(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
f(f(a, a), x) → f(a, f(b, f(a, x)))
f(x, f(y, z)) → f(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(f(a, a), x) → F(a, f(b, f(a, x)))
F(f(a, a), x) → F(b, f(a, x))
F(f(a, a), x) → F(a, x)
F(x, f(y, z)) → F(f(x, y), z)
F(x, f(y, z)) → F(x, y)
The TRS R consists of the following rules:
f(f(a, a), x) → f(a, f(b, f(a, x)))
f(x, f(y, z)) → f(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(f(a, a), x) → F(b, f(a, x))
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
F(
x1,
x2) =
x1
f(
x1,
x2) =
x1
a =
a
b =
b
Lexicographic Path Order [LPO].
Precedence:
a > b
The following usable rules [FROCOS05] were oriented:
f(x, f(y, z)) → f(f(x, y), z)
f(f(a, a), x) → f(a, f(b, f(a, x)))
(4) Obligation:
Q DP problem:
The TRS P consists of the following rules:
F(f(a, a), x) → F(a, f(b, f(a, x)))
F(f(a, a), x) → F(a, x)
F(x, f(y, z)) → F(f(x, y), z)
F(x, f(y, z)) → F(x, y)
The TRS R consists of the following rules:
f(f(a, a), x) → f(a, f(b, f(a, x)))
f(x, f(y, z)) → f(f(x, y), z)
Q is empty.
We have to consider all minimal (P,Q,R)-chains.