(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
a(f, a(f, x)) → a(x, x)
a(h, x) → a(f, a(g, a(f, x)))
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:
A(f, a(f, x)) → A(x, x)
A(h, x) → A(f, a(g, a(f, x)))
A(h, x) → A(g, a(f, x))
A(h, x) → A(f, x)
The TRS R consists of the following rules:
a(f, a(f, x)) → a(x, x)
a(h, x) → a(f, a(g, a(f, x)))
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 2 less nodes.
(4) Obligation:
Q DP problem:
The TRS P consists of the following rules:
A(h, x) → A(f, x)
A(f, a(f, x)) → A(x, x)
The TRS R consists of the following rules:
a(f, a(f, x)) → a(x, x)
a(h, x) → a(f, a(g, a(f, x)))
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.
A(h, x) → A(f, x)
A(f, a(f, x)) → A(x, x)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
A(
x1,
x2) =
A(
x1,
x2)
h =
h
f =
f
a(
x1,
x2) =
a(
x2)
Lexicographic Path Order [LPO].
Precedence:
h > f > A2
a1 > A2
The following usable rules [FROCOS05] were oriented:
none
(6) Obligation:
Q DP problem:
P is empty.
The TRS R consists of the following rules:
a(f, a(f, x)) → a(x, x)
a(h, x) → a(f, a(g, a(f, x)))
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