Precedence:
G2 > h2
The following usable rules [FROCOS05] were oriented:
g(f(x, y), z) → f(x, g(y, z))
g(h(x, y), z) → g(x, f(y, z))
g(x, h(y, z)) → h(g(x, y), z)
(4) Obligation:
Q DP problem:
The TRS P consists of the following rules:
G(f(x, y), z) → G(y, z)
The TRS R consists of the following rules:
g(f(x, y), z) → f(x, g(y, z))
g(h(x, y), z) → g(x, f(y, z))
g(x, h(y, z)) → h(g(x, y), 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.
G(f(x, y), z) → G(y, z)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
G(
x1,
x2) =
G(
x1,
x2)
f(
x1,
x2) =
f(
x2)
g(
x1,
x2) =
g(
x1)
h(
x1,
x2) =
x1
Recursive Path Order [RPO].
Precedence:
G2 > f1
g1 > f1
The following usable rules [FROCOS05] were oriented:
g(f(x, y), z) → f(x, g(y, z))
g(h(x, y), z) → g(x, f(y, z))
g(x, h(y, z)) → h(g(x, y), z)
(6) Obligation:
Q DP problem:
P is empty.
The TRS R consists of the following rules:
g(f(x, y), z) → f(x, g(y, z))
g(h(x, y), z) → g(x, f(y, z))
g(x, h(y, z)) → h(g(x, y), z)
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(7) PisEmptyProof (EQUIVALENT transformation)
The TRS P is empty. Hence, there is no (P,Q,R) chain.
(8) TRUE