(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
f(0, 1, x) → f(g(x), g(x), x)
f(g(x), y, z) → g(f(x, y, z))
f(x, g(y), z) → g(f(x, y, z))
f(x, y, g(z)) → g(f(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:
F(0, 1, x) → F(g(x), g(x), x)
F(g(x), y, z) → F(x, y, z)
F(x, g(y), z) → F(x, y, z)
F(x, y, g(z)) → F(x, y, z)
The TRS R consists of the following rules:
f(0, 1, x) → f(g(x), g(x), x)
f(g(x), y, z) → g(f(x, y, z))
f(x, g(y), z) → g(f(x, y, z))
f(x, y, g(z)) → g(f(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.
F(x, y, g(z)) → F(x, y, z)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
F(
x1,
x2,
x3) =
F(
x3)
0 =
0
1 =
1
g(
x1) =
g(
x1)
Recursive Path Order [RPO].
Precedence:
0 > F1
0 > g1
1 > F1
1 > g1
The following usable rules [FROCOS05] were oriented:
none
(4) Obligation:
Q DP problem:
The TRS P consists of the following rules:
F(0, 1, x) → F(g(x), g(x), x)
F(g(x), y, z) → F(x, y, z)
F(x, g(y), z) → F(x, y, z)
The TRS R consists of the following rules:
f(0, 1, x) → f(g(x), g(x), x)
f(g(x), y, z) → g(f(x, y, z))
f(x, g(y), z) → g(f(x, y, z))
f(x, y, g(z)) → g(f(x, y, z))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.